Alexander of Aphrodisias: On Aristotle Prior Analytics: 1.8-13 (with 1.17, 36b35-37a31) 9781472551641, 9780715628553

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Editor’s Note This text, translated in two volumes, is a very important one because Alexander’s is the main commentary on the chapters in which Aristotle invented modal logic, i.e. the logic of necessary (possible) and contingent propositions. Because it is more technical than the other texts in this series, Ian Mueller explains the modal logic in his masterly introduction, which takes an exceptional form, being couched in logical symbols partly of his own devising. All symbols are explained on first occurrence. Symbols are entirely excluded from the translation itself, and this can be consulted freely by those who do not wish to master the entire modal system. In this volume, Alexander reaches the chapter of Aristotle’s Prior Analytics (1.13) where Aristotle discusses the notion of contingency, and we have added Alexander’s commentary on that part of 1.17 where the conversion of contingent propositions is handled, the contingent being what may or may not happen. Aristotle also invented the theory of the syllogism, and in the present chapters he discusses syllogisms consisting of two necessary propositions, as well as the more controversial syllogisms containing one necessary and one non-modal premiss. The discussion of syllogisms containing contingent propositions is reserved for the companion volume. December 1998

R.R.K.S.

Preface This translation has been literally decades (two) in the making. Josiah Gould, acting on a suggestion of Ian Mueller, prepared a first draft of the translation. Mueller produced a second draft and, then, in consultation with Gould, a third and final version with introduction, notes, appendices, and indices. We are certain that errors remain, but know that there would have been many more without the advice of Tad Brennan, Glenn Most, Richard Patterson, Robin Smith, and several anonymous readers whose friendly but stern admonitions turned us from some paths. We take full responsibility for remaining on other paths despite their counsel. December 1998

I.M. J.G.

Introduction We offer here a translation of Alexander of Aphrodisias’ commentary on chapters 8-13 and most of 17 of book 1 of Aristotle’s Prior Analytics. In chapters 8-12 Aristotle presents what we call his modal logic as applied to necessary and to what we call unqualified propositions. In chapter 13 Aristotle discusses the notion of contingency, and in the part of 17 translated here he treats the so-called conversion of contingent propositions. In a separate volume we have translated Alexander’s commentary on chapters 14-22, in which Aristotle treats arguments involving contingent propositions. Chapters 1, 2, and 4-7 of the Prior Analytics constitute a self-contained presentation of what we will call non-modal or assertoric syllogistic. Alexander’s commentary on this material (and on chapter 3) has been admirably translated and discussed by Barnes et al. We refer the reader to their introduction for information about Alexander, ancient commentaries, and the general character of Alexander’s commentary on the whole Prior Analytics. In making choices for how to deal with our task, we have always begun by consulting Barnes et al. for guidance, and in many cases have followed their practices. But the greater difficulty of the material we have to present here has led us to diverge from them in some significant ways. First of all, in our notes and discussions we have relied on a quasi-formal symbolism. We hope that the symbolism is enlightening; we are confident that a full exposition of our text not using some formalism would run to much greater length. To put this another way, if one used a formal symbolism one could encapsulate the full content of Alexander’s commentary in many fewer pages than Alexander has used. We have, however, not thought it a good idea to introduce formal symbolism into the translation itself. Our major departure from Alexander’s text is that we have used considerably more variables than Alexander uses; we have sometimes done the same thing in our translations of Aristotle. To give one example, at 121,4-6, where Alexander writes, In both cases the conclusion proved is a particular negative necessary proposition of which the opposite is ‘It is contingent of all’ (endekhetai panti).

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Introduction

we have translated, In both cases the conclusion proved is a particular negative necessary proposition 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., the translation ‘A holds of all B’ is literal or corresponds to something like ‘holds of all’ or ‘universal affirmative’ without consulting the Greek text.1 The modern literature on Aristotle’s modal logic is substantial and itself difficult. The interpretations offered have been quite diverse, and a number of them have connected the modal logic with Aristotelian metaphysics. We did not see any way to enter into these various interpretations, and we have thought it best to focus on what we would call logical content, which seems to us also to be the focus of Alexander’s commentary. In fact, it seems to us that Alexander’s frequently expressed perplexities about what Aristotle says are a more accurate reflection of Aristotle’s presentation of modal logic than is the work of many subsequent interpreters who have attempted to turn the modal logic into a coherent system. In our notes and discussions we have primarily tried to extract the logical content of Alexander’s tortured prose. Just as we have not devoted much attention to modern interpretations of Aristotle’s logic, so we have not devoted much to parallel passages in other ancient texts. In both cases limitations of our time, of our knowledge, and of the space allotted by the publisher have constrained us. We have included a complete translation of the text of Aristotle as read by Alexander (insofar as we can infer that) in the lemmas; we have inserted texts as lemmas in places where there is no lemma in the edition of Wallies, on which our translation is based; and we have sometimes produced a stretch of Aristotelian text more than once. The reader can identify the exact extent of the lemmas in Wallies’ text, since material not in the lemmas is enclosed in square brackets. Our own judgment is that the lemmas are matters of convenience; they tell us more about the practice of scribes and later teachers than about the practices of ancient commentators. In our translation we have adopted the unusual practice of placing note references at the beginning of paragraphs which we judge to be especially difficult to follow. We believe that some readers will find it useful to have an account of what Alexander is going to say before trying to follow his own words. Those who prefer to make their own way through the text can simply ignore those notes initially and recur to them as they deem necessary. In section I of this introduction we give a brief and schematic presentation of non-modal syllogistic to familiarize the reader with

Introduction

3

terminology and with some of our apparatus for representing Alexander’s discussions. In section II we try to give at least a partial overview of chapters 8-12 and the part of chapter 3 relevant to it. But we shall postpone the discussion of some of that material because it presupposes the discussion of contingency, which we take up in section III. On occasion the reader may find it useful to refer to the formal Summary which follows this introduction. Here is an outline of the contents of the first 22 chapters of the Prior Analytics: 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 1.20-22 The third figure 1.20 Both premisses contingent 1.21 One premiss unqualified 1.22 One premiss necessary

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

For the most part it suffices for understanding Aristotelian modal syllogistic to have only a schematic understanding of the non-modal or assertoric syllogistic as it is developed in the first six chapters of the Prior Analytics. We present such a schematic representation here in quasi-formal terms which we will also rely on in our commentary. Qualifications of this schematic representation will be introduced only as 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 XeY XiY XoY

(read X holds of all Y or All Y are X) (read X holds of no Y or No Y are X) (read X holds of some Y or Some Y are X) (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 the subject of the proposition).2 These propositions are sometimes referred to as a-propositions, e-propositions, etc. Propositions of the first two kinds are called universal, those of the last two particular; a- and i-propositions are called affirmative, e- and o- negative. Universality and particularity are called quantity, affirmativeness and negativeness quality. When we wish to represent a proposition in abstraction from its 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 Second figure Third figure

XZ ZX XZ

ZY ZY YZ

As members of combinations propositions are called premisses. In the schemata given, Z is called the middle term of the combination, and X and Y are called extremes or extreme terms. In addition X is called the major term, Y the minor term; and the premiss containing the major term is called the major premiss, the one containing the minor term the minor premiss. D. The major problem for Aristotle is to determine which combinations are syllogistic, that is imply a proposition (called the conclusion) with

Introduction

5

the major term as predicate and the minor as subject.3 Aristotle restricts himself to considering the strongest conclusion implied by a syllogistic combination. In the first figure he recognizes the following syllogistic combinations:4 1. 2. 3. 4.

AaB AeB AaB AeB

BaC BaC BiC BiC

AaC AeC AiC AoC

(1.4, 25b37-40) (1.4, 25b40-26a2) (1.4, 26a23-5) (1.4, 26a25-7)

As a preliminary notation for these syllogisms (which will be complicated when we take up modal syllogistic) we introduce 1. 2. 3. 4.

AAA1 EAE1 AII1 EIO1

where the letters give the quality and quantity of the propositions involved and the subscripted number gives the figure. When we wish to represent 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 premisses, but after some hesitation, we have decided also to use the medieval names for the categorical syllogisms in the belief that most people who work on syllogistic will find them easier to read than the more abstract symbolism. Those unfamiliar with the names need only remember that the sequence of vowels in the medieval names reproduces the sequence of letters in the symbolism we have introduced; for further clarity we will add to the names numerical subscripts indicating the figure.5 Thus we will refer to the four first-figure syllogisms as 1. 2. 3. 4.

Barbara1 Celarent1 Darii1 Ferio1

Aristotle calls these four syllogisms complete (teleios, rendered ‘perfect’

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Introduction

by Barnes et al). Aristotle says that a syllogism is complete if it ‘needs nothing apart from the assumptions in order for the necessity to be evident’ (1.1, 24b22-4). Modern scholars have disputed what Aristotle means here,6 but Alexander clearly thinks that the complete syllogisms receive a kind of justification from the so-called dictum de omni et nullo, which he takes to give an account of the relations expressed by ‘a’ and ‘e’:7 For one thing to be in another as in a whole and for the other to be predicated of all of the one are the same thing. We say that one thing is predicated of all of another when it is not possible to take any of it of which 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 the dictum 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 possible to take any C of which B is not said. Again, if B is in A as in a whole, A is said of every B. Hence it is not possible to take any of B of which A is not said. Now, if nothing of B can be taken of which A is not said, and C is something 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 not hold of some C. For something of C is under B; but nothing of B can be taken of which A is said. Hence A will not be said of that item of C which is something of B. (60,27-61,1)

Whether one thinks that for Aristotle complete assertoric syllogisms are simply self-evident or – in agreement with Alexander – that their validity depends on the dictum de omni et nullo, affects one’s understanding of Aristotle’s conception of logic, but it does not affect one’s understanding of which assertoric combinations are syllogistic. In the case of modal syllogistic the situation changes. At least in antiquity the dictum played a role in disputes about whether certain combinations are syllogistic. We will say more about the issue in section II.C. E. At this point it is convenient to describe the principal procedure by which Aristotle shows that a combination is non-syllogistic. We would normally show that a given first-figure combination XY, YZ does not yield a specific conclusion XZ by specifying concrete terms which, when

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substituted for X, Y, and Z, make XY and YZ true and XZ false. Thus to show 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 are animals’ is false. Aristotle shows that a first-figure combination of specific premisses XY and YZ yields no conclusion (of the relevant kind with X as predicate and Z as subject) by giving two interpretations, one which makes XY, YZ, and XaZ true, the other of which makes XY, YZ, and XeY true. This procedure works because of the following relations among 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 out any negative conclusion XZ and an interpretation making XeZ true rules out any affirmative conclusion. On pp. 12-14 Barnes et al. discuss Alexander’s understanding of this method of rejecting non-syllogistic pairs and say, ‘He always misunderstands 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 reasonable to think that he has succeeded by mistake.’ We agree with this assessment of Alexander. He consistently treats the method of rejection as a matter of showing that both XaY and XeY (or their analogues in modal syllogistic, ‘X holds of all Y by necessity’ and ‘X holds of no Y by necessity’) follow from a pair of premisses. We have signalled Alexander’s misapprehension in cases where – if we have understood him correctly – it has led him to express a false opinion or made his discussion less cogent than it might be, and sometimes we have done so in the many more numerous passages where Alexander’s misdescription of what is going on is harmless. But we have frequently left it to the 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 all three propositions being true.10 F. Aristotle shows that second- and third-figure assertoric combinations are syllogistic by completing them or reducing them to first-figure syllogisms. Reductions are either direct or indirect. Direct reductions make use of the following rules of conversion enunciated and discussed by Aristotle in the second chapter of the Prior Analytics:

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Introduction EE-conversion: AI-conversion: II-conversion:

XeY  YeX XaY  YiX XiY  YiX

(25a14-17) (25a17-19) (25a20-2)

Aristotle uses terms to reject the possibility of any kind of O -conversion at 25a22-6. The second-figure syllogisms are: 1. Cesare2 2. Camestres2 3. Festino2 4. Baroco2

AeB AaB AeB AaB

AaC AeC AiC AoC

BeC BeC BoC BoC

(1.5, 27a5-9) (1.5, 27a9-15) (1.5, 27a32-6) (1.5, 27a36-b3)

The first three of these are completed directly. We indicate the way in which we will describe their reductions or proofs (deixeis), as Alexander most frequently calls them, in the Summary. Baroco2 is justified indirectly by reductio ad absurdum: from the contradictory of the conclusion and one of the premisses, one uses a first-figure syllogism to infer the contradictory of the other premiss. For our representation of the argument see the Summary, which gives similar representations for the third figure. These derivations for modally unqualified propositions are worth learning since in general Aristotle tries to adapt them to modally qualified propositions. In the directly derivable cases he faces few problems so that many of the main issues for them arise already in connection with the first figure. However, the addition of the modal operators causes special problems in the indirect cases. G. Aristotle seems to assume the completeness of his reduction procedures, that is, he assumes that any combination can either be refuted by a counterinterpretation or reduced to a first-figure syllogism. He also assumes that the system is consistent in the sense that one cannot give both a counterinterpretation and a reduction for a given syllogism. These assumptions are correct for assertoric syllogistic, and they make possible another method of showing a combination non-syllogistic: show that the rules do not allow the combination to be reduced to a first-figure syllogism.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), and Alexander does it even more frequently. The applications of this method are not up to the standards of modern proof theory, but they are generally corrrect. A more important point is that the modal syllogistic is not consistent, so that a derivation does not suffice to show that a counterinterpretation is impossible, and a counterinterpretation does not suffice to show that a derivation is impossible. Alexander is aware of some of the cases in which this is true,12 but – as is frequently the case in the commentary

Introduction

9

– he does not seem to be aware of either the depth of the problem created by this situation or its devastating effect on Aristotle’s modal syllogistic. II. Modal syllogistic without contingency (1.3, 25a27-36 and 8-12) As a first approximation modal syllogistic can be understood as an extension of assertoric syllogistic brought about by adding for every proposition P of assertoric syllogistic the propositions ‘It is necessary that P’ and ‘It is contingent that P’. The issues which arise in connection with the notion of contingency are considerably more complex than those which arise in connection with necessity. Unfortunately some of the issues which arise in connection with necessity are inextricably bound up with contingency. We are going to try to abstract from those issues here, and return to them after we have discussed contingency. We shall adopt the abbreviation NEC(P) for various Greek expressions which we take to have the sense of ‘It is necessary that P’. Ultimately we will use abbreviation CON(P) for ‘It is contingent (usually endekhetai) that P’, using the word ‘possible’ informally (and in the translation of such expressions as dunaton, dunatai, enkhôrei, hoion, estai). We shall call a proposition NEC(P) a necessary proposition, CON(P) a contingent proposition; if NEC(P) (CON(P)) is true we will say that P is necessary (contingent). To be explicit we shall call a proposition of assertoric syllogistic an unqualified proposition.13 We will define various formal notions in the same way as before, but we will extend our representation of syllogisms and combinations. Assertoric Barbara1 now 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) Bocardo3(NCU) EA_2(CU_)

NEC(AaB) NEC(AoC) CON(AeB)

BaC NEC(AaC) CON(BaC) AoB AaC

II.A. Conversion of necessary propositions (1.3, 25a27-36) Aristotle accepts the same conversion laws for necessary propositions as for unqualified ones, that is, he accepts:

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Introduction EE-conversionn: AI-conversionn: II-conversionn:

NEC(XeY)  NEC(YeX) NEC(XaY)  NEC(YiX) NEC(XiY)  NEC(YiX)

(25a29-31) (25a32-4) (25a32-4)

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 A holds 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 appears to justify by citing EE-conversionn. For AI-conversionn and II-conversionn Aristotle writes, If A holds of all or some B by necessity, it is necessary that B holds of some A. 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, would seem to involve some such reasoning as the following. Assume NEC(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 relies on laws concerning contingency which he has not yet discussed, but (i) and (ii) are laws which Aristotle rejects at 1.17, 36b35-37a31. In the

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11

course of doing so he denies that an indirect argument works by denying an instance of:  CON(P)  NEC( P)

( C  N )

which is equivalent to (i). Aristotle is, however, committed to C   N and its equivalent: NEC(P)   CON( P)

(N   C )

Since we cannot hope to clarify this situation without looking at Aristotle’s treatment of contingency and Alexander’s understanding of it, we shall for now simply take for granted the conversion laws for necessary propositions and turn to Aristotle’s application of them. However, before doing 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 and necessary propositions greatly simplifies the treatment of NN combinations in chapter 8, and Aristotle’s discussion is very succinct. The principal value of Alexander’s commentary on chapter 8 is its scholasticism, the concrete filling out of what Aristotle describes in outline. We here follow Alexander’s account. Aristotle assumes that an NN combination is syllogistic if the corresponding UU combination is, and that the former will yield the conclusion NEC(P) if the latter yields the conclusion P. The argument that the converses of these assumptions holds has three steps. The first two are stated briefly in the following passage: For, if the terms are posited in the same way in the case of holding and in that of holding by necessity – or in the case of not holding – there either will or there won’t be a syllogism , except that they will differ 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 same account of ‘being in as a whole’ and ‘said of all’. (29b37-30a3)

Alexander points out that Aristotle means to include all conversion rules in this remark (120,20-5), and he applies the reference to the dictum de omni et nullo to the first figure (120,13-15), a sure sign that he takes Aristotle to be treating the first-figure NNN syllogisms as complete. Thus the argument is that the parallel first-figure combinations are

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Introduction

syllogistic of parallel conclusions and that conversion will generate the parallel directly verified syllogisms in the second and third figures. The only remaining problem concerns: Baroco2(NNN) NEC(AaB) Bocardo3(NNN) NEC(AoC)

NEC(AoC) NEC(BaC)

NEC(BoC) 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 to decide which modal analogues of the complete first-figure assertoric syllogisms are syllogistic14 and then to ask whether the second- and third-figure analogues of syllogisms can be derived in ways analogous to those in which the first-figure ones were. Only when a derivation cannot be provided does Aristotle look for counterinterpretations. In other words, Aristotle does not appear to first raise the question whether a second- or third-figure combination is syllogistic, but first asks what, if any conclusion can be derived from the combination by a derivation 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 alternative derivation. If we try to copy the indirect derivations of Baroco2(UUU) and Bocardo3(UUU) for the corresponding NNN cases we run into the same kind of problems we encountered with Aristotle’s indirect arguments for the conversion laws for necessary propositions. We here give the indirect 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 could infer CON(AaC), which implies (C   N )  NEC(AoC), contradicting 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 true that Aristotle sometimes uses the equivalent of this rule, namely  C  N . Indeed, he uses it without acknowledgement in arguing that Bar-

Introduction

13

bara1(NC ) yields a contingent conclusion.15 Alexander is quite clear that 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 involves what we will call Theophrastean contingency because it was the notion of contingency highlighted by Theophrastus.16 The situation is sufficiently fluid that we might choose to allow Aristotle the use of  N  C in arguing for Baroco2(NNN) and Bocardo3(NNN). By itself this would take care of Bocardo3(NNN), since Aristotle takes Barbara1(CNC) to be complete at 1.16, 36a2-7. However, Barbara1(NC‘C’)17 is not complete for Aristotle and requires an argument which invokes the 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 a part D of C of which A does not hold by necessity. Substituting NEC(AeD) for NEC(AoC), we have in the case of Baroco2(NNN) an instance of Camestres2(NNN) with the conclusion NEC(BeD); but D is part of C, so NEC(BoC). For Bocardo3(NNN), one changes the second premiss to NEC(BaD) to get an instance of Felapton3(NNN). (In both cases Alexander carries out the reduction to the first figure.) Alexander discusses the character of the ekthetic arguments starting at 123,3-24, drawing a contrast between them and the ekthesis arguments of assertoric syllogistic. At 123,18-24 he provides the important historical information that Theophrastus preferred to postpone the treatment of Baroco2(NNN) and Bocardo3(NNN) until he could establish them indirectly, that is, use some version of the argument we have just sketched. We discuss the question of how Theophrastus might have done this in the introduction to the second volume (section IV). II.C. N+U combinations (1.9-11) In chapters 9-11 Aristotle takes up the N+U cases, devoting a chapter to each of the three figures. In 9 he takes as complete all the NUN and UNU analogues of the complete UUU first-figure syllogisms. Given these syllogisms, the direct derivations for the second- and third-figure N+U combinations are straightforward. The indirect cases are again problematic. Aristotle decides that each of the four N+U cases of Baroco2 and Bocardo3 yields only an unqualified conclusion. He gives no positive argument for any of the four, but only uses terms to show that none of the four combinations yield a necessary conclusion. We shall discuss his use of terms to show that certain N+U combinations yield an unqualified conclusion in a moment. For now we simply remark that all four cases accepted by Aristotle have simple indirect derivations. Alexander points out at 144,23-145,20 and 151,22-30 that the kind of ekthesis argument which Aristotle used to establish Baroco2(NNN) could be used for

14

Introduction

Baroco2(UNN) and Bocardo3(NUN). Unfortunately, Alexander’s discussion of the implications of this situation in which a proof and a counterinterpretation conflict (145,4-20 and 151,22-30) is very indecisive, to say the least. We shall approach Aristotle’s treatment of the complete combinations in terms of the two cases of Barbara1. For Barbara1(UNU) Aristotle takes for granted that Barbara1(UN ) yields either an unqualified or a necessary conclusion and offers two kinds of arguments to show that the conclusion cannot be necessary. One is a specification of terms, which, indeed, work if one assumes the truth of the following propositions: (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 Barbara1(NUN) since – to use an example of Theophrastus mentioned by Alexander 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 a necessary truth. (a) and (a’) are typical problematic examples of an unqualified truth: they are not, in fact, true, but they are taken to be true for the sake of making an argument, in Alexander’s terminology, they are ‘hypotheses’.19 Unfortunately, this way of interpreting unqualified statements makes it very difficult to see that there is any difference between unqualified and contingent propositions. Alexander raises this issue in connection with Aristotle’s remarks at 1.15, 34b7-18 in the context of an apparent counterinterpretation to Barbara1(UC‘C’); see volume 2. A modern way of making a distinction between (c) and (c’) invokes the distinction between what are called de re and de dicto necessity. To say that NEC(XaY) is true de dicto is to say that there is some lawlike connection between the notion of being a Y and the notion of being an X, so that just knowing that something is a Y is enough to know it is an X. Both (c) and (c’) are true de dicto because there is no such connection between being an animal and being in motion or between the latter and being a human; knowing that something is an animal does not suffice to tell us that it is in motion and knowing that something is in motion does not suffice to tell us it is a human. We find the notion of de re necessity hard to grasp, but perhaps the following will do. We must

Introduction

15

imagine that individuals have necessary properties, that, for example, Socrates is necessarily a human being and an animal. Socrates has those properties no matter how he is described, e.g., as the anathema of the politicians. Now we say that NEC(XaY) is true de re if each of the Y’s has the property of being necessarily X. If (a’) is true, then each of the things in motion is necessarily an animal, even though there is no lawlike 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 other hand, (c) is true de re because no individual human being is necessarily in motion. The issues surrounding the de re/de dicto distinction and the interpretation of Aristotle’s modal syllogistic have received a great deal of discussion, which we cannot recapitulate here.20 We shall occasionally invoke the distinction in our notes, but on the whole we shall leave it out of account since it does not come to the surface in Alexander’s remarks. In the Appendix on conditional necessity we discuss another distinction which he does sometimes invoke, namely the distinction between what is necessary without qualification and what is necessary on a condition. Aristotle’s brief remarks about the validity of Barbara1(NUN) have been 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 all B by necessity and C is some of the B’s, it is evident that [A will hold] of C by necessity. (30a17-23)

Alexander’s paraphrase of this passage shows that he takes it to involve an application of the dictum de omni et nullo and hence to be an argument for completeness: For since A is said of all B by necessity, and C is under B and is some of B, A is also said of C by necessity. For what is said of all B by necessity will also be predicated of what is under B by necessity – at least if being said of all is ‘when nothing of the subject can be taken of which the predicate will not be said’.22 But C is some of the B’s. For being said of all by necessity is taken in the same way , as he said before in the case of necessary things: ‘For the privative converts in the same way, and we will give the same account of “to be in as a whole” and “said of all” ’ . (126,1-8)

For Alexander, then, the validity of Barbara1(NUN) depends on interpreting NEC(AaB) as saying that no B can be taken of which A does not hold by necessity (to which we might add, ‘no matter how the B is described’). Alexander explicitly refrains from committing himself on the correctness of Aristotle’s position, but it is clear that he is quite impressed by the arguments of Theophrastus and Eudemus,23 who, as

16

Introduction

Alexander tells us, rejected Barbara1(NUN) in favour of Barbara1(NUU), and adopted what Bochenski (1947, p. 79) called ‘la règle du peiorem’ and we will call the peiorem rule, according to which the conclusion of a combination can be no stronger than its strongest premiss.24 Throughout the commentary Alexander signals when a move of Aristotle’s depends or appears to depend on his acceptance of first-figure NUN syllogisms, a clear indication that he thinks the move is problematic.25 It may be that his ultimate position is that the notion of necessity is ambiguous. Commenting on a passage (1.13, 32b25-32) in which Aristotle says that contingency can be taken in two ways, Alexander writes: But if ‘It is contingent that A holds of that of which B is said’ has two meanings, so will ‘By necessity A holds of that of which B is said’ have two meanings; for it will mean either ‘A holds by necessity of all of that of which B is said unqualifiedly’ or ‘A holds by necessity of all of that of which 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 all of that of which B is said’, as is said by some of those who show that it is true that the conclusion of a necessary major and an unqualified minor is necessary. (166,19-25)

Before he gives terms for rejecting Barbara1(UNN), Aristotle offers the following argument against it: But if the proposition AB is not necessary, but BC is necessary, the conclusion will not be necessary. For, if it is, it will result that A holds of some B by necessity – through the first and through the third figure. But this is false. But it is possible that B is such that A can hold of none of it. (30a23-28)

After giving terms Aristotle says that the proof that Celarent1(UNN) fails will be the same. Later, having affirmed Darii1(NUN) and Ferio1(NUN), Aristotle rejects Darii1(UNN) and Ferio1(UNN): But if the particular premiss is necessary, the conclusion will not be necessary; for nothing impossible results, just as in the universal syllogisms. Similarly in the case of privatives. Terms: motion, animal, white. (30b2-6)

It seems reasonably clear that Alexander is right to interpret Aristotle’s first rejection of Barbara1(UNN) as something like the following correct argument: Assume that AaB and NEC(BaC) yield NEC(AaC). But NEC(AaC) and NEC(BaC) yield (Darapti3(NNN)) NEC(AiB). However, we ought to be able to make AaB true while making NEC(AiB) false. Hence, the assumption that Barbara1(UNN) holds is wrong.

Introduction

17

We prefer the following paraphrase of this argument: Assume, as is possible, that AaB,  NEC(AiB), NEC(BaC), and assume that Barbara1(UNN) is valid. Then NEC(AaC), which with NEC(BaC) implies (Darapti3(NNN)) NEC(AiB), contradicting  NEC(AiB). Hence Barbara1(UNN) is not valid.

We shall call such an argument against a rule of inference an incompatibility rejection argument, meaning an argument which shows that acceptance of a proposed rule of inference would allow one to derive an inconsistency from a set of compatible premisses, and we shall call an argument against the possibility of an incompatibility rejection argument an incompatibility acceptance argument. In his remarks on Darii1(UN ) and Ferio1(UN ) Aristotle claims that he has incompatibility acceptance arguments for all four first-figure UNU cases as well as incompatibility rejection arguments for the UNN cases. The former claim is incorrect in the case of Barbara1(UNU), since – once the complete Darii1(NUN) (or Darapti3(UNN)) is available – the argument we have given above could be formulated as a rejection of Barbara1(UNU).26 On the other hand, the claim is correct for the other three cases. 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 with AeB.27 For: Darii1(UNU)

AaB

NEC(BiC)

AiC

AeB

NEC(BiC)

AoC

and Ferio1(UNU)

the conclusion and either premiss entail nothing. However, in the case of these two the situation is exactly the same if the conclusion is taken to be NEC(AiC) or NEC(AoC), as Alexander points out at 134,32-135,6 and 135,12-19. Hence Aristotle cannot give incompatibility rejection arguments for either Darii1(UNN) or Ferio1(UNN). At 129,9-22 Alexander more or less shows that there is no incompatibility rejection argument for Barbara1(NUN). The same is true for the other first-figure NUN cases.28 In commenting on the rejection of Barbara 1(UNN) (128,3-129,7) and Celarent1(UNN) (130,27-131,4) Alexander contents himself with showing that incompatibility arguments work for rejecting these. However, as we have seen, when he gets

18

Introduction

to Aristotle’s specification of terms, he points out (129,23-130,24) that very similar terms would suffice for the rejection of Barbara1(NUN), and offers essentially Theophrastean considerations against Aristotle’s position. He subsequently (131,8-21) tries to explain the difference between incompatibility rejection arguments and reductios, and then says that Aristotle doesn’t seem to be entirely confident about these rejection arguments. This remark might seem out of place, given what Alexander has said up to this point, but it is not if we realize the complications which we have already outlined. Alexander goes on to give his own method (132,5-7), which involves the attempt to produce a reductio on the denial of a purported conclusion; if one is produced the purported conclusion follows, if it isn’t, the purported conclusion does not. Application of the method requires Alexander to look ahead not only to third-figure N+U (and UU) combinations, which is all right since these combinations reduce to first-figure ones, but – because the denial of a necessary proposition is a ‘contingent’ one – also to N+C (and U+C) combinations. The method appears to work for accepting Barbara1(UNU) and rejecting Barbara1(UNN), but it would commit Aristotle to acceptance of Celarent1(UNN).29 Alexander is obviously in difficulty when he gets to Aristotle’s rejection of Darii1(UNN) and Ferio1(UNN), since what Aristotle says or clearly implies is false: we cannot give incompatibility rejection arguments for these cases. Essentially Alexander considers various alternatives without clearly espousing any one of them. We describe the text, since it offers some difficulty. Alexander considers three alternative interpretations. He first suggests (133,20-9) that Aristotle is intending to apply his method of incompatibility argumentation to Darii1(UNN) and Darii1(NUN). But now he claims that the method would not generate a contradiction if applied to Barbara1(NUN). This claim is, of course, false, and in trying to defend it Alexander uses Darapti3(UNU) rather than the stronger Darapti3(UNN) which is accepted 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 incompatibility arguments will not work to reject either Darii1(UNN) or Ferio1(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 will suffice to confirm Darii1(NUU) but not Darii1(UNN). He does not point out that it also confirms Darii1(NUN). Nor does he say anything about Ferio1. 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 concrete counterinterpretations. This has the benefit of putting Aristotle on logically sound ground, but it is hard to believe that this is what the text means.

Introduction

19

III. Modal syllogistic with contingent propositions (1.13-22) We have seen that full treatment of Aristotle’s discussion of the conversion of necessary propositions requires reference to his treatment of conversion for contingent propositions. In III.A we say something about Alexander’s understanding of the notion of contingency and the rules for converting contingent propositions. In III.B we go into more detail on Alexander’s interpretation of the three modal notions, and in III.C and D we look in more detail at his treatment of conversion for necessity and contingency. III.A. Strict contingency and its transformation rules (1.13, 32a18-32b1) At the beginning of his discussion of the transformation rules for contingency in 1.3 Aristotle says that ‘to be contingent is said in many ways, since we say that the necessary and the non-necessary and the possible are contingent’ (25a37-9). Commenting on this remark, Alexander writes: He (sc. Aristotle) showed us the homonymy of ‘contingent’ in On Interpretation too. For we apply ‘It is contingent’ to what is necessary 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 contingent 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 while sharing with it the fact of holding at the present time. (Note the expression: what holds contingently is the same as what is signified by an unqualified 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 most part and by nature – this is the way in which we determine the contingent ’ . (37,28-38,10)

We discuss the reference to On Interpretation in Appendix 4 (On Interpretation, chapters 12 and 13). At this point what is important is that Alexander understands Aristotle to hold that we use ‘It is contingent that’ in three different senses when we apply it to a proposition expressing a necessary truth, a proposition expressing something which holds but is not necessary, and a proposition which expresses a mere possibility. For Alexander it is only the third sense which gives the strict meaning of contingency, the one which is central to Aristotle’s syllogistic. We also wish to signal the curious sentence in parenthesis calling attention to the notion of holding contingently (endekhomenôs). In the next sections we shall emphasize occurrences of this word by including the transliterated Greek. In his comment on 25b14, Alexander says:

20

Introduction He set down only this sort of contingency – what holds for the most part and is by nature (for what is by nature is for the most part), since only this sort is useful in the employment of syllogisms. The possible also covers what holds in equal part and what holds infrequently, but syllogisms 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 of contingency. Aristotle specifies the defining feature toward the beginning of chapter 13 when he announces what Alexander calls (on the basis of 1.14, 33b21-3, 1.15, 33b25-31, and 1.15, 34b27-9) the diorismos of contingency: I call P contingent or say it is contingent that P if P is not necessary and if, when P is posited to hold, nothing impossible will be because of it. For we 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 his specious justifications of certain first-figure UC and NC syllogisms, notably Barbara1(UC_) and Celarent1(UC_).32 Commenting on the diorismos Alexander argues that for Aristotle CON(P) rules out P as well as NEC(P): Since he is going to discuss syllogisms from contingent premisses, he first defines the contingent. He does not define it in its homonymous use since it is not possible to define something as it is used homonymously. Rather he isolates contingency as said of the necessary and the unqualified from the contingent. For he showed that the contingent is also predicated of these things. By saying ‘when P is posited to hold’ he indicates that, in addition to not being necessary, the contingent is not unqualified either. For what is contingent according to the third adjunct33 is of this kind and it differs from what is necessary and what is unqualified because if P is said to be possible (dunasthai), P is not yet (mêdepô) the case. So, P would be contingent in the strict sense if P is not the case and if when P is posited to be the case it has nothing impossible as a consequent. And he would have spoken more strictly about the contingent if he said ‘P is not the case and when P is posited to hold’. For although what is not the case is not necessary, what is not necessary is not ipso facto not the case. (156,1122)34

Thus we may state ‘Alexander’s diorismos’ as: CON(P) iff (i)  P, and (ii) no impossibility follows from P35

Introduction

21

Alexander frequently refers to this strict sense of contingency as contingency in the way specified (kata ton diorismon). At 32a29-35 Aristotle announces rules of transformation for contingent propositions: It results that all contingent propositions convert with one another. I do not mean that the affirmative converts with the negative, but rather that whatever has an affirmative form converts with respect to its antithesis, e.g., that ‘It is contingent that X holds’ converts with ‘It is contingent that X 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 does not hold of all B’, and ‘It is contingent that A holds of some B’ converts with ‘It is contingent that A does not hold of some B’, and the same way in the other cases.

If one understands ‘ “It is contingent that X holds,” converts with “It is contingent that X does not hold” ’ to mean that CON(P) is equivalent to CON( P) and applies that understanding to modal syllogistic, the result, taken in conjunction with other equivalences accepted by Aristotle is to make all contingent statements involving two terms A and B equivalent and so to render syllogistic with contingency more or less bankrupt. It seems certain that Aristotle does not intend this, and the thought that he might doesn’t even enter Alexander’s head.36 He takes Aristotle’s point to apply only to so-called indeterminate propositions, that is, propositions which are ambiguous with respect to quantity.37 This means that the relevant transformations for syllogistic are simply: AE-transformationc:38 EA-transformationc: IO-transformationc: OI-transformationc:

CON(AaB)  CON(AeB)  CON(AiB)  CON(AoB) 

CON(AeB) CON(AaB) CON(AoB) CON(AiB)

Unfortunately, Aristotle does not offer any argument for any of these rules, 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 that it holds of all, it is also contingent that it does not hold of all. And similarly in the case of particular affirmations. (32a36-40)

Alexander does not choose to expand significantly on these remarks, telling us only that this position is ‘reasonable’ (eikotôs) given the diorismos of contingency. When we add to these transformation rules the conversion rules announced at 1.3, 25a37-b3:

22

Introduction AI-conversionc: II-conversionc:

CON(AaB)  CON(BiA) 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). On the other hand, as we have already mentioned, Aristotle denies EEconversionc at 1.17, 36b35-37a31. The equivalences Aristotle does accept have the effect of generating what we will call waste cases of syllogistic validity. For example, since Aristotle accepts: Barbara1(CCC)

CON(AaB)

CON(BaC)

CON(AaC)

the equivalence of (ia) and (ib) would also commit him to EAA1(CCC) AEA1(CCC) EEA1(CCC)

CON(AeB) CON(AaB) CON(AeB)

CON(BaC) CON(BeC) CON(BeC)

CON(AaC) CON(AaC) CON(AaC)

to give only examples with an a-conclusion. Aristotle’s handling of the waste cases is not always perspicuous. He mentions some and not others, and, for example, he chooses to endorse Celarent1(CCC) without mentioning EAA1(CCC). For the most part the waste cases are of no interest, and we shall not worry about them. But in some places, particularly after Aristotle loses sight of – or perhaps interest in – the various notions of contingency which he has brought into play, Alexander addresses difficulties implicit in determining exactly what waste case Aristotle is espousing. The diorismos of contingency appears to commit Aristotle to the following instances of CON(P)   NEC(P) (C   N): (i) CON(AaB)  (ii) CON(AeB)  (iii) CON(AiB)  (iv) CON(AoB) 

 NEC(AaB)  NEC(AeB)  NEC(AiB)  NEC(AoB)

Introduction

23

The first two of these propositions are clearly Aristotelian, but the last two cause some difficulty. One can see in a rough way that if sense could be made of a de dicto reading of particular propositions these two would be true de dicto, but false de re, since, for example, there might be some animals, e.g., humans, for which it is contingent that they are white and other 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 note that 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 be true. The last pair seems reasonable enough on a de re reading, but the first pair seems to be false on such a reading. Whatever Aristotle may have thought about (iii) and (iv), Alexander is 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 taking CON(White i Walking) and CON(White o Walking) to be true. This choice is equally problematic on the intuitive de re reading which lies behind 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 of modality: preliminary remarks At the beginning of chapter 2 Aristotle announces that ‘every proposition says either that something holds or that it holds by necessity or that it is contingent that it holds’ (25a1-2). Alexander’s comment on this passage helps to fill out our understanding of his conception of the three modalities: It is necessary to understand the word ‘categorical’ added to the words ‘every proposition’, since he is now talking about such propositions and syllogisms .  Now in every categorical proposition one term is predicated of another either affirmatively or negatively, i.e., as holding or not holding of the subject; and if X holds of Y, it either holds always or holds at some time and doesn’t hold at another. 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 negative true proposition is one which takes what by nature never holds of something as never holding of it. But if X does not always hold of Y, if it holds at the present moment, the proposition which indicates this is an unqualified true affirmative; and similarly a proposition which says that what 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) hold of it and is taken in this way – i.e., as being able to hold – the proposition is a true contingent (endekhomenon) affirmative; and a proposition which says of what holds or does not hold but can (hoion) both hold and not hold

24

Introduction that 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 whether Alexander is speaking about (in our formulations) the assertion that ‘Animal a Human’ is a (true) necessary proposition, the assertion that NEC(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 Alexander commits himself to at least a partial temporal interpretation of necessity, contingency, and unqualified holding. Part of the difficulty in construing what Alexander has in mind here arises from his attempt to distinguish between affirmative propositions, which we shall temporarily represent as XaffY, and negative ones, which we shall represent as XnegY. We can construe Alexander’s account of the modalities as follows: 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 of contingency for affirmative and negative statements. We can see Alexander’s difficulty by considering the two possible ways of making the definitions 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 Aristotle is committed to AE-, EA-, IO- and OI-transformationc. However, it is relatively certain that Alexander thinks of (ii) as something like a feature of contingency, whereas (i) is closer to a genuine analysis of it. For we have seen that for him the primary account of contingency is given by the diorismos, which he takes to imply that what is contingent does not hold. For this reason we take (i) instead of (ii) as the relevant account of contingency. We can then drop the distinction between aff and neg, and write the three accounts as

Introduction

25

(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 in unpacking the notion of contingency than the assertion that nothing impossible follows from the assumption that P. In both cases we are using the notion of possibility to explain the notion of possibility. Unfortunately, Alexander does not seem to have any non-circular way of explaining what ‘can be true’ means. However, it is useful to have in mind a strictly temporal version of (C*), since Alexander sometimes seems 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 conversion laws for necessary propositions and that Ct allows one to do the same for not only AI-conversionc and II-conversionc, but also EE-conversionc. In order to indicate Alexander’s apparent flirtation with Ct we shall look at his account of Aristotle’s justification of the conversion laws for necessary propositions, which as we explained in section II.A, seem to rely on claims about contingency which Aristotle hasn’t proved or – worse yet – ultimately decides are false. However, before doing so, we should mention that, insofar as Alexander equates contingency with possibility, he explicitly assigns C* rather than Ct to Aristotle at 184,9-11. III.C. Conversion of necessary propositions (1.3, 25a27-36) The laws in question are: EE-conversionn: AI-conversionn: II-conversionn:

NEC(AeB)  NEC(BeA) NEC(AaB)  NEC(BiA) NEC(AiB)  NEC(BiA)

We recall Aristotle’s justification of EE-conversionn: 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 A holds of some B. (25a29-32)

Here is Alexander’s comment: Here again he seems to have used the conversion of particular contingent

26

Introduction affirmative propositions in his proof for necessary universal negative ones, even though he has not yet discussed conversions of contingent propositions. Or should we rather say this? He holds it to be agreed that (i) particular affirmative contingent propositions are opposite to universal necessary negative ones since they are contradictories, and therefore assumes this. Having assumed it, then, (ii) since if B holds of some A but not by necessity, it is said that it is contingent that B holds of some A and that it holds contingently (endekhomenôs) of some A, and (iii) since he has proved that particular unqualified propositions convert with themselves, he makes use of propositions of this kind. Thus he does away with the necessity by saying that it is contingent that A holds of some 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 holds of A or it is possible (hoion) that it will hold of it at some time. (vi) In this way 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 distinguishes kinds of contingency that what holds but not by necessity is said to be contingent. (viii) For he says that contingency signifies both what is necessary and also what is not necessary but holds – and he now uses it in application to the latter case. (ix) And what holds contingently (endekhomenôs) of some or will hold of some is the opposite of what holds of none by 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 contingent that XiY’ (i). Hence (ii) he assumes  NEC(BeA) and infers ‘It is contingent that (BiA)’ and so (v) either BiA or it is contingent that B will hold of A at some time. But (iv) at the time BiA holds, AiB holds by II-conversionu. But this conflicts with the assumption NEC(AeB) (vi and ix). Hence we see that Aristotle uses only II-conversionu. When in his argument he seems to invoke II-conversionc he is using ‘contingent’ in the sense 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 to this argument: It is clear from this that in the previous proof too he used ‘It is contingent that B holds of some A’ in connection with something unqualified; for there ‘for if it is contingent that B holds of some’ should be understood to mean ‘For if B holds contingently (endekhomenôs) of some A’. (37,17-21; cf. 149,5-7)

Clearly (vi) and (ix) presuppose Nt, but Alexander’s vocabulary shows the same wavering between (C*) and (Ct) to which we have already called attention. There is a perhaps more serious problem raised by (i). Alexander offers no justification for how Aristotle can take this for granted when he himself holds that CON(XiY) does not follow from  NEC(XeY), since  NEC(XeY) is compatible with NEC(XaY), which is

Introduction

27

incompatible with CON(XiY). Perhaps when Alexander says that Aristotle takes (i) to be something agreed, he means that Aristotle is taking (i) as an endoxon, albeit one which he does not accept. Alexander’s discussion of AI- and II-conversionn, to which we now turn, throws some further light on his treatment of EE-conversionn. Alexander’s summary of the argument involves another (to us approximate) use of temporal considerations and the same assertion of the equivalence of  NEC(P) and ‘It is contingent that  P’. He proves that particular affirmative necessary propositions convert from both universal affirmative necessary and particular affirmative necessary ones in the same way as he did in the case of privative universal ones. For if A holds of all or some B by necessity, but B does not hold of some 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 not necessary that B holds of some A’, which is equivalent to ‘It is contingent that 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 of no A, A will hold of no B (for this has been proved). Hence, A will not hold of all or some B by necessity. (37,3-13)

Insofar as there is anything new in Alexander’s discussion of AI- and II-conversionn, it comes when he tries to defend Aristotle against the charge of using EE-conversionc: It is clear that he has not conducted the proof with contingent negative propositions; for he thinks that they do not convert. Rather he reduces to an unqualified one, subtracting necessity 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 is assuming that unqualified propositions convert. (37,14-17)

Here Alexander lights on the fact that in the justification of AI- and II-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 A holds of no B and so  NEC(AiB)’, but simply ‘if  NEC(BiA) then  NEC(AiB)’. III.D. The conversion of contingent propositions III.D.1 Conversion of affirmative contingent propositions (1.3, 25a37-b3); more on Alexander and the temporal interpretation of modality Aristotle argues for AI- and II-conversionc simultaneously. We wish to consider what he says as an alternative to a simple argument which one might have expected him to use. Suppose  CON(YiX). Then ( C  N ) NEC(YeX). But (EE-conversionn) NEC(XeY), contradicting CON(XaY) or CON(XiY). Aristotle avoids such an argument because of the use of

28

Introduction

 C  N ; he later (1.17, 37a9-31) rejects the analogous argument for EE-conversionc: assume  CON(YeX); then ( C  N ) NEC(YiX), so that (II-conversionn) NEC(XiY), contradicting CON(XeY). But Aristotle’s own argument for AI- and II-conversionc is very problematic: Since to be contingent is said in many ways (since we say that the necessary and the non-necessary and the possible are contingent) in the case of contingent propositions, the situation with respect to conversion will be the same in all cases of affirmative propositions. For if it is contingent that A holds of all or of some B, then it will be contingent that B holds of some A. For if of none, then A of no B; this has been proved earlier. (25a37-b3)

Alexander takes for granted that Aristotle’s argument must turn on the three ways in which contingency is said, and that it will proceed indirectly 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 and A is the subject, and then to: (iii) a universal negative statement in which A is the predicate and B is the subject, which contradicts: (iv) ‘It is contingent that A holds of some B’. To try to work out an interpretation satisfying these conditions Alexander 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 to find an inconsistency: (iva) CON(AiB) (ivb) AiB (ivc) NEC(AiB)

Introduction

29

Case (b) is easy since  (BiA), i.e., BeA, yields (EE-conversionu) AeB, contradicting AiB. Similarly, given  C  N , which Alexander presumably again takes as ‘agreed’, case (c) reduces to EE-conversionn. For case (a) Alexander takes for granted  N  C and gives his most straightforward temporal argument: if  NEC(BiA) then ( N  C ) CON(BeA), so that (Ct) at some time BeA, so that at that time AeB, so CON(AeB), contradicting NEC(AiB). He does not seem to notice that if this argument were correct it would establish EE-conversionc. Alexander preserves for us something like such an argument of Theophrastus and Eudemus for a version of EE-conversion for contingent propositions, although it too shows an unclear handling of temporal considerations: If it is contingent that A holds of no B, it is also contingent that B holds of no A. For since it is contingent that A holds of no B, when it is contingent that it holds of none, it is then contingent that A is disjoined from all the things of B. But if this is so, B will then also have been disjoined from A, and, if this is so, it is also contingent that B holds of no A. (220,12-16)

Alexander defends Aristotle against this argument: It seems that Aristotle expresses a better view than they do when he says that a universal negative which is contingent in the way specified does not convert with itself. For if X is disjoined from Y it is not thereby contingently (endekhomenôs) disjoined from it. Consequently it is not sufficient to show that when it is contingent that A is disjoined from B, then B is also disjoined from A; in addition that B is contingently disjoined from A. But if this is not shown, then it has not been shown that a contingent proposition converts, since what is separated from something by necessity is also disjoined from it, but not contingently. (220,16-23; cf. 221,1-2)

Alexander here seems to concede that if CON(AeB), then at some time AeB and therefore BeA. But he insists that one cannot infer CON(BeA) because one doesn’t know that BeA holds contingently if we have inferred BeA from AeB, where AeB holds contingently.45 It seems clear that Alexander is invoking a distinction between the ways in which things hold. We cannot infer CON(P) from P unless we know that P holds contingently. Alexander uses the words ‘necessarily’ (anankaiôs) and ‘unqualifiedly’ (huparkhontôs) as well as ‘contingently’ in the commentary.46 Although Aristotle never uses any of these words in a logical context, they are also found in the other commentaries on his logical works. For the most part they are simply variants of expressions such as ‘It is contingent that’, but we are convinced that Alexander wishes to put special weight on the ideas of holding contingently and of holding but

30

Introduction

not holding necessarily. By insisting on the latter notion Alexander is able to maintain the position that unqualified propositions for Aristotle do not signify holding necessarily or eternally. But he has much more difficulty with what the difference is between a contingent and an unqualified proposition. Indeed, his assertion at 38,5-7 that holding contingently correlates with ‘what is signified by an unqualified proposition’ is probably intended to justify the application of II-conversionu which Alexander detects in Aristotle’s justification of AI-conversionc. Similarly in his account of the justification of EE-conversionn Alexander wants to stress that  NEC(BeA) implies that BiA holds contingently to justify the alleged application of the same rule. If Alexander were willing to use the temporal reading of the modal operators straightforwardly, he would have no difficulty, but, as we have seen, he instead mixes the temporal reading with the idea of something holding contingently. But using that idea depends on blurring the distinction between what holds now and what holds at some time. To put this point another way, for Alexander’s reasoning to work, one has to assume that Aristotle proves II-conversionu not just for propositions which hold now, but for proposition which hold at some time. But, on the temporal reading of the modalities, that is to say that II-conversionu is or includes II-conversionc. Although Alexander makes no such claim, it seems to us that his handling of the modal conversion rules more or less commits him to some such idea. Moreover the lumping together of unqualified and contingent propositions is quite in keeping with Aristotle’s use of false but possible truths, e.g., ‘All animals are moving’ to interpret unqualified propositions and with his willingness to use the same terms to verify corresponding contingent and unqualified propositions.47 As Alexander explains in connection with the proposition ‘No horse is white’: For if someone requires that we take as universal what holds always but not what holds at some time, he will be requiring nothing else than that the unqualified be necessary, since the necessary does always hold. Furthermore, he himself, when he is considering an unqualified proposition with respect to terms does not ever consider it with respect to terms of this kind. (232,32-6; cf. 130,23-4)

If Alexander adhered to a strict temporal interpretation of contingency what he says here would implicitly commit him to the identification of unqualified truths with propositions true at some time, that is, with contingent propositions. He, of course, never makes this identification. If he had, he might have seen problems which face any interpreter trying to understand why Aristotle accepts certain U+C combinations while rejecting their CC analogues. There are many reasons why Alexander never offers a strict temporal interpretation. Perhaps the most impor-

Introduction

31

tant is that for him the meaning of contingency is determined by the diorismos, not by any temporal account. III.D.2. Non-convertibility of negative contingent propositions (1.3, 25b3-19; 1.17, 36b35-37a31) Aristotle’s denial of EE-conversionc is controversial. Alexander’s discussion of it is dense, but is largely a scholastic defence of Aristotle’s position. We will mention a few points in it, but we will mainly content ourselves with describing Aristotle’s text. In the case of negative propositions, it is not the same. With those which are said to be contingent inasmuch as they do not hold by necessity or they hold but not by necessity, the case is similar, e.g., if someone were to say that it is contingent that what is human is not a horse or that white holds of 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 converts in the same way; for, if it is contingent that horse holds of no human, it will be possible (enkhôrei) that human holds of no horse, and if it is possible that white holds of no cloak, it is possible that cloak holds of nothing white – for if it is necessary that it holds of some, then white will also hold of some cloak by necessity (for this was proved earlier). And similarly in the case of particular negatives. (25b3-14)48

Alexander understands Aristotle to be dealing here with the situation in 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 proposition ‘It is contingent that horse holds of no human’ and unqualified holding with ‘It is possible that white holds of no cloak’. Aristotle’s argument that the latter converts seems to be a straightforward indirect argument moving from ‘It is not possible that cloak holds of nothing white’ to ( C  N ) NEC(Cloak i White) to (II-conversionn) NEC(White i Cloak). Alexander insists on reparsing what Aristotle says to make it fit the case of contingency as holding: He says ‘for if it is necessary that it holds of some, then white will also hold of some cloak by necessity’ since a particular affirmative necessary proposition must be the opposite of a contingent universal negative one, and the unqualified proposition was assumed as contingent in its verbal formulation. And the verbal opposite will contain necessity, although what is signified by it will be particular affirmative unqualified. For this is the opposite of a universal negative unqualified proposition. (39,4-11, our italics)

That is to say, according to Alexander, Aristotle uses the vocabulary of necessity although he expects us to understand that he is talking about

32

Introduction

unqualified propositions. Aristotle has little to say about the third case. He remarks that EE-conversionc fails and OO-conversionc works, but defers discussion until chapter 17: But those things which are said to be contingent inasmuch as they are for the most part and by nature – and this is the way we specify contingency – will not be similar in the case of negative conversions. Rather a universal negative proposition does not convert, and the particular does convert. This will be evident when we discuss contingency. (25b14-19)49

Aristotle’s actual argument for rejecting EE-conversionc is confusing for a number of reasons, one of which is his tacit reliance on the equivalence of CON(XaY) and CON(XeY). He begins the rejection, which is what we have called an incompatibility rejection argument, as follows: It should first be shown that a privative contingent proposition does not convert; that is, if it is contingent that A holds of no B, it is not necessary that it is also contingent that B holds of no A. For let this be assumed and let it be contingent that B holds of no A. Then, since contingent affirmations convert with negations – both contraries and opposites – and it is contingent that B holds of no A, it is evident that it will also be contingent that B holds of all A. But this is false. For it is not the case that if it is contingent that X holds of all Y, it is necessary that it be contingent that Y holds of all X. So the privative does not convert. (36b35-37a3)

Here Aristotle takes for granted the equivalence of CON(XeY) and CON(XaY) and the compatibility of CON(XaY) (or equivalently CON(XeY)) and  CON(YaX). We may represent his argument as follows. Assume that EE-conversionc holds and that CON(AeB) (or equivalently CON(AaB)) and, what is possible,  CON(BaA). Then (EE-conversionc) CON(BeA) and (EA-transformationc) CON(BaA), contradicting  CON(BaA). Therefore EE-conversionc cannot be correct.50 Aristotle goes on to give terms for rejecting EE-conversionc:51 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 is contingent that white does not hold of any human being – for it is also contingent that it holds of every human being –, but it is not true to say that it is contingent that human holds of nothing white. For it does not hold of many white things by necessity, but what is necessary is not contingent. (37a4-9)

Using our symbols we represent what Aristotle says as follows: furthermore, in some cases, CON(AeB) and NEC(BoA) (i.e., NEC  (BaA)). For example, 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 rejection

Introduction

33

of 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), contradicting CON(AeB). Aristotle rejects the transition from  CON(BeA) to NEC  (BeA) or, equivalently, NEC(BiA). Underlying his rejection is the idea that, even if  NEC(BiA), one might have  CON(BeA) because NEC(BoA). That is, 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 discussion it 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 he accepts 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 the Appendix on weak two-sided Theophrastean contingency. We now look at Aristotle’s rejection of *. It begins at 37a14: It is not the case that if it is not contingent that B holds of no A, it is necessary that B holds of some A. For ‘It is not contingent that B holds of no A’ is said in two ways; it is said if B holds of some A by necessity and if it does not hold of some by necessity. (37a14-17)

We take Aristotle to here be asserting (NCe) and not ( CeN). He goes on 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 it is contingent that it does not hold of all, just as if B does hold of some A by necessity, it is not true to say that it is contingent that it holds of all. (37a17-20)

That is,

34

Introduction (NCe’) NEC(BoA)   CON(BeA) (NCa’) NEC(BiA)   CON(BaA)

He now goes on to deny the analogue of * for a- and o-propositions, and insist that we might have  CON(BaA),  NEC(BoA) and NEC(BiA): So, if someone were to maintain that, since it is not contingent that B holds of all A,52 it does not hold of some by necessity, he would take things falsely. For it holds of all,53 but we say that it is not contingent that it holds of all because it holds of certain of them by necessity. (37a20-24)

Aristotle now says: Consequently both ‘X holds of some Y by necessity’ and ‘X does not hold of some Y by necessity’ are opposite to ‘It is contingent that X holds of all Y’. 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) and CON(BeA). What is not clear is whether he is simply asserting (NCa) and (NCe) or also ( CaN) and ( CeN). That he intends to make the stronger assertion is suggested by what he goes on to say about the alleged indirect proof of EE-conversionc: It is clear then that with respect to things which are contingent and not contingent in the way which we have specified initially it is necessary to take ‘B does not hold of some A by necessity’ and not ‘B holds of some A by necessity’. But if this is taken, nothing impossible results, so there is no syllogism. (37a26-30)

We are inclined to think that Aristotle should say simply that we have no right to infer NEC(BiA) from  CON(BeA). But instead he says that we must infer NEC(BoA). Alexander54 understands Aristotle’s claim to be based on the idea that NEC(BiA), i.e., NEC(AiB), is incompatible with the assumed CON(AeB). This interpretation seems to presuppose the truth of ( CeN). That is, the interpretation assumes that if  CON(BeA), then either NEC(BiA) or NEC(BoA) and rules out the former option. If this interpretation is correct, then Aristotle presumably also accepts ( CaN). Notes 1. The reader can be sure that any variable letter other than ‘A’, ‘B’, ‘C’, ‘D’ and ‘E’ has no correspondent in the Greek original. 2. In the Introduction and Summary we ignore Aristotle’s treatment of so-called indeterminate propositions, ‘X holds of Y’ and ‘X does not hold of Y’. 3. We also use the word ‘syllogism’ to mean roughly ‘valid inference’. If the

Notes to pp. 5-13

35

premisses P1 and P2 are syllogistic, Alexander says things such as ‘There is (or will be) a syllogism’, and if the conclusion yielded is P3, he often says there is a syllogism of P3. We frequently render the former words as ‘The result is a syllogism’ and the latter ‘There is a syllogism with the conclusion P3’. 4. We adopt the convention of writing the conclusions of syllogistic combinations after the premisses. 5. We will also frequently write out the propositions involved in a combination or syllogism. The order in which we list the syllogisms correponds to the way Alexander orders them. He occasionally refers to, e.g., the third syllogism in the first figure, meaning Darii1. See, for example 120,25-7. 6. For discussion see Patzig (1968), pp. 43-87. 7. See the note on 32,11 in Barnes et al., p. 87. 8. On this understanding of BiC see the notes on 49,22 (p. 111) and 32,20 (p. 88) of Barnes et al. 9. On Alexander’s terminology for contradictories and contraries, see Barnes et al., pp. 26-7. We have followed them in rendering antikeimenon ‘opposite’ and enantios ‘contrary’, saving antiphasis and antiphatikos for ‘contradictory’. In some passages (e.g. 195,18-22, 237,29-32) Alexander uses antikeimenon as a general term of which contraries and contradictories are species. But most often, e.g., in representations of reductio proofs, he uses antikeimenon to refer to the contradictory of a proposition. The reader is well advised to learn the equivalences expressed by a and b, since both Alexander and Aristotle by and large take them for granted. 10. We remark here that in the introduction and summary we pay virtually no attention to Aristotle’s uniform rejection of combinations which do not include a universal premiss. 11. Generally speaking it is not feasible to show that a combination is syllogistic by showing directly that it admits no counterinterpretation because it is not feasible to survey all possible interpretations. 12. See especially 238,22-38. 13. We do not, however, say that if P is an unqualified proposition and true, P is unqualified, because if NEC(P), then P, but P is necessary, not unqualified. The notation we have adopted represents necessity and contingency as operators on sentences. Many interpreters prefer to represent them as operators on predicates or the copula joining predicate to subject. See, e.g., Patterson (1995). Our view is that no uniform representation, i.e., one in which the same words of Aristole are always or almost always represented by the same formula, is fully satisfactory, and that the notation we have adopted is simple and by and large adequate to capture Alexander’s perspective. For the most part, notation becomes significant when one is concerned with the question of truth, e.g., whether or not it is the case that a certain combination is syllogistic or a conversion rule correct. When one is concerned, as Alexander for the most part is, with the overall coherence of what Aristotle says, the interpretation of a formalism is much less significant: roughly speaking one can interpret the formalism however one wants as long as one interprets it consistently. 14. And also – except in the UC and NC cases – complete. The situation changes somewhat when contingent premisses are introduced because the conversion rules allow for the justification of syllogisms with no analogue among combinations not containing a contingent premiss. 15. More precisely, Aristotle uses the equivalent in his argument at 1.15, 34a34-b2 that Barbara1(UC_) yields a contingent conclusion and claims at 1.16,

36

Notes to pp. 13-20

35b37-36a2 that the fact that Barbara1(NC_) also yields such a conclusion ‘will be proved in the same way as in the preceding cases’. 16. See, e.g., 174,13-19. 17. We here begin a practice of writing ‘C’ or ‘CON’ where there is some unclarity about the specific character of an allegedly contingent propostion. 18. Aristotle’s formulation at 30a30-2 is slightly different. 19. It appears that some people tried to reject (a’) by saying that Aristotle does not interpret unqualified propositions as hypotheses. Alexander shows the untenability of this position; see 126,9-22 and 130,23-4. 20. See Patterson (1995). 21. See the textual note on 30a21-2 (Appendix 6). 22. This is the way Alexander expresses 1.1, 24b29-30. When applied to the notion of holding of all by necessity it provides one of the clearest expressions of the idea of de re necessity: A holds of all B by necessity if A holds by necessity of whatever is under B. Cf. 129,34-130,1 and 167,14-18. 23. Alexander most frequently refers to Theophrastus and Eudemus with some such phrase as ‘his [i.e., Aristotle’s] associates’; sometimes he names them both, and sometimes he names only Theophrastus. At no point does he distinguish between their views, and we see no basis for trying to do so. We shall follow most modern scholarship by talking only about Theophrastus. 24. Alexander’s fullest discussion is at 123,28-127,16; cf. 129,21-130,24 and 132,23-34. The crucial applications of the rule come in connection with the first-figure NUN cases (and their consequences) and first-figure NC_ cases which Aristotle says imply unqualified conclusions. See, e.g., 1.16, 36a7-17 with Alexander’s discussion at 208,8-209,32. 25. See, e.g., 247,39-248,3. 26. Assume, as is possible, that AaB,  NEC(AiB), NEC(BaC), and assume that Barbara1(UNU) is valid. Then AaC, which with NEC(BaC) implies (Darapti3(UNN)) NEC(AiB), contradicting  NEC(AiB). Hence Barbara1(UNU) is not valid. This argument is a demonstration of the incoherence of Aristotle’s treatment of combinations with a necessary and an unqualified premiss. 27. Alexander gives the incompatibility rejection argument for Celarent1(UNN) at 130,25-131,4. 28. We give the arguments. For Celarent1(NUN) NEC(AeB) BaC NEC(AeC) NEC(AeC) and NEC(AeB) entail nothing, and NEC(AeC) and BaC entail (Ferison3(NUN)) NEC(AoC), which is implied by NEC(AeC). In the case of: Darii1(NUN) NEC(AaB) BiC NEC(AiC) and: Ferio1(NUN) NEC(AeB) BiC NEC(AoC), nothing is entailed by the conclusion and either of the other premisses. 29. See the notes on 132,8 and 17. 30. See the note on 133,20. 31. For discussion see the note on 132,29. 32. See chapter 1.16 and volume 2, Introduction, section III.E.2.a. 33. The third adjunct (prosrhêsis) is ‘It is contingent that’. See 1.2, 25a2-3 with Alexander’s explanation at 26,29-27,1. 34. At 156,26 Alexander mentions a second consideration: that an unqualified proposition is ‘necessary on a condition’ and so ruled out by the words ‘if P is not necessary’. For discussion see Appendix 3 on conditional necessity. 35. Alexander’s interpretation is very problematic. For, as we will see shortly, Aristotle is committed to the idea that, e.g., CON(AaB) implies CON(AoB). But

Notes to pp. 21-32

37

then, if CON(AaB) is true, so is CON(AoB), and hence so are  (AaB) and  (AoB), i.e., AoB and  (AoB). Alexander attempts unsuccessfully to wriggle out of these difficulties at 161,3-26; see also 222,16-35. 36. We note that this means that, at least within the context of syllogistic, neither of them is committed to two-sided contingency, if that means the equivalence of CON(P) and CON(  P) for any proposition P. 37. See 159,22-4. 38. Here and elsewhere Aristotle speaks of conversion. Modern scholars sometimes speak of complementary conversion. In our discussion we use the word ‘transformation’ to bring out that the order of terms is preserved when the rule is applied. 39. For an incisive account of the difficulties involved in what Alexander says here see Barnes et al., pp. 79-80, n. 157. Although we do not claim to be able to eliminate these difficulties, we hope to give some sense of what Alexander has in mind. 40. See especially 38,23-6. 41. We use the future tense ‘will be’ because Alexander says things such as that a contingent proposition does ‘not yet’ (mêdepô) hold (e.g. at 156,18). Alexander never considers the possibility of a proposition which held at some time in the past but never thereafter, but it does not seem unreasonable for logical purposes to take his references to the future in such contexts to include the past, so that a contingent proposition is understood to be one that holds at some time but not at the present. For a discussion of this whole topic see Hintikka (1973). For a discussion of Alexander’s conception of possibility see Sharples (1982). 42. Here we depart significantly from the translation of Barnes et al. And this is one of the many places in which we have inserted variables where Alexander has none. In the present case the insertion requires interpretation of the text. One might choose to interchange the B’s and A’s in sentence (v). 43. Accepting the reading endekhetai of some manuscripts adopted by Barnes et al. 44. Here we follow the manuscripts rather than accepting the emendation of Barnes et al.; see their note 51 on 37,16 (p. 94); nothing significant turns on this difference. 45. We note that in Alexander’s argument for EE-conversionn, the question of how AiB holds is irrelevant since, no matter how it holds, AiB contradicts NEC(AeB). 46. See the Greek-English Index. 47. Compare, e.g., 1.14, 33b3-8 with 1.15, 35b11-19. 48. For deviations of Alexander’s text of this passage from Ross’s see Barnes et al., pp. 200-1. 49. Most of Alexander’s discussion of this passage (39,17-40,4) is devoted to explaining that, although what is contingent may not hold for the most part, Aristotle mentions only what holds for the most part – which, according to Alexander, is the same as what holds by nature – because there is no scientific value in arguments about what holds no more often than it fails to hold. Aristotle has something further to say on this subject at 1.13, 32b4-22, and in connection with this material Alexander discusses the subject in more detail (161,29165,14). 50. As always, Aristotle and so Alexander present these arguments in what we think is a less satisfactory way. They assume EE-conversionc and

38

Notes to pp. 32-34

CON(AeB), infer CON(BaA) and then point out that CON(AeB) is compatible with  CON(BaA). 51. Alexander (221,7-13) shows uncertainty about whether what follows is an independent argument. 52. Aristotle’s text actually says ‘C holds of all D’, but the change in letters is irrelevant. 53. Aristotle does not need BaA only  NEC(BoA), i.e.,  NEC  (BaA); see the note on 225,21. 54. See especially 226,13-31. Immediately after this passage at 226,32-227,9 Alexander gives the correct explanation of the illegitimacy of the inference.

Summary

Symbols and rules Our symbols are all explained in the introduction. We here give brief explications of the less usual ones. NEC(P) is read ‘It is necessary that P’. CON(P) is read ‘It is contingent that P’. In the introduction we have tried to ‘unfold’ our understanding of the relevant notion of contingency. Because Aristotle wavers in his understanding we sometimes write ‘CON’(P) to indicate that the notion of contingency is uncertain in one way or another. We frequently write  NEC  (P) to stand for ‘It is contingent (in another sense) that P’; this sense is so-called Theophrastean contingency; we sometimes use CONt(P) as an abbreviation for  NEC  (P). Finally, Aristotle sometimes infers ‘It is contingent that P’ from P; in these cases we write CONu(P). We also recall the following abbreviations: XaY XeY XiY XoY

for ‘X holds of all Y’ or ‘All Y are X’ for ‘X holds of no Y’ or ‘No Y are X’ for ‘X holds of some Y’ or ‘Some Y are X’ for ‘X does not hold of some Y’ or ‘There are some Y which are not X’

The following three equivalences are frequently taken for granted:   P if and only if P XaY if and only if  (XoY) (so that also XoY if and only if  (XaY)) XeY if and only if  (XiY) (so that also XiY if and only if  (XeY)) The relations among the different modal notions are given by the following rules: U   N C   N

P   NEC (P) CON(P)   NEC (P) P  CONu(P)

Aristotle accepts the following transformation rules: EE-conversionu: AI-conversionu: II-conversionu:

XeY  YeX XaY  YiX XiY  YiX

(1.2, 25a14-17) (1.2, 25a17-19) (1.2, 25a20-2)

40

Summary EE-conversionn: AI-conversionn: II-conversionn:

NEC(XeY)  NEC(YeX) (1.3, 25a29-31) NEC(XaY)  NEC(YiX) (1.3, 25a32-4) NEC(XiY)  NEC(YiX) (1.3, 25a32-4)

AI-conversionc: II-conversionc:

CON(XaY)  CON(YiX) (1.3, 25a37-b3) CON(XiY)  CON(YiX) (1.3, 25a37-b3)

EA-transformationc: AE-transformationc: IO-transformationc: OI-transformationc:

CON(XeY)  CON(XaY)  CON(XiY)  CON(XoY) 

CON(XaY) CON(XeY) CON(XoY) CON(XiY)

(1.13, 32a34) (1.13, 32a34) (1.13, 32a35) (1.13, 32a35)

He rejects: OO-conversionu: XoY  YoX (1.2, 25a22-6) OO-conversionn: NEC(XoY)  NEC(YoX) (1.3, 25a34-6) *EE-conversionc: CON(XeY)  CON(YeX).1 (1.17, 36b35-37a31). Theophrastus apparently followed Aristotle on the conversion of unqualified and necessary propositions, but he accepted analogues of the same rules for Ct propositions, i.e., AI-conversionCt: II-conversionCt: EE-conversionCt:

CONt(XaY)  CONt(YiX) CONt(XiY)  CONt(YiX) CONt(XeY)  CONt(YeX)

but not OO-conversionCt:

CONt(XoY)  CONt(YoX)

These principles are consequences of Aristotelian assumptions about NEC and the definition of CONt as  NEC  .

Assertoric Syllogistic (Chapters 4-6) First figure (Chapter 4) Complete Barbara1(UUU) Celarent1(UUU) Darii1(UUU) Ferio1(UUU)

AaB AeB AaB AeB

BaC BaC BiC BiC

AaC AeC AiC AoC

(25b37-40) (25b40-26a2) (26a23-5) (26a25-7)

Summary

41

Second figure (Chapter 5) Direct reductions AeB AaC BeC (27a5-9) Cesare2(UUU) Since AeB (EE-conversionu) BeA. So (Celarent1(UUU)) BeC. Camestres2(UUU) AaB AeC BeC (27a9-15) Since AeC (EE-conversionu) CeA. So (Celarent1(UUU)) CeB, and so (EE-conversionu) BeC. Festino2(UUU) AeB AiC BoC (27a32-6) Since AeB (EE-conversionu) BeA. So (Ferio1(UUU)) BoC. Indirect reduction Baroco2(UUU) AaB AoC BoC (27a36-b3) Assume  (BoC), i.e., BaC. So (Barbara1(UUU)) AaC, contradicting AoC. Third figure (Chapter 6) Direct reductions Darapti3(UUU) AaC BaC AiB (28a17-26) Since BaC (AI-conversionu) CiB. So (Darii1(UUU)) AiB. Felapton3(UUU) AeC BaC AoB (28a26-30) Since BaC (AI-conversionu) CiB. So (Ferio1(UUU)) AoB. Datisi3(UUU) AaC BiC AiB (28b7-11) Since BiC (II-conversionu) CiB. So (Darii1(UUU)) AiB. Disamis3(UUU) AiC BaC AiB (28b11-15) Since AiC (II-conversionu) CiA. So (Darii1(UUU)) BiA, and (II-conversionu) AiB. Ferison3(UUU) AeC BiC AoB (28b33-5) Since BiC (II-conversionu) CiB. So (Ferio1(UUU)) AoB. Indirect reduction AoC BaC AoB (28b17-21) Bocardo3(UUU) Assume  (AoB), i.e., AaB. So Barbara1(UUU)) AaC, contradicting AoC. All other premiss combinations rejected. Modal syllogistic without contingency NNN (Chapter 8) First figure Complete Barbara1(NNN) Celarent1(NNN)

NEC(AaB) NEC(AeB)

NEC(BaC) NEC(BaC)

NEC(AaC) NEC(AeC)

42

Summary

Darii1(NNN) Ferio1(NNN)

NEC(AaB) NEC(AeB)

NEC(BiC) NEC(BiC)

NEC(AiC) NEC(AoC)

Second figure Direct reductions (cf. the corresponding UUU cases) NEC(AeB) Cesare2(NNN) Camestres2(NNN) NEC(AaB) Festino2(NNN) NEC(AeB)

NEC(AaC) NEC(AeC) NEC(AiC)

NEC(BeC) NEC(BeC) NEC(BoC)

Proof by ekthesis (not accepted by Theophrastus) *Baroco2(NNN) NEC(AaB) NEC(AoC) NEC(BoC) (30a6-14) Take D to be a part of C such that NEC(AeD). Then (Camestres2(NNN)) NEC(BeD). But D is part of C. So NEC(BoC). Third figure Direct reductions (cp. the corresponding UUU cases) Darapti3(NNN) Felapton3(NNN) Datisi3(NNN) Disamis3(NNN) Ferison3(NNN)

NEC(AaC) NEC(AeC) NEC(AaC) NEC(AiC) NEC(AeC)

NEC(BaC) NEC(BaC) NEC(BiC) NEC(BaC) NEC(BiC)

NEC(AiB) NEC(AoB) NEC(AiB) NEC(AiB) NEC(AoB)

Proof by ekthesis (not accepted by Theophrastus) *Bocardo3(NNN) NEC(AoC) NEC(BaC) NEC(AoB) (30a6-14) Take D to be a part of C such that NEC(AeD). Then, since by necessity all C are B and D is a part of B, NEC(BaD) and (Felapton3(NNN)) NEC(AoB). All other NN combinations (tacitly) rejected. N+U (Chapters 9-11) First figure (Chapter 9) Complete NUN (held to be NUU by Theophrastus)2 *Barbara1(NUN) *Celarent1(NUN) *Darii1(NUN) *Ferio1(NUN)

NEC(AaB) NEC(AeB) NEC(AaB) NEC(AeB)

BaC BaC BiC BiC

NEC(AaC) NEC(AeC) NEC(AiC) NEC(AoC)

(30a17-23) (30a17-23) (30a37-b2) (30a37-b2)

Summary

43

UNU Barbara1(UNU) Celarent1(UNU) Darii1(UNU) Ferio1(UNU)

AaB AeB AaB AeB

NEC(BaC) NEC(BaC) NEC(BiC) NEC(BiC)

AaC AeC AiC AoC

(30a23-33) (30a23-33) (30b2-6) (30b2-6)

Second figure (Chapter 10) Direct reductions (cf. the corresponding UUU cases) Cesare2(NUN) Cesare2(UNU) Camestres2(UNN) Camestres2(NUU) Festino2(NUN) Festino2(UNU)

NEC(AeB) AeB AaB NEC(AaB) NEC(AeB) AeB

AaC NEC(AaC) NEC(AeC) AeC AiC NEC(AiC)

NEC(BeC) BeC NEC(BeC) BeC NEC(BoC) BoC

(30b9-13) (30b18-19) (30b14-18) (30b18-40) (31a5-10) absent

Indirect reductions *Baroco2(NUU) *Baroco2(UNU)

NEC(AaB) AoC BoC AaB NEC(AoC) BoC

(31a10-15) (31a15-17)

Third figure (Chapter 11) Direct reductions (cf. the corresponding UUU cases) Darapti3(NUN) Darapti3(UNN) Felapton3(NUN) Felapton3(UNU) Datisi3(NUN) Datisi3(UNU) Disamis3(UNN) Disamis3(NUU) Ferison3(NUN) Ferison3(UNU)

NEC(AaC) AaC NEC(AeC) AeC NEC(AaC) AaC AiC NEC(AiC) NEC(AeC) AeC

BaC NEC(BaC) BaC NEC(BaC) BiC NEC(BiC) NEC(BaC) BaC BiC NEC(BiC)

NEC(AiB) NEC(AiB) NEC(AoB) AoB NEC(AiB) AiB NEC(AiB) AiB NEC(AoB) AoB

(31a24-30) (31a31-3) (31a33-7) (31a37-b10) (31b19-20) (31b20-31) (31b12-19) (31b31-3) (31b35-7) (32a1-4)

Indirect reductions *Bocardo3(UNU) *Bocardo3(NUU)

AoC NEC(BaC) AoB NEC(AoC) BaC AoB

All other N+U combinations (tacitly) rejected.

(31b40-32a1) (32a4-5)

44

Summary Syllogistic with contingency (Chapters 13-22) CCC (Chapters 14, 17, 20) First figure (Chapter 14) Complete

Barbara1(CCC) Celarent1(CCC) Darii1(CCC) Ferio1(CCC)

CON(AaB) CON(AeB) CON(AaB) CON(AeB)

CON(BaC) CON(BaC) CON(BiC) CON(BiC)

CON(AaC) CON(AeC) CON(AiC) CON(AoC)

(32b38-33a1) (33a1-5) (33a23-5) (33a25-7)

Waste cases justifiable by transformationc rules (but not by Theophrastus’ transformationCt rules) AEA1(CCC) EEA1(CCC) AOI1(CCC) EOO1(CCC)

CON(AaB) CON(AeB) CON(AaB) CON(AeB)

CON(BeC) CON(BeC) CON(BoC) CON(BoC)

CON(AaC) (33a5-12) CON(AaC) (33a12-20) CON(AiC) (33a27-34) CON(AoC) (included in general statement at 33a21-3)

Aristotle rejects all forms with a particular major and either a universal or a particular minor premiss at 33a34-b17. Second figure (Chapter 17) *Aristotle rejects all second-figure CC combinations. Third figure (Chapter 20) Direct reductions (cf. the corresponding UUU cases) Darapti3(CCC) CON(AaC) CON(BaC) CON(AiB) Felapton3(CCC) CON(AeC) CON(BaC) CON(AoB) Datisi3(CCC) CON(AaC) CON(BiC) CON(AiB) Disamis3(CCC) CON(AiC) CON(BaC) CON(AiB) Ferison3(CCC) CON(AeC) CON(BiC) CON(AoB)

(39a14-19) (39a19-23) (39a31-5) (39a35-6) (39a36-8)

Waste cases justifiable by transformationc rules EEI3(CCC) AEI3(CCC) AOI3(CCC) OAI3(CCC) OEO3(CCC) EOO3(CCC) IEO3(CCC)

CON(AeC) CON(AaC) CON(AaC) CON(AoC) CON(AoC) CON(AeC) CON(AiC)

CON(BeC) CON(BeC) CON(BoC) CON(BaC) CON(BeC) CON(BoC) CON(BeC)

CON(AiB) (39a26-8) CON(AiB) (not mentioned) CON(AiB) (not mentioned) CON(AiB) (39a36-8?) CON(AiB) (39a38-b2) CON(AiB) (39a38-b2) CON(AoB) (not mentioned)

Summary

45

Aristotle rejects CC combinations with no universal premisses at 39b2-6. U+C (Chapters 15, 18, 21) First figure (Chapter 15) Complete (CUC) Barbara1(CUC) Celarent1(CUC) Darii1(CUC) Ferio1(CUC)

CON(AaB) CON(AeB) CON(AaB) CON(AeB)

BaC BaC BiC BiC

CON(AaC) CON(AeC) CON(AiC) CON(AoC)

(33b33-6) (33b36-40) (35a30-5) (35a30-5)

Incomplete (UC‘C’)3 *Barbara1(UC‘C’) *Celarent1(UC N ) Darii1(UC‘C’) Ferio1(UC N )

AaB AeB AaB AeB

CON(BaC)  NEC  (AaC) (34a34-b2) CON(BaC)  NEC  (AeC) (34b19-35a2) CON(BiC)  NEC  (AiC) (35a35-b8) CON(BiC)  NEC  (AoC) (35a35-b8)

Waste cases justifiable by transformationc rules AEA1(UC‘C’) EEE1(UC  N  ) AOI1(UC‘C’) EOO1(UC  N  )

AaB AeB AaB AeB

CON(BeC)  NEC  (AaC) CON(BeC)  NEC  (AeC) CON(BoC)  NEC  (AiC) CON(BoC)  NEC  (AoC)

(35a3-11) (35a11-18) (35a35-b8) (35a35-b8)

Aristotle rejects EE_1(CU_) and AE_1(CU_) at 35a20-4, AO_1(CU_) and EO_1(CU_) at 35b8-11, all forms with a particular major and either a universal or a particular minor premiss at 35b11-19. Second figure (Chapter 18) Direct reductions (cf. the corresponding UUU cases) Cesare2(UC‘C’) *Camestres2(CU‘C’) Festino2(UC‘C’) *IEO2(CU?)

AeB CON(AaC) CON(AaB) AeC AeB CON(AiC) CON(AiB) AeC

 NEC  (BeC) (37b23-8)  NEC  (BeC) (37b29)  NEC  (BoC) (38a3-4)  NEC  (CoB)? (38a3-4?)

Waste cases justifiable by transformationc rules EEE2(UC‘C’) EEE2(CU‘C’) EOO2(UC‘C’) OEO2(CU?)

AeB CON(AeB) AeB CON(AoB)

CON(AeC) AeC CON(AoC) AeC

 NEC  (BeC)  NEC  (BeC)  NEC  (BoC)  NEC  (CoB)?

(37b29-35) (37b29-35) (38a4-7) (38a4-7?)

46

Summary Rejected standard cases

Cesare2(CU_) Camestres2(UC_) Festino2(CU_) Baroco2(UC_) Baroco2(CU_)

CON(AeB) AaB CON(AeB) AaB CON(AaB)

AaC CON(AeC) AiC CON(AoC) AoC

(37b19-23) (37b19-23) (37b39-38a2) (37b39-38a2) (38a8-10)

Aristotle rejects the waste cases AA_2(CU_) and AA_2(UC_) at 37b35-8, and he rejects OA_2(UC_), OE_2(UC_), and EO_2(CU_) at 38a8-10. He apparently rejects IE_2(UC_) and OA_2(CU_) at 37b39, and he rejects cases without a universal premiss at 38a10-12. He does not mention the waste cases AI_2(CU_), AI_2(UC_), IA_2(UC_), and IA_2(CU_), which are presumably rejected. Third figure (Chapter 21) Direct reductions (cf. the corresponding UUU cases) Darapti3(UC‘C’) Darapti3(CUC) Felapton3(CUC) Felapton3(UC‘C’) *AE?3(CU?) *EE?3(CU?) Datisi3(CUC) Datisi3(UC‘C’) Disamis3(UCC) 31) Disamis3(CU‘C’) Ferison3(CUC) Ferison3(UC‘C’) IEO3(UCC) 31) IE?3(CU?)

AaC CON(AaC) CON(AeC) AeC CON(AaC) CON(AeC) CON(AaC) AaC AiC

CON(BaC) BaC BaC CON(BaC) BeC BeC BiC CON(BiC) CON(BaC)

 NEC  (AiB) (39b10-16) CON(AiB) (39b16-22) CON(AoB) (39b16-22)  NEC  (AoB) (39b16-22)  NEC  (AoB)? (39b22-5)  NEC  (AoB)? (39b22-5) CON(AiB) (39b26-31)  NEC  (AiB) (39b26-31) CON(AiB) (39b26-

CON(AiC) CON(AeC) AeC AiC

BaC BiC CON(BiC) CON(BeC)

 NEC  (AiB) (39b26-31) CON(AoB) (39b26-31)  NEC  (AoB) (39b26-31) CON(AoB) (39b26-

CON(AiC) BeC

 NEC  (AoB)? (39b26-31)

Waste cases justifiable by transformationc rules CON(BeC) AEI3(UC‘C’) AaC EEO3(UC‘C’) AeC CON(BeC) EOO3(UC‘C’) AeC CON(BoC) tioned) OE?3(CU?) CON(AiC) BeC

 NEC  (AiB) (39b22-5)  NEC  (AoB) (39b22-5)  NEC  (AoB) (not men NEC  (AoB)? (not mentioned)

Indirect reductions

Summary *Bocardo3(CU‘C’) AO?3(UC?) AO?3(CU?) OA?3(UC?)

47

CON(AoC) BaC  NEC  (AoB) (39b31-9) AaC CON(BoC) ? (39b31-9?) CON(AaC) BoC ? (39b31-9?) AoC CON(BaC) ? (39b31-9?) Further waste cases

EO?3(CU?) OA?3(CU?)

AeC CON(AoC)

CON(BoC) BaC

? ?

(not mentioned) (not mentioned)

These are all the third-figure U+C combinations. N+C (Chapters 16, 19, 22) First figure (Chapter 16) Complete (CNC) Barbara1(CNC) Celarent1(CNC) Darii1(CNC) Ferio1(CNC)

CON(AaB) CON(AeB) CON(AaB) CON(AeB)

NEC(BaC) NEC(BaC) NEC(BiC) NEC(BiC)

CON(AaC) CON(AeC) CON(AiC) CON(AoC)

(36a2-7) (36a17-24) (not mentioned) (36a39-b2)

Incomplete (NC‘C’)4 *Barbara1(NC‘C’) NEC(AaB) 36a2) *Celarent1(NCCu) NEC(AeB) *Darii1(NC‘C’) NEC(AaB) Ferio1(NCCu) NEC(AeB)

CON(BaC)

 NEC (AaC) (35b37-

CON(BaC) CONu(AeC) (36a7-17) CON(BiC)  NEC  (AiC) (36a39-b2) CON(BiC) CONu(AoC) (36a34-9)

Waste cases justifiable by transformationc rules AEA1(NC‘C’) EEE1(NCCu) AOI1(NC‘C’) EOO1(NCCu)

NEC(AaB) NEC(AeB) NEC(AaB) NEC(AeB)

CON(BeC)  NEC  (AaC) (36a25-27) CON(BeC) CONu(AeC) (not mentioned) CON(BoC)  NEC  (AiC) (35b28-30?) CON(BoC) CONu(AoC) (35b30-31?)

Aristotle rejects AE_1(CN_) and EE_1(CN_) at 36a27-31, IA_1(NC_), OA_1(NC_), IE_1(NC_), and OE_1(NC_) at 36b3-7, IE_1(CN_), OE_1(CN_), IA_1(CN_), and OA_1(CN_) at 36b7-12, and all combinations with two particular premisses at 36b12-18. He tacitly rejects AO_1(CN_) and EO_1(CN_). Second figure (Chapter 19) Direct reductions (cf. the corresponding UUU cases) NEC(AeB) CON(AaC) CONu(BeC) Cesare2(NCCu) u Camestres2(CNC ) CON(AaB) NEC(AeC) CONu(BeC)

(38a16-25) (38a25-6)

48

Summary

Festino2(NCCu) IEO2(CN?)

NEC(AeB) CON(AiC) CONu(BoC) (38b24-7) CON(AiB) NEC(AeC) CONu(CoB)? (38b24-7?)

Waste cases justifiable by transformationc rules EEE2(NCC ) EEE2(CNCu) EOO2(NCCu) OEO2(CN?)

NEC(AeB) CON(AeB) NEC(AeB) CON(AoB)

u

CON(AeC) NEC(AeC) CON(AoC) NEC(AeC)

CONu(BoC) (38b6-12) CONu(BoC) (38b12-13) CONu(BoC) (38b31-5) CONu(CoB)? (38b31-5?)

Rejected standard cases CON(AeB) *Cesare2(CN_) *Camestres2(NC_) NEC(AaB) Festino2(CN_) CON(AeB) *Baroco2(NC_) NEC(AaB) Baroco2(CN_) CON(AaB)

NEC(AaC) CON(AeC) NEC(AiC) CON(AoC) NEC(AoC)

(38a26-b4) (38b4-5) (not mentioned) (38b27-9) (not mentioned)

The rejected standard cases generate the following equally problematic waste cases: AA_2(CN_) and AA_2(NC) (both rejected at 38b13-23), AI_2(CN_) and AI_2(NC_) (both rejected at 38b29-31, where Aristotle also rejects IA_2(CN_) and IA_2(NC_)) and EO_2(CN_), which Aristotle does not discuss. Aristotle also rejects OA_2(CN_) at 38b27-29, and the rejection of IA_2(NC_) carries with it the rejection of IE_2(NC_). He rejects all forms with two particulars at 38b35-7. He tacitly rejects OA_2(NC_) and OE_2(NC_). Third-figure (Chapter 22) Direct reductions (cf. the corresponding UUU cases) Darapti3(NC‘C’) Darapti3(CNC) Felapton3(CNC) Felapton3(NCCu) Datisi3(CNC) Datisi3(NC‘C’) Disamis3(NCC) Disamis3(CN‘C’) Ferison3(CNC) Ferison3(NCCu) *Bocardo3(CN‘C’) Bocardo3(NCCu)

NEC(AaC) CON(BaC)  NEC  (AiB) CON(AaC) NEC(BaC) CON(AiB) CON(AeC) NEC(BaC) CON(AoB) NEC(AeC) CON(BaC) CONu(AoB) CON(AaC) NEC(BiC) CON(AiB) NEC(AaC) CON(BiC)  NEC (AiB) NEC(AiC) CON(BaC) CON(AiB) CON(AiC) NEC(BaC)  NEC  (AiB) CON(AeC) NEC(BiC) CON(AoB) NEC(AeC) CON(BiC) CONu(AoB) CON(AoC) NEC(BaC)  NEC  (AoB) NEC(AoC) CON(BaC) CONu(AoB)

(40a12-16) (40a16-18) (40a18-25) (40a25-32) (40a40-b2) (40a40-b2) (40a40-b2) (40a40-b2) (40b2-3) (40b3-4) (40b2-3)5 (40b3-4?)

Waste cases justifiable by transformationc rules AE_3(NC‘C’)

NEC(AaC) CON(BeC)  NEC  (AiB)? (40a33-5)

EEO3(NCCu) AOI3(NC‘C’) IEO3(NCC) EOO3(NCCu) OEO3(NCCu)

NEC(AeC) NEC(AaC) NEC(AiC) CON(AeC) NEC(AeC)

Summary

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CON(BeC) CON(BoC) CON(BeC) NEC(BoC) CON(BiC)

CONu(AoB) (not discussed)  NEC  (AiB) (40b2-3?) CON(AoB) (40b8-12)6 CONu(AoB) (not discussed) CONu(AoB) (not discussed)

Rejected Cases *AE_3(CN_) *IE_3(CN_)

CON(AaC) NEC(BeC) CON(AiC) NEC(BeC)

(40a35-8) (40b8-12)

These two rejections imply rejection of their equivalents, EE_3(CN_) and OE_3(CN_). Aristotle tacitly rejects AO_3(CN_), EO_3(CN_), and all third-figure N+C combinations with two particular premisses.

Notes 1. Asterisks indicate places of difficulty in the modal syllogistic on which Alexander has an interesting discussion. 2. The controversy concerning these four syllogisms transfers to any N+U combination held by Aristotle to have a necessary conclusion. 3. These cases are very problematic, especially Barbara and Celarent; their problematic nature transmits itself to combinations reduced to them. 4. The difficulties attaching to Barbara1(UC‘C’) transfer to Barbara1(NC‘C’). New difficulties arise with Celarent1(NCCu). 5. Alexander wavers between thinking Aristotle espouses Bocardo3(CN‘C’) and OAI3(CN‘C’), the waste case of Disamis3(CN‘C’). 6. The waste case (of Disamis3(NCC)) would actually be: IEI3(NCC) NEC(AiC) CON(BeC) CON(AiB) but Aristotle implies a derivation of the syllogism we have given, and Alexander carries it out at 253,23-7, perhaps in order to keep the conclusion of a syllogism with a negative premiss negative.

Alexander of Aphrodisias On Aristotle Prior Analytics 1.8-13 (with 1.17, 36b35-37a31) Translation

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1.8 Combinations with two necessary premisses. All figures1 29b29 Since holding, holding by necessity and it being contingent that something holds are distinct [for many things hold, but not by necessity; others hold neither by necessity nor at all, but it is contingent that they hold – it is clear that a syllogism of each of these things will also be distinct and its terms will not be related in the same way, but one syllogism will be from necessary things, a second from unqualified ones, a third from contingent ones.]2 It is reasonable that he next gives his account of necessary syllogisms, which are from necessary premisses, and, after treating them, his account of contingent syllogisms, since holding and holding by necessity and it being contingent that something holds are distinct. And, since these are distinct, it is clear that the corresponding propositions and syllogisms will differ from one another in accordance with these differences. Consequently, since they do differ, it is necessary for anyone who discusses syllogisms and their construction and production to speak about them also. For in this way he will have spoken about the construction and production of all categorical syllogisms if he goes through all the modalities with respect to which such syllogisms differ. He indicates to us very clearly the way in which the unqualified proposition differs from the necessary and the contingent one when he says: 29b30 For many things hold, but not by necessity; others hold neither by necessity nor at all, but it is contingent that they hold. For in this way unqualified propositions will be indicative of holding but not necessarily holding, necessary propositions of necessarily holding, and contingent propositions of not holding now and being capable of holding;3 and these propositions truly indicate what they indicate if they are true, falsely indicate it if they are false. It is clear that the difference of propositions and of the syllogisms with respect to the modalities can be recognized by the addition of the modalities;

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for the appropriate modality will be co-predicated for each proposition and each syllogism. 120,1

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29b33 [it is clear that a syllogism of each of these things will be distinct] and its terms will not be related in the same way, [but one syllogism will be from necessary things, a second from unqualified ones, a third from contingent ones.] He says that the ‘syllogism of each of these things’ (i.e., of what is necessary or unqualified or contingent) ‘will be distinct’, and he indicates how they are distinct: they are not from similar propositions and terms, but some syllogisms will be ‘from necessary things’, some ‘from unqualified ones’, and some ‘from contingent ones’. 29b364 They are related in much the same way in the case of necessary and of unqualified things. [For, if the terms are posited in the same way in the case of holding and in that of holding by necessity – or in the case of not holding – there either will or there won’t be a syllogism , except that they will differ 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 same account of ‘being in as a whole’ and ‘said of all’.]

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He says that the syllogisms with a necessary conclusion will exist in the same way in each figure as they were proved to occur in the case of the unqualified. For in each figure the same combination of premisses along with the addition of necessity will make necessary syllogisms. And there will be four syllogistic combinations in the first figure, four in the second, and six in the third. The reason for this is, , that ‘said of all’ and ‘said of none’ – through which the syllogisms in the first figure are proved – are taken in the same way in the case of what is necessary and of what is unqualified, and, , that the conversions of necessary propositions – through which three syllogisms in the second figure and five in the third are proved to yield a conclusion – have been shown to be the same in the case of unqualified things and of necessary ones. 30a2 For the privative converts in the same way, [and we will give the same account of ‘being in as a whole’ and ‘said of all’. In the other cases the conclusion will be shown to be necessary by conversion in the same way as in the case of unqualified holding. But in the middle figure, when the universal is affirmative and the particular privative,5 and again in the third figure when

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the universal is affirmative and the particular is privative,6 the demonstration will not be the same, but it is necessary to set out something of which each does not hold and make the syllogism with respect to this. For the syllogism will be necessary in the case of these things. But if the syllogism is necessary with respect to what is set out, it is necessary with respect to some of what is set out. For what is set out is just what some of the original term is. And each of the syllogisms comes in the appropriate figure.] The words ‘for the privative converts in the same way’ should not be understood as meaning ‘only the privative converts similarly’. For affirmative propositions also convert similarly. He mentions only negative propositions because the affirmatives convert in the same way in all modalities, but the negatives do not, since they do so differently in the case of contingent propositions.7 Since conversion is not thought to be the same in the case of all negatives, he reminds us that these also behave in the same way in these modalities. However, in the case of syllogisms composed of unqualified propositions, neither the fourth syllogism in the second figure, which has a negative particular minor, nor the sixth in the third, which has a particular negative major, was proved by conversion, but by reductio ad impossibile.8 As will be clear, he does not use the same proof in the case of these syllogisms when they have necessary premisses. he adds the words ‘in much the same way’.9 The others are proved in the same way. Consequently he doesn’t mention them further, thinking that what he said about is also sufficient for understanding them. But he does mention the syllogisms which are not still proved in the same way, and he tells us how in their case we should give the proof and make them syllogistic. In their case he does not use reductio ad impossibile – as he did when the premisses were unqualified – because the opposite of what is necessary is contingent. In both cases the conclusion proved is a particular negative necessary proposition of which the opposite is ‘It is contingent that X holds of all Y’. But if this is taken as hypothesis and the other premiss, which is necessary universal affirmative, is added, this combination is a mixture of a contingent universal affirmative proposition and a necessary universal affirmative one. But in the case of such mixtures it is not yet known what follows.10 Furthermore, it is necessary to understand the simple cases11 first, and then to understand the composite ones. Therefore, he rejects the indirect method of proof as unclear and dependent on posterior things. And he speaks about both of them12 with the following words:

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Translation 30a9 [But in the middle figure, when the universal is affirmative and the particular privative, and again in the third figure when the universal is affirmative and the particular is privative, the demonstration will not be the same,] but it is necessary to set out something of which each does not hold and make the syllogism with respect to this. [For the syllogism will be necessary in the case of these things. But if the syllogism is necessary with respect to what is set out, it is necessary with respect to some of what is set out. For what is set out is just what some of the original term is. And each of the syllogisms comes in the appropriate figure.]

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He is speaking about the particular negative premisses in the two figures at the same time. And he is not saying about both of them that it is necessary for something of which they do not hold to be set out; for, if he said that, he would be speaking about two negative premisses and he would not even say that they compose the combinations under consideration.13 Rather, he is maintaining the following. In both the combinations, that in the second figure and that in the third, there is a particular negative premiss; therefore, taking – in the case of each combination of premisses all that of which does not hold,14 one makes the syllogism with respect to that by transforming the particular negative premiss into a universal negative one and proving by conversion that the combination with the universal negative premiss is syllogistic – just as was done in the case of the other combinations in which the universal premiss was negative; what has been proved in this way is that by necessity it does not hold of some of that of which the other is a part.15 For example, if A holds of all B by necessity and A does not hold of some C by necessity (for this is the second-figure combination which he is discussing), he says that one should take from C that of it of which A does not hold by necessity and because of which A was said not to hold of some C by necessity, and, taking that, make the necessary premiss universal negative with respect to it. Let D be some part of C of which A does not hold by necessity. Then the whole combination will be the following: A holds of all B by necessity, and A holds of no D (which is some of C) by necessity. This combination is proved to be syllogistic by conversion of the universal negative premiss. For if A holds of no D by necessity, D holds of no A by necessity as well; but also A holds of all B by necessity; therefore, D holds of no B by necessity, so that B also holds of no D by necessity – for to prove the proposed conclusion a combination of this kind in the second figure requires conversion of the conclusion. Thus, if B holds of no D by necessity and D is some of C, B also does not hold of some C by necessity. 16 The proof is the same in the case of the sixth syllogism in the

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third figure. For let A not hold of some C by necessity and B hold of all C by necessity. And again let some part of C of which A does not hold by necessity be taken, and let it be D. (For it was assumed that A does not hold of some C by necessity.)17 Then A holds of no D by necessity and B of all D; for if B holds of all C by necessity, it would also hold of part of it by necessity. If then, we convert the universal affirmative BD, which is necessary, the result will be that D holds of some B by necessity. But also A holds of no D by necessity. Therefore, A does not hold of some B by necessity. So, if the proof relative to a part of C is sound, there will also be a sound proof relative to some of C. It should be noted that this form of ekthesis is not the same as the one which he mentioned in the third figure when his topic was unqualified syllogisms.18 There what is set out and taken was simply some sensible which did not need proof, and consequently its being taken was sufficient to make the inference evident. But here what is taken is not still taken in this way, nor is he satisfied with sense perception; rather he makes the syllogism with respect to the thing set out. This is why he here adds the phrase ‘and make the syllogism with respect to this’, but there he did not make use of any proof after taking the ekthesis. And this was reasonable. For he would have been doing the same thing with respect to some other subject which, being equivalent to and the same as the original subject, would prove nothing about it. He also indicates the nature of what is taken in ekthesis in the present case by saying, ‘For what is set out is just what some of the original term is.’19 For this will be a certain part or species of the term. Therefore, if holds of none of what is set out by necessity, then it also does not hold of some of that from which what is set out is taken by necessity. For what is set out is such as to be just what some of is. And even if the ekthesis is a matter of perception, the same thing will also be proved in this way. For example, in the second figure, if D is some of C and an individual, none of A will be said of it,20 so that neither will B be. For B is some of A, since A is said of all B. The proof is the same in the case of the third figure. For if A does not hold of some C by necessity, but B holds of all C by necessity and some individual of C of which A does not hold is taken none of A will be said of it. But that individual is itself also some of B, since all C is under B. Therefore, A will not hold of some B by necessity. In both combinations it is reasonable for him to take in the ekthesis that thing of some of which something does not hold.21 For by necessity what holds of all of something holds of that part of it of which something is assumed not to hold. But it is not the case that by

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necessity what is assumed not to hold of some of something will not hold of a random part taken of that of which the thing was assumed to hold of all; for it is possible for it to belong to this part of it, but not to some other part. (He also used this method a short while ago22 when he did an ekthesis with respect to a particular negative premiss.) Having said how one should carry out the proof in the case of these combinations, he adds that ‘And each of the syllogisms comes in the appropriate figure’. He means that in the case of the syllogisms involving the term set out, the one for the second figure will be in the second figure and that for the third in the third. For in each combination the universal negative premiss constructed in relation to the term set out does not alter the figure; rather the first is in the second figure and the second in the third, as we have set forth.23 For they are proved by a reduction to the first figure using conversion, and what is proved is in the second figure in one case and in the third in the other. 24 However, Theophrastus, discussing these things in the first book of his Prior Analytics, does not use the method of ekthesis for proving that the combinations under consideration are syllogistic. Rather he postpones discussion of them because it requires reductio ad impossibile, but what results is not prima facie clear because a mixture of premisses results and it is not yet known what follows from mixtures. 1.9-11 Combinations with one necessary and one unqualified premiss 1.9 The first figure25

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30a1526 It sometimes results that the syllogism27 is necessary when just one of the premisses is necessary – not either one of the premisses, but only that with respect to the major extreme term, [for example, if A has been taken to hold or not to hold of B by necessity and B just to hold of C. If the premisses have been taken in this way, A will hold or not hold of C by necessity. For since A is assumed28 to hold or not to hold of all B by necessity and C is some of the B’s, it is evident that one or the other of these will be in the case of C by necessity.] Having first discussed the syllogisms in each figure from two unqualified premisses and then those from necessary ones, he speaks next about the syllogisms in each figure from a mixture of a necessary and an unqualified premiss, saying what they are, what quality of conclu-

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sion results in their mixtures, and what the differences in the resulting syllogisms corresponding to the mixtures of these modalities are. Subsequently he will speak about syllogisms involving contingency. He says that sometimes a necessary conclusion results from the mixture of a necessary and an unqualified premiss, and he says in addition when a conclusion of this kind results. For he says that in the first figure, when both premisses are universal, if the major premiss is necessary and either affirmative (as in the first indemonstrable29) or negative (as in the second), the conclusion is necessary. This is what he says, but his associates Eudemus and Theophrastus30 do not agree. They say that in all the combinations of a necessary and an unqualified premiss which are put together syllogistically, the conclusion is unqualified. They take this from the that in every combination the conclusion is always like the less and weaker of the premisses assumed.31 For the conclusion which follows from an affirmative and a negative premiss is negative, and the conclusion which follows from a universal and a particular premiss is particular. And, it is the same way in the case of mixed premisses: in the case of combinations of a necessary and an unqualified premiss the conclusion is unqualified because the unqualified is less than the necessary. They also prove this by argument. For if B holds of all C but not by necessity, it is contingent that B sometime be disjoined32 from C. But when B has been disjoined from C, A will also be disjoined from it. And if this is so, A will not hold of C by necessity. And they also prove that this is so using material terms. For they take the major premiss to be universal necessary and either affirmative or negative, and the minor to be universal affirmative unqualified, and they prove that the conclusion is unqualified. For animal holds of every human by necessity; let human hold of all that moves; it is not true that animal holds of all that moves by necessity. Furthermore, if having knowledge is said of everything literate by necessity, and literate is said of every human unqualifiedly, it is not true that having knowledge is said of every human by necessity. And moving by means of legs is said of all that walks by necessity; let walking hold of every human; it is not true that moving holds of every human by necessity.33 And this seems to be reasonable.34 For if the major extreme is applied to the minor by means of the middle term, the major is related to the last term as the middle is to the minor; for the major is applied to the last term by means of the middle. Thus, as that because of which the major is attached to the last term is to the last term, so will the major be to the last term. 35 to maintain that Aristotle has not here said that the conclusion of mixtures of this kind is necessary, but

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only when certain material terms are used and that this is confirmed by his having said, ‘It sometimes results when just one of the premisses is necessary’. For the word ‘sometimes’ does not apply to such a mixture as if the conclusion of such a mixture is sometimes necessary and sometimes not; rather the word applies to the mixture generally. For the conclusion of a mixture of a necessary and an unqualified premiss is necessary sometimes. For it is not always necessary, since he also says that nothing necessary follows if the minor premiss is taken to be necessary . Thus the word ‘sometimes’ indicates the character of the mixture and not that with the same character and same combination the conclusion is sometimes necessary and sometimes unqualified. And he himself indicates this clearly by adding ‘not either one of the premisses, but only that which concerns the major extreme term’, thereby indicating the point of adding the word ‘sometimes’. And it is ridiculous to think that he says ‘sometimes’ because the conclusion of such a combination is necessary in the case of certain material terms. For on this way nothing would prevent one from saying that even non-syllogistic combinations are syllogistic ‘sometimes’; for they will be found to yield a conclusion in the case of certain material terms. For example, in the second figure the combination with two universal affirmatives will imply a universal affirmative conclusion, if the extremes are co-extensive with one another and terms of this kind have been taken. For if every human is an animal but also everything that laughs is an animal, it follows that every human is a thing that laughs. But the combination is not syllogistic just because something follows sometimes in the case of certain material terms. Furthermore if he wanted to indicate this, he would have proved in the case of which material terms it holds.36 For this would be appropriate for someone who added the word ‘sometimes’ for this reason. But he does not do this; he carries out his discussion using letters; he gives universal proofs using these because they cannot apply more to these material terms than to those. Consequently one ought to reject this as totally empty. We ought also to decline an extensive investigation of what he says, since we have discussed it in our work On the difference between Aristotle and his associates concerning mixtures .37 We shall set down here both the things which Aristotle uses to make what he says credible and the things someone would use to support that what he says is sound. He himself uses said of all. 38For since A is said of all B by necessity, and C is under B and is some of B, A is also said of C by necessity. For what is said of all B by necessity will also be predicated of what is under B by necessity – at least if being said of all is ‘when nothing of

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the subject can be taken of which the predicate will not be said’.39 But C is some of the B’s. For being said of all by necessity is taken in the same way , as he said before in the case of necessary things: ‘For the privative converts in the same way, and we will give the same account of “to be in as a whole” and “said of all”.’40 Of those who support his view some maintain that he takes the unqualified universal affirmative in such a mixture to be true, i.e., as holding truly and not relative to a hypothesis; for in the latter way it becomes a postulate and is no longer true.41 For if a proposition is taken to be true in actuality42 and universally, what it says will not be established as false in the case of certain material terms. For ‘Human holds of all that moves’ is not true in the sense of holding universally. Nor is ‘Literate holds of every human’. Nor ‘Walking holds of every human’. The conclusion is not necessary when the premisses are taken as holding in this way, because they are not true when taken as holding universally. However, what will these people say when the minor unqualified premiss is no longer taken as universal but as particular? For in this case he says that if the major premiss is necessary the conclusion is necessary; and it is no longer possible to say that the particular unqualified premiss is not true, e.g., ‘Human holds of something that is moving’ or ‘Literate holds of some human.’ Others say that if the proposition which says that A is said of all B is the same as the proposition which says that A is said of all of that of all of which B is said – as he43 says several times – then, also, the proposition that A is said of all B by necessity will be the same as the proposition that A is said by necessity of all of that of all of which B is said. And if this is what a universal necessary proposition signifies, the conclusion is always necessary even if the minor premiss is taken as unqualified.44 45 There are also some people who try to prove that what Aristotle says is sound by reductio ad impossibile. For let the mixture under consideration hold; let A hold of all B by necessity; and let B just hold of all C. I say that A holds of all C by necessity. For, if not, the opposite : it is contingent that A does not hold of some C. But it is assumed that A holds of all B by necessity. The result is a combination in the second figure of a major premiss which is necessary universal affirmative, and a minor which is particular negative contingent. This combination implies a particular negative contingent conclusion, according to what both Theophrastus and Eudemus think. Therefore, it is contingent that B does not hold of some C. But it was hypothesized to hold of all. 46 That what Aristotle says is sound will be best confirmed by reductio ad impossibile using the third figure. For let it be assumed

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that A holds of all B by necessity and that B holds of all C. I say that A holds of all C by necessity. For if not, the opposite : it is contingent that A does not hold of some C. But it is assumed that B holds of all C. The result is a combination in the third figure of a minor premiss which is unqualified universal affirmative and a major which is contingent particular negative. These imply a particular negative contingent conclusion. Therefore, it is contingent that A does not hold of some B, which is impossible, since it is assumed that it holds of all by necessity. Both Aristotle and his associates47 think that in such a combination in the third figure, the conclusion is particular contingent negative. Such and so many are the considerations which a supporter of Aristotle’s view on these matters might use. As I said,48 we have said in detail elsewhere which of the considerations seem to be sound and which not. 30a17 for example, if A has been taken to hold or not to hold of B by necessity.

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He is talking about the first two combinations in the first figure.49 For if the major premiss is affirmative and universal necessary and the minor is unqualified universal affirmative, and if, again, the major is negative and similarly necessary and the minor is unqualified universal affirmative, he says that the conclusion is necessary and affirmative in one combination, negative in the other. 30a21 For since A is assumed to hold or not to hold of all B50 by necessity and C is some of the B’s, it is evident that one or the other of these will be in the case of C by necessity. He has said ‘one or the other of these’ instead of ‘ affirmatively or negatively’. And he has stated the reason why he thinks that the conclusion will be necessary when the terms are related in this way: because C is some of the B’s and A is assumed to hold or not to hold of all B by necessity. 30a2351 But if the proposition AB is not necessary, but BC is necessary, the conclusion will not be necessary. [For, if it is, it will result that A holds of some B by necessity – through the first and through the third figure. But this is false. But it is possible52 that B is such that A can hold of none of it. It is also evident from terms that the conclusion will not be necessary, for example, if A is motion, B animal, C human. For a human is an

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animal by necessity, but an animal does not move by necessity, nor does a human.] He says that if the minor premiss is taken to be necessary universal, the major to be unqualified universal, the conclusion – he says – will not be necessary. With these words he indicates why he said earlier ‘It sometimes results that the conclusion is necessary when just one of the premisses is necessary’.53 Showing that the conclusion of such a combination is not necessary, he says that if the conclusion were necessary it will result that A holds of some B by necessity – through the first and the third figure, which will be false because it is assumed that A holds of all B, but not by necessity. It will be proved in the first figure that A holds of some B by necessity in the following way. Let the combination be assumed: A holds of all B, B holds of all C by necessity. Let the conclusion ‘A holds of all C by necessity’ be taken as necessary. But C also holds of some B by necessity. For since B holds of all C by necessity but a particular affirmative necessary proposition converts with respect to a universal affirmative necessary one, C will also hold of some B by necessity. But it is also assumed that A holds of all C by necessity. Therefore A holds of some B by necessity; for there are two necessary propositions, one universal and one particular. He says that this is false because A was assumed to hold of all B simply and it is possible that what holds of all holds of none at some time. The same thing could be proved through the third figure in the following way. Let the conclusion of the combination under consideration be taken to be that A holds of all C by necessity, and let the minor necessary premiss, ‘B holds of all C by necessity’, be added. The result is a combination in the third figure of two universal affirmative necessary propositions, which implies a particular affirmative necessary conclusion. Therefore, A holds of some B by necessity. Which is false. For it was assumed to hold of all of it simply, but what holds of all of a thing simply might also hold of none of it. (He indicates this when he says ‘But it is possible that B is such that A can hold of none of it’,54 thereby indicating to us again what sort of thing he means an unqualified proposition to be.) 55 It should be noted that he does not say that it is impossible for A to hold of some B by necessity. For nothing prevents what holds of all of something from also holding of some of it by necessity. But since the holding of all is not necessary, holding of some of something by necessity is not directly contained in holding of all of it.56 For it is possible that it holds of all of it in such a way that it is also possible that it holds of none. For a universal affirmative unqualified proposition is not prevented from being true in this way. Consequently, the proposition that what holds of all of something holds of some of it by

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necessity is false in the case of a proposition which is unqualified universal true in this way. 30a27 But it is possible that B is such a thing.57

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He says ‘but’ instead of ‘for’.58 This is the congruous construal of what is said. 59 It is not the case that, if someone were to say that the conclusion is necessary when the major is necessary and the minor is unqualified, something false of this kind follows. For if A holds of all B by necessity and B just holds of all C, and one takes it that A holds of all C by necessity, a falsehood similar to the preceding follows neither in the first figure nor in the third. For it follows either that A holds of some B by necessity (if it had been agreed that if the major premiss is necessary so is the conclusion), or that A just holds of some B (if someone were to say that the conclusion of such a mixture is also unqualified); neither of these is false when it is assumed that A holds of all B by necessity. Thus it is also reasonable60 for him, being moved by this difference, to lay down that the conclusion is necessary in the mixture in which the major premiss is necessary, but unqualified in that in which the minor is necessary. 30a2861 It is also evident from terms that the conclusion will not be necessary, [for example, if A is motion, B animal, C human. For a human is an animal by necessity, but an animal does not move by necessity, nor does a human.]

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He also gives a refutation and proof using terms. For if A is motion, B animal, and C human, it will be the case that motion holds of all animals unqualifiedly, animal of all humans by necessity, and motion will hold of all humans, but not by necessity. It is worth observing here how it is that, even though he establishes through terms that no necessary conclusion follows in the case of this combination, he does not recognize that this same thing can also be proved in the case of combinations having their major universal and necessary. For setting down the same terms proves that the conclusion is not necessary in their case either. For, if we take it that animal holds of all humans and human of all that moves, it will follow that animal holds of all that moves.62 But it seems to follow only if ‘said of all by necessity’ as signifying ‘when nothing of the subject can be taken of which the predicate will not be said’ by necessity.63 If one takes what is under B64 as being some things of B, he takes it that A is predicated of what is under B by necessity. And this would be true if everything under B were a part of B and some things of B in the sense of being in its substance. But if some things

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under B can also be separated from it, A will not hold of the things under B in this way by necessity. The cause of the mistake concerning the universal unqualified proposition :65 since in the case of such a proposition, A must always hold of the things under B, it seems to follow that, if A holds necessarily of all B, it will hold of what is under B necessarily, the reason being that the things under B are some things of B. But ‘being some things of B’ is a quite general expression, and it does not convey the impression that the things are in the substance of B. But if they are not in the substance, A will not hold of them by necessity. For what holds by necessity holds not just at the present but also in the future; therefore, it does not posit an unqualified proposition. For in the case of these things what is predicated of all B by necessity will also hold by necessity of the things under B in which B is by necessity. In the case of what is predicated of all B unqualifiedly it is true that nothing is predicated of the things under B of which what is said of all B will not be predicated. For it is true to say that what is predicated of all of B holds of what B is predicated of either necessarily or unqualifiedly. But it is not true that what is predicated of all B by necessity will be predicated by necessity of that of which B is predicated unqualifiedly. For it is true to say that what holds by necessity holds, but it is not true to say that what simply holds holds by necessity. And he also indicated just now66 that the unqualified universal premiss is a hypothesis through the terms he set down.

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He says that it will be proved similarly that the conclusion is not necessary either if the major premiss is universal negative unqualified. For again we will hypothesize68 that the conclusion is universal negative necessary. And if we make the proof in the first figure, we will convert the universal affirmative necessary premiss BC and we will have that A holds of no C by necessity and C of some B by necessity. From these it will follow69 that A does not hold of some B by necessity, which is false, since it was hypothesized to hold of none simply. And if we make the proof in the third figure, we will have that A holds of no C by necessity and B of all C by necessity (since this was assumed). From these again it will follow70 that A does not hold of some B by necessity. It will also be proved again through the same terms that the conclusion is not necessary. For let motion hold of no animal and animal of all humans by necessity; motion will hold of no human, but not by necessity. 71 It is necessary to understand that, although his proof resembles

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a reductio ad impossibile, it is not the same as it. For he did not hypothesize the opposite of the conclusion. For A’s holding of all C by necessity is not the opposite of A’s holding of all C, nor is its holding of none by necessity the opposite of its holding of none. But these are what he hypothesized. 72Nor was what was proved from the hypotheses impossible, as he himself indicated, but false.73 For what was hypothesized was also false since it was not impossible.74 And it is not impossible either that what holds of all also holds of some by necessity; and what holds of none is not prevented from not holding of some by necessity. Thus a falsehood followed from a false hypothesis; for an impossibility follows from an impossibity, a falsehood from a falsehood. 75Furthermore, insofar as the proof is concerned, the proposed conclusion76 was not established. For just because holding of all by necessity is done away with as false, it is not thereby necessary that holding of all is true. For only the opposite of a thing is posited by necessity when the thing is denied and the denial refuted. But these things are not opposites. Nor does he seem to be entirely confident about this method of proof.77 However he uses it to prove that if something false follows when the conclusion is taken to be necessary but not when it is taken to be unqualified, then it is necessary for the conclusion to be unqualified; and since the premisses78 are of this kind, the conclusion would be proved to be unqualified. For he has not applied reductio ad impossibile to what is proved , but taking another proposition, a necessary instead of an unqualified one, and making a syllogistic combination from this and the other assumed premiss, the minor, he finds that what follows is false and uses this fact to establish that the conclusion is not necessary. Someone might also ask this: what is the necessity that there is no necessary conclusion, if something false follows when the unqualified proposition is transformed into a necessary one? Isn’t it because a falsehood does not follow from things which are true? He then has added this from outside: if the conclusion is hypothesized to be necessary, something false follows when the minor premiss, which is itself also necessary, is added. 79 If someone refused to agree that the conclusion is unqualified, one could also prove that what follows is not necessary but unqualified by positing it to be unqualified, using80 reductio ad impossibile, and encountering something impossible, but, on the other hand, not encountering something impossible when assuming it necessary. 81 For let it be assumed that A holds of all B unqualifiedly, but B of all C by necessity. I say that A holds of all C. For if not, the opposite : A does not hold of all C. And let there be added the premiss

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‘B holds of all C by necessity’. From these it follows that A does not hold of some B, which is impossible, since it was assumed to hold of all of it. But if in the case of the combination under consideration ‘A holds of all C by necessity’ is taken , and the opposite of holding of all by necessity (‘it is contingent that A does not hold of some C’) is taken , and if ‘B holds of all C by necessity’ is added, it follows that it is contingent that A does not hold of some B, which is not impossible. For if A holds of all B it is not impossible that it is contingent that A does not hold of some B. 82 The proof is again the same if the major premiss is universal negative unqualified, the minor universal affirmative necessary. For also in this case if someone making a proof by reductio ad impossibile using the third figure posits that the conclusion is unqualified negative, something impossible will follow, but not if he posits that the conclusion is necessary. Consequently, if the conclusion can be proved to be unqualified but cannot be proved to be necessary by means of reductio ad impossibile, the conclusion will be unqualified. 83 Theophrastus expresses in this way the view that the conclusion in this combination84 is not necessary: ‘For if B holds of C by necessity, A of B not by necessity, and what is not by necessity might be separated, it is evident that A will be separated from C when it is separated from B, so that by necessity because of the premisses assumed.’ This being shown, he adds, ‘Likewise too, if the major is necessary. For since the middle is not by necessity, it might be separated , but if this is separated, so will the major be. For if someone takes the major premiss as if it said “what B is said of, A is said of by necessity”, he is taking both premisses as if they were necessary. For if he does not take them in this way, it is false’.85 In this way Theophrastus shows that in mixtures in which there is an unqualified and a necessary premiss, the conclusion is unqualified, whichever of the premisses is necessary.

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30a3386 In the case of the particular syllogisms, if the universal premiss is necessary, the conclusion will also be necessary. [But it will not be necessary if the particular premiss is necessary, whether the universal premiss is privative or affirmative. For first let the universal be necessary, and let A hold of all B by necessity and B just hold of some C. Then it is necessary that A hold of some C by necessity. For C is under B and A held of every B by necessity. Similarly, if the syllogism is privative; for the demonstration will be the same. 30b287 But if the particular premiss is necessary, the conclu-

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Translation sion will not be necessary; for nothing impossible results, just as in the universal syllogisms.88 Similarly in the case of privatives. 30b5 Terms: motion, animal, white.]

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He has spoken about the two combinations in the first figure in which a universal conclusion follows from two universal premisses, and said that if the premisses are a mixture of an unqualified and a necessary proposition the conclusion sometimes will be necessary (and he said which premiss is taken to be necessary) and sometimes unqualified. Now he also speaks about two combinations in which the minor premiss is particular affirmative, the major universal and either affirmative or negative, and he says that also in these combinations, if the major is necessary and either affirmative or negative, the conclusion will be necessary, as in the case of the first combinations, but, if the minor is necessary, the conclusion will be unqualified. Using the same in these cases he thinks89 that he proves that the conclusion is necessary when the major premiss is necessary universal. For he thinks it will follow from A holding of all B by necessity and some of C being under B that A holds of that C which is under B by necessity. The argument is the same if the major is universal negative necessary. (30b2) He says that, if the particular premiss is necessary, the conclusion will not be necessary, and he adds the reason when he says, ‘for nothing impossible results, just as in the universal syllogisms’. 90One thing he might mean is this. 91If the conclusion is hypothesized to be unqualified, nothing impossible (that is ‘false’)92 follows, just as when both premisses were universal and the necessary one was the minor. For in the case of them, when the conclusion was hypothesized to be necessary something false followed, but nothing false followed when it was hypothesized to be unqualified; for if we hypothesize in the case of the universal premisses that A holds of all C unqualifiedly and also take it that B holds of all C by necessity, it follows in the third figure that A holds of some B, which is true, since it was hypothesized to hold of all. As, then, in that case, when the conclusion was hypothesized to be necessary something false followed, but not when the conclusion was hypothesized to be unqualified, so, he says, it will be the same in the case of particular syllogisms, if the minor premiss is particular affirmative necessary. 93 Alternatively the words ‘for nothing impossible results – just as in the universal syllogisms’ might mean the following. As in the universal cases nothing impossible (just something false)94 followed when the conclusion was posited to be necessary and we used reductio ad impossibile to try to establish it so that a necessary conclusion is not proved (this was shown by the preceding95), so too if the minor is posited to be particular necessary affirmative. For if A held of all B and B of some C by necessity, and someone wanted to prove by reductio ad impossibile that A holds of some C by necessity, he would take the opposite of this: ‘it is contingent that A holds of no C’; if ‘B holds of some C by necessity’ is added, it would follow in the third figure that it is contingent that A does not hold of some B, which is not impossible, since A was assumed to hold of all B. Thus it is not established . But if we want to prove that the conclusion is particular affirmative and not necessary but unqualified by reductio ad impossibile, the argument will go through and something impossible will follow from the hypothesis. For the opposite of ‘A holds of some C’ is ‘A holds of no C’. And if ‘B holds of some C by necessity’ is added, it will follow that A does not hold of some B, which is impossible, since it holds of all of it. Thus the hypothesis which says that A holds of no C is impossible; therefore A holds of some. As, then, in the case of universal syllogisms the conclusion was not necessary because nothing impossible followed when we applied reductio ad impossibile to them, so too in the case of particular syllogisms. He did not set out the proof by impossibility which we have mentioned in the case of either universal or particular syllogisms, but only indicated it because it involved a mixture of a contingent and a necessary premiss and he has not yet discussed this mixture. 96 It is possible that he has said ‘for nothing impossible results’ not to suggest to us either the proof which he used to show that something false follows when is hypothesized to be necessary nor the reductio ad impossibile about which I have just spoken, but rather the evidence provided by terms. For the proof from terms also fits what is said. For if the propositions in this combination and mixture are taken, the conclusion is unqualified. But what we said would be impossible if refutation using terms were at issue. (30b5) That he sets down terms immediately may be an indication that this is what is meant. For motion holds of every animal (or of none) unqualifiedly, and animal holds of something white (e.g., swan) by necessity, and motion holds of something white, but not by necessity (or does not hold of something white unqualifiedly). 97 It is necessary to understand that in the case of these syllogisms one cannot use the proof which derives a falsehood and which he used in the case of the universal syllogisms. For if it is assumed that A holds of some C by necessity, whichever of the assumed premisses we add to this conclusion, the resulting combination is non-syllogistic. For if we add that B holds of some C by necessity, we will have two particular affirmatives in the third figure; and similarly in the first figure if we convert ‘B holds of some C’ . But if we assume that A holds of all B

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and of some C by necessity, again we will have two affirmatives in the second figure, and a proof will not be possible. 98 It is possible for us to maintain that we produce a justification for the particular syllogisms from that for the universal ones. For since the particular syllogisms differ from the universal ones only by yielding a particular conclusion, and nothing impossible, i.e., false, followed in the case of universal syllogisms when the conclusion was posited to be unqualified, so in the case of these particular syllogisms, as in the case of those universal ones, the conclusion will be unqualified, but it will be particular ; for only this difference was assumed in the premisses. 99 Furthermore, ‘nothing impossible results’ if the conclusion is assumed to be unqualified because no syllogistic combination (through which an impossibility is proved) results. And100 this is so even if the conclusion is hypothesized to be necessary. Nor does the proof go through if the universal unqualified major is taken to be negative. For, again, both premisses become particular and in the first or third figure, one negative and one affirmative; or both become negative in the second figure. The terms which he sets down also fit this combination.101 1.10 The second figure

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30b7102 In the case of the second figure, if the privative premiss is necessary, the conclusion will also be necessary; but if the affirmative premiss is necessary, the conclusion will not be necessary. It is necessary to understand ‘universal’ along with the words ‘if the privative premiss is necessary’. For he speaks first about syllogisms from universal premisses. For in the combination in which the particular premiss is privative – and the negative is particular in the fourth combination103 – when this premiss is necessary, the conclusion is not necessary. Therefore one should understand the words ‘if the privative premiss is necessary’ as applying to the universal syllogisms . For he discusses these things first; and he says later, ‘Things will also be the same in the case of the particular syllogisms’.104 The things which he says in the case of the second figure follow from what he proved in the case of the first. in the first figure with mixed premisses, when the major premiss is necessary, the conclusion was necessary; and the second figure is produced from the first (for the second figure was produced when the major premiss in the first figure was converted); and

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most105 of the syllogisms in the second figure are reduced to the first figure by conversion (for when a universal privative premiss in the second figure is converted, it becomes a major premiss in the first figure); consequently, when the universal privative premiss in a second-figure syllogism is necessary, the conclusion will also be necessary. For in the first figure it is necessary for the major to be universal if there is to be a syllogism; so it is necessary for something in the second figure to become the major premiss when it is converted, since the second figure was generated in this way, and this converted premiss will be one which both is universal and remains universal when it is converted. But only the universal negative is of this kind. Therefore, the reduction of the second figure to the first occurs when it is converted. And when it is necessary, the conclusion will be necessary, since it remains necessary when it is converted and becomes a major premiss in the first figure. But it is not true that the conclusion is necessary when an affirmative premiss in the second figure is necessary. For in the reduction of combinations in the second figure to the first figure, this is the premiss which becomes the minor; but when the minor premiss in the first figure was necessary, the conclusion was unqualified.

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30b9106 For first, let the privative premiss be necessary, [and let it be not contingent that A holds of any B and let it just hold of C. Then, since the privative converts, it is not contingent that B holds of any A; but A holds of all C, so that it is not contingent that B holds of any C. For C is under A.] The proof is clear. He converts the universal negative premiss, which is necessary, and produces the first figure; since in the first figure when the major premiss is necessary, the conclusion is necessary, he says that the situation will be the same also in the case of this figure. 107It is known that the phrase ‘It is not contingent that X holds of any Y’, which has been applied to the premiss AB, being the negation of contingency, indicates necessity. For if it is not contingent that A holds of any B, A holds of no B by necessity. For the proposition has not been taken as contingent but as the negation of a contingent proposition and is equivalent to ‘A holds of no B by necessity’. And the same holds for ‘so that it is not contingent that B holds of any C’. For what is said is the same as ‘it is not contingent that B holds of some C’, which is equivalent to ‘B holds of no C by necessity’. And he uses the same consideration again108 when he says ‘For C is under A’. For since A was assumed to hold of all C and B holds of no A by necessity, B will hold of no C, which is some of the A, of none of which B holds by necessity.

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Translation 30b14109 Likewise, if the privative is posited with respect to C. [For if it is not contingent that A holds of any C, then it is not possible that C holds of any A. But A holds of all B, so that it is not contingent that C holds of any B. For, again, the first figure results. Therefore, it is not contingent that B holds of any C; for the conclusion converts in the same way.]

There were two syllogisms from universal premisses in the second figure,110 one when the major is universal negative and one when the minor is; and both were reduced to the first figure when the universal negative was converted and became a major premiss in the first figure. Their difference from one another was that the one having the major universal negative was proved by inferring the proposed conclusion through a single conversion whereas the other having the minor universal negative was proved through two conversions; for in the latter case not only was the negative premiss converted; the conclusion – a universal negative – was converted as well. Since, then, also in the second figure, when the minor premiss was universal negative and it was converted, the first figure with it as major resulted, he says that if the minor premiss in the second figure is universal negative necessary, the conclusion will be necessary. For whatever was said about the combination before this one will also apply to it, except that in this case it will also be necessary to convert the conclusion, as he indicates when he adds: 30b17 Therefore, it is not contingent that B holds of any C; for the conclusion converts in the same way.

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The addition of ‘in the same way’ indicates that if the converted proposition is necessary the result of converting it is necessary, and if it is unqualified the result will be unqualified. Or perhaps in the same way to necessary and to unqualified propositions.111 30b18112 But if the affirmative premiss is necessary, the conclusion will not be necessary. [For let A hold of all B by necessity, and let it just hold of no C. If the privative premiss is converted, the first figure results. But it has been shown in the case of the first figure that the conclusion will not be necessary, if the privative premiss relating to the major term is not necessary, so that neither will the conclusion be necessary in the case under consideration. 30b24 Furthermore, if the conclusion is necessary, it results that C does not hold of some A by necessity. For if B holds of no

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C by necessity, C will also hold of no B by necessity. But it is necessary for B to hold of some A, since A held of every B by necessity. Thus it is necessary that C does not hold of some A. But nothing prevents A from being taken in such a way that it is contingent that C holds of all of it. 30b31 Furthermore it would be possible to show, by setting out terms, that the conclusion is not necessary without qualification, but only if certain things are the case. For example, let A be animal, B human, and C white, and let the premisses be taken in the same way – for it is contingent that animal holds of nothing white. Then, human will also hold of nothing white, but not by necessity; for it is contingent that a human be white, although not so long as animal holds of nothing white. So the conclusion will be necessary if certain things are the case, but it will not be necessary without qualification.] He proves that when in the second figure the affirmative premiss is universal and necessary and the other premiss is universal negative unqualified, the conclusion is not necessary; he reduces the combination under consideration to the first figure by conversion , and he shows that in the first figure the minor premiss is the necessary one. For in the syllogistic combinations in the first figure the negative proposition is the major, because the minor must always be affirmative. But if the negative premiss was not necessary, the conclusion was not necessary.

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30b24113 Furthermore, if the conclusion is necessary, it results that C does not hold of some A by necessity. [For if B holds of no C by necessity, C will also hold of no B by necessity. But it is necessary for B to hold of some A, since A held of every B by necessity. Thus it is necessary that C does not hold of some A. But nothing prevents A from being taken in such a way that it is contingent that C holds of all of it.] He doesn’t prove that the combination with the universal affirmative premiss necessary and universal negative one unqualified in the second figure does not imply a necessary conclusion just by reduction to the first figure; he also proves it by showing that something false follows if the conclusion is hypothesized to be necessary. (He also did a proof in this way in connection with the first figure when the minor premiss was universal affirmative necessary.114) For let the combination be assumed, and let A hold of all B by necessity, and let it just hold of no C. If someone should say that it follows that B holds of no C by necessity, it will also be the case that C holds of no B by necessity, since the proposition converts. But since A was assumed

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to hold of all B by necessity, B will also hold of some of the A’s by necessity. From these premisses in the first figure with two necessary premisses, the major universal negative and the minor particular affirmative, a particular negative necessary conclusion will follow, namely ‘C does not hold of some A by necessity’. He says this conclusion is false because when A is assumed to hold of no C nothing prevents A from then being such that it can hold of all C – even though C holds of no A (because it is assumed that A holds of no C, and a universal negative converts to a universal negative). For the negative proposition which says that C holds of no A is not prevented from being true, if it is contingent that it holds of all of it. Thus, when the negative premiss ‘A holds of no C’ has been taken in such a way that it is contingent that A holds of all C and C of all A115 and it is inferred that C does not hold of some A by necessity, what is inferred will be false. Thus, when the conclusion of the combination under consideration is hypothesized to be necessary, something false follows. Therefore, the hypothesis is false, since something false cannot be inferred from truths. He also set down terms116 for which it is contingent117 that the universal negative premiss is true and contingent that what is truly denied of the subject holds of all of it. For if A is animal, C moving, A, i.e. animal, may hold of nothing moving. But this does not preclude that it is also contingent that it holds of everything moving. Similarly, too, even if moving were to hold of no animal, it is nevertheless also contingent that it holds of all. For moving can hold of no animal, that is, C can hold of no A – because the universal negative converts. But in fact it is also contingent that it holds of all. For also, even if someone were not to agree that animal can hold of all that moves,118 nevertheless he would agree that it is contingent that moving holds of all animal. Thus if it is contingent for things to be this way in some cases, but the person who makes the conclusion negative particular necessary does away with this contingency (for he proves that what was assumed to hold simply of none does not hold of some by necessity), he would be shown to be wrong in taking the conclusion to be necessary in the case of this mixture and combination of premisses. 119 That the conclusion is not necessary when the combination is of this kind is also proved by reductio ad impossibile. For if, when the conclusion is posited to be unqualified and the opposite of the conclusion is hypothesized, an impossibility follows but if the conclusion is hypothesized to be necessary, nothing impossible is proved by reductio ad impossibile, it is clear that the conclusion will be proved to be unqualified, not necessary. For, let it be assumed that A holds of all B by necessity and of no C. I say that B will hold of no C. For if not, it holds of some. But A also holds of all B by necessity. Therefore, A

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holds of some C by necessity, which is impossible, since it was assumed to hold of none of it. But suppose someone were to say that in the case of the combination under consideration a necessary negative universal conclusion follows and B holds of no C by necessity. If we take the opposite of this, which is ‘It is contingent that B holds of some C’ and we add ‘A holds of all B by necessity’, the conclusion will be ‘It is contingent that A holds of some C’, which is not impossible, since it is not impossible that A holds of no C and it is contingent that it holds of some.

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30b31120 Furthermore it would be possible to show, by setting out terms, that the conclusion is not necessary without qualification, but only if certain things are the case. [For example, let A be animal, B human, and C white, and let the premisses be taken in the same way – for it is contingent that animal holds of nothing white. Then, human will also hold of nothing white, but not by necessity; for it is contingent that a human be white, although not so long as animal holds of nothing white. So the conclusion will be necessary if certain things are the case, but it will not be necessary without qualification.] He also proves by setting down terms that the conclusion of a combination of this kind is not necessary. For if A is animal, B human, and C white, A, animal, will hold of all B, human, by necessity and of no C, white, simply. He posits this together with an explanation when he says, ‘For it is contingent that animal holds of nothing white.’ (For if this is not the case, something else for which it is the case can be taken.121) Human, then, will hold of nothing white, but not by necessity. He makes clear both the words ‘not by necessity’ and what the proposition which says ‘neither will human hold of anything white’ signifies by adding that ‘it is contingent that a human be white’. For the person who says ‘human holds of nothing white’ says ‘nothing white is human’, which is equivalent to ‘no human is white’. But even if the proposition saying no human is white were true, that would not mean that some human will not be white by necessity. For as long as animal does not hold of white, a human will not be white, but this is not always so, nor does it hold by necessity. By adding the words:122 although not so long as animal holds of nothing white. So the conclusion will be necessary if certain things are the case, but it will not be necessary without qualification he indicates that when he says, in connection with mixtures, that the

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conclusion is necessary, he means ‘necessary without qualification’ and not ‘necessary on a condition’, as some of the interpreters of the topic of mixture of premisses say, thinking that they strengthen his position; they assert that he does not speak about inferring necessity without qualification, but about inferring necessity on a condition. For they say that when animal holds of every human by necessity and – as in the first figure – human of all that moves or walks, the conclusion is necessary on a condition; for animal holds of all that moves or walks as long as the middle, human, holds of it. For it is not the case that if the minor premiss is necessary, the conclusion is necessary; for it is not the case that if moving holds of every animal and animal of every human by necessity, moving holds of every human by necessity as long as animal holds of every human – for that is false – but for as long as moving holds of every animal. He has indicated that he does not intend the conclusion to be necessary in this way when, in showing that the conclusion is of this kind and necessary in this way in the second figure if the affirmative premiss, whether it is the major or the minor, is necessary, he did not say that the conclusion of such a mixture is necessary without qualification.123 Alternatively, if he had said that this conclusion is necessary in a way similar to that one,124 he would have added to that one the words ‘It is not necessary without qualification, but necessary on a condition’,125 as he also does in this case. At the same time he has also indicated by the addition that he is aware of the division of necessity which his associates126 have made, and which he has also already established in On Interpretation, where, discussing contradiction of propositions about the future and individual things, he says, ‘It is necessary that what is is when it is, and that what is not is not when it is not.’127 For the necessary on a hypothesis is of this kind. What he has said would be more evident if we were to take instead truer terms. For the premisses were not true in the case of the terms with which he tried to carry out his proof. For the premiss ‘animal holds of nothing white’ is not true since it holds of swan by necessity. Let us either posit moving instead of white or take other terms. Let A be being awake or moving, B walking, C human. Then, being awake holds of all that walks by necessity, and so does moving. But let being awake hold of no human. Then walking will hold of no human, but it will not hold by necessity of either no human or some human. For it is contingent that a human walks, but not in a situation in which it is true that being awake or moving holds of no human. 128 If the minor premiss is universal affirmative necessary and the

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major negative universal unqualified, then, if the conclusion BC is hypothesized to be necessary, then it would not seem that in this case the conclusion is proved to be false, either in the first or the third figure. This is so in the case of the first figure because two negative premisses result. For A is assumed to hold of no B and of all C by necessity. If ‘B holds of no C by necessity’ is taken as the conclusion of these propositions and converted, it will be the case that C also holds of no B by necessity. But it is also the case that B holds of no A, since A was assumed to hold of no B. The result is two negative propositions, and the combination is non-syllogistic. This is also true in the third figure. For the minor negative premiss is ‘B holds of no C’ by necessity; and A holds of all C by necessity. However, perhaps such a combination is non-syllogistic with respect to the proposed conclusion (just like the combination just discussed when the major was necessary affirmative).129 The conclusion ‘B does not hold of some A by necessity’ does indeed follow syllogistically through the combination by conversion of the affirmative premiss (just as in the other case ‘C does not hold of some A by necessity’ followed, although this was thought impossible since it was contingent that C sometime holds of A).130 For B holds of no C by necessity and C of some A by necessity; from these it follows that B does not hold of some A by necessity. But again it is false to say that B does not hold of some A by necessity when it is only assumed that A holds of no B simply, as was proved a little while ago.131 For because of conversion it is contingent that what holds of no A does not hold in such a way that it is also contingent that it holds of all.132 Consequently the conclusion would be true, but it would not be, in addition, necessary. The same thing133 would also be proved to be false in the second figure if both propositions were converted. For if, by conversion of the conclusion and of the assumed affirmative necessary premiss, one takes it that C holds of no B by necessity and C holds of some A by necessity, it follows that B does not hold of some A by necessity, which is false. For it was assumed to not hold simply, so that it is also possible for it to hold. It is also prima facie clear through terms that the conclusion is not necessary in the case of this combination either. For animal holds of nothing that moves, and it holds of every human by necessity, and moving holds of no human, but not by necessity.

31a1134 Things will also be the same in the case of the particular syllogisms. [For when the privative premiss is universal and necessary, the conclusion will also be necessary. But when the

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Translation affirmative premiss is universal, the privative particular, the conclusion will not be necessary.]

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He says that things will be the same in the case of the particular combinations which yield a conclusion. For if the universal premiss is negative necessary, the conclusion will be necessary, but if the affirmative is necessary, whether it is particular (as in the third syllogism, which consists of a universal negative major and a particular affirmative minor) or universal (as in the fourth, which consists of a universal affirmative major and a particular negative minor) the conclusion will not be necessary. 31a5135 First let the privative premiss be universal and necessary, [and let it not be contingent that A holds of any B, and let it hold of some C. Since the privative premiss converts, it will not be contingent that B holds of any A. But A holds of some C, so that B will not hold of some C by necessity.]

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He proves in the third combination among the syllogisms of the second figure how, if the universal negative premiss is necessary, the conclusion is necessary. For since, when the universal negative premiss is converted, there results the first figure and in that figure the fourth syllogism, which has a universal negative necessary major premiss, and since in a combination of this kind the conclusion is necessary, it is clear that also in the second figure, when the premisses are related in this way, the conclusion will be necessary particular negative. 31a10136 Again, let the affirmative premiss be universal and necessary, [and let the affirmative be assumed in relation to B. Then, if A holds of all B by necessity, but does not hold of some C, it is evident that B will not hold of some C, but not by necessity. For the same terms as were used in the case of the universal syllogisms will serve for the demonstration. 31a15 Neither will the conclusion be necessary if the privative is necessary and taken as particular. The demonstration is through the same terms.137]

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138 He does not mention the fact that in the combination just discussed the conclusion is not necessary if the particular premiss is affirmative necessary because it is known. For by the same conversion the minor necessary particular affirmative premiss of the third syllogism in the second figure is produced in the first figure. And would be proved through terms if animal were taken to hold of nothing that moves and of something white by necessity (since

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animal holds of swan by necessity); for moving will not hold of something white, but not by necessity. He has turned to the fourth combination, in which the major premiss is universal affirmative, the minor particular negative. His proof in the case of unqualified premisses was not by conversion but by reductio ad impossibile.139 In the case of necessary premisses his proof was by ekthesis and taking something of which does not hold.140 He proves in the case of the present combination that the conclusion is not necessary no matter which of the two premisses is taken to be necessary; and first he proves why the conclusion is not necessary if the universal affirmative is necessary. He proves this by setting down terms. He himself says that when the universal affirmative premiss is necessary the conclusion will be proved not to be necessary by the same terms as were also used in the case of the combination with two universal premisses, the affirmative being necessary. These terms were animal, human, and white.141 For animal holds of every human by necessity, it does not hold of something white simply, and human will not hold of something white simply. But since these propositions are not true, just as they were not true when they were universal142 – for the proposition ‘Animal does not hold of something white’ is not unqualified but necessary –, we will do the proof with other terms. Moving holds of all that walks by necessity; let it just not hold of some human; then also some human will not be walking, but not by necessity. 143 (31a15) Nor will the conclusion be necessary if the privative particular premiss is taken to be necessary.  144For being awake holds of everything literate unqualifiedly; let it necessarily not hold of some human (viz, a sleeping one); it will not be the case that some human will not be literate by necessity. Again, animal holds of all that moves, and animal does not hold of something white by necessity; moving will not hold of something white, but not by necessity. Furthermore, two-footed holds of everything which is awake, it does not hold of some animal by necessity, and being awake does not hold of every animal,145 but not by necessity. He says, ‘The demonstration will be by means of the same terms if just one of them is changed.’146 For this is also added in some texts, and may seem to be unsound. For in the case of the universal terms were animal, human, white.147 With these premisses the universal affirmative was necessary, but not the particular negative, which is now under consideration. Nor if we interchange white and human, on the grounds that animal holds of everything white unqualifiedly; for this is also false.148 Nor if we change white to A, human to B, and animal to C, on the grounds that white holds of everything human and does not hold of some animal by necessity. For then the

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first premiss is false, and the conclusion is necessary.149 The reason is that these terms were not true in the case of the universal premisses either, as we indicated when we took moving instead of white.150 And just as we there took moving instead of white, so here, if moving is taken instead of human and white retained, what has been said will be sound. For what results is that animal holds of all that moves unqualifiedly, animal does not hold of some white thing (snow) by necessity, and moving will not hold of something white, but not by necessity. Thus, a proof has been given with one term being changed.151 (It is possible that the words ‘The demonstration will be through the same terms’ have been used instead of ‘The demonstration is through the same things’. But what is the sense of ‘through the same things’? For the proof is through terms and the setting down of terms.)152 153 In the case of this combination someone might reasonably ask why, according to him, the conclusion isn’t necessary when the particular negative premiss is. For when both premisses were necessary, then by ekthesis one premiss was proved to turn into a universal negative; for when A held of all B by necessity and not of some C by necessity, he took some of C, namely D, of which A did not hold by necessity and made AD universal negative necessary; then, converting it, he took the proposition ‘D holds of no A by necessity’. But also A was assumed to hold of all B by necessity; therefore, D also holds of no B by necessity; therefore, B also holds of no D in the same way;154 but, since D was some of C, B was proved not to hold of some C by necessity. So if the particular necessary negative premiss becomes by ekthesis universal negative necessary, then, when both155 of the premisses are converted, there results a combination in the first figure of a universal negative necessary premiss and a universal affirmative unqualified one, of which the conclusion was assumed to be156 necessary. For A holds of no D by necessity,157 A of all B unqualifiedly; therefore, D holds of no B by necessity, and if this is so, B also holds of no D by necessity; but since D is some of C, B also does not hold of some C by necessity. 158 But he proved using terms that the conclusion is not necessary. 159 For animal holds of all that moves, it does not hold of some white thing by necessity, and moving does not hold of some white thing, but not by necessity. One should take this as an indication that according to him the conclusion is not necessary in the case of a mixture of a necessary and an unqualified premiss when he investigates them using material terms and does not also invoke being said of none by necessity.160 Invoking that leads to mistake. 161 For it is found that the conclusion is unqualified and not necessary. However, if what was said earlier162 were true, it would be necessary for the conclusion to be always necessary in the present combination,

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as I proved. 163For that by using material terms the conclusion is found to be unqualified when one of the premisses is unqualified is clear from the fact that if the two premisses are taken to be necessary in this combination it will no longer be possible for us to find, using material terms, that the conclusion is unqualified. For let it be assumed that animal holds of every human by necessity, and let animal not hold of something white (e.g., snow) by necessity; human will also not hold of something white by necessity since animal doesn’t either. For this is the conclusion corresponding to the premisses assumed.

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31a18164 In the last figure if the terms are universally related to the middle and both premisses are affirmative, [if either is necessary, the conclusion will also be necessary. But if one is privative and the other affirmative, when the privative is necessary the conclusion will also be necessary, but when the affirmative is necessary, it will not be necessary.] He has turned to the third figure. He proves also in this case when the conclusion is necessary and when it is unqualified, if one premiss is necessary and the other unqualified. And in this figure, too, both the necessary and the unqualified conclusion are proved by a reduction to the first figure using conversion. For, if a premiss in the third figure is assumed necessary and when the combination is reduced to the first figure by conversion, this premiss is the major in that figure, the conclusion will be necessary, but if it is the minor the conclusion will be unqualified.

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[31a24165 For first let both premisses be affirmative, and let A and B hold of all C, and let AC be necessary. Since, then, B holds of all C, C will also hold of some B because the universal converts with the particular; thus, if A holds of all C by necessity and C holds of some B, it is also necessary that A holds of some B; for B is under C. Thus the first figure results. 31a31166 It will be proved in the same way if BC is necessary. For C converts with some A, so that, if B holds of all C by necessity it will also hold of some A by necessity.] When both affirmative premisses are universal, whichever is taken to be necessary, the conclusion will be necessary. For if we keep the universal affirmative necessary premiss fixed and convert the other,

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unqualified, one and make it particular unqualified, we will have in the first figure the major premiss affirmative universal necessary and the minor particular affirmative unqualified;167 and when there is such a combination the conclusion is necessary. For if A holds of all C by necessity, B of all C unqualifiedly, if we convert BC, C will hold of some B. But it is also assumed that A holds of all C by necessity. From these propositions it will follow that A holds of some B by necessity. He adds the words ‘for B is under C’ to show that the first figure results and that CB is the minor premiss in it because, since A holds of all C by necessity, it will also hold of B . (31a31) If, conversely, B is taken to hold of all C by necessity and A holds of all C, the conclusion will be necessary. For we will convert the unqualified premiss AC and C will hold of some A. But B was also assumed to hold of all C by necessity. From these things it will follow that B holds of some A by necessity. And since a particular affirmative necessary proposition converts, A will also hold by necessity of some B. For it is necessary that A be predicated in the conclusion since it was assumed to be the major term. Therefore, in the case of a combination of this kind we need to convert the conclusion also. However, he does not mention converting the conclusion, perhaps because it is clear. He only proves that B will also hold of some A by necessity, since it was assumed to hold of all C by necessity. But again, as I said, it is also necessary to convert BA in order that the proof not be of something other than the proposed conclusion. For A is assumed as the major term, so that it must be predicated in the conclusion. 31a33168 Again, let AC be privative, BC affirmative, and let the privative premiss be necessary. [Then, since C converts with some B and A holds of no C by necessity, A will also not hold of some B by necessity. For B is under C.] He proves that also in the combination of a universal negative necessary major and a universal affirmative unqualified minor the conclusion is necessary, because, when the affirmative is converted there results the first figure having the major universal negative necessary. For let A hold of no C by necessity, and let B hold of all C. If BC is converted it results that C holds of some B. But A was also assumed to hold of no C by necessity. Therefore, A will not hold of some B by necessity. [31a37169 But if the affirmative is necessary, the conclusion will not be necessary. For let BC be affirmative and necessary, AC privative and not necessary. Then, since the affirmative con-

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verts, C will also hold of some B by necessity, so that if A holds of no C and C of some B, A will not hold of some B; but not by necessity. For it has been shown in the first figure that if the privative premiss is not necessary the conclusion will not be necessary either.] But if the privative AC is not necessary and the affirmative BC is universal and necessary, the conclusion will not be necessary. For the necessary affirmative premiss BC converts, and when it is converted and becomes particular affirmative necessary, the result is a combination in the first figure having the minor necessary and the major unqualified, namely the following: A holds of no C, C holds of some B by necessity, and A will not hold of some B by necessity.

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[31b4170 Furthermore, this would be evident through terms. For let A be good, B animal, C horse. Then it is contingent that good holds of no horse, but it is necessary that animal holds of all horses; however, it is not necessary that some animal is not good, since it is contingent that all are good. (Or, if this is not possible, one should posit as a term being awake or asleep; for these can apply to every animal.)] Having proved by conversion that the conclusion is not necessary because the minor premiss in the first figure is necessary, he also sets down terms through which he also shows that the conclusion of these premisses is not necessary. The terms which he takes are: A: good; B: animal; C: horse. Good holds of no horse, animal of all horse by necessity; therefore, good will not hold of some animal, but not by necessity. Wanting to show that he has taken the negative premiss to be unqualified and not necessary, he says, ‘it is contingent that good holds of no horse’ to indicate that good does not hold of horse contingently, not by necessity. For he is not taking a contingent premiss. Rather for him the proposition that it is contingent that good holds of no horse and the proposition ‘however, it is not necessary that some animal is not good’ (which is equivalent to ‘the good will not hold of some animal, but it will not not hold by necessity’) are both indicative of non-necessity. He shows why this second proposition is sound by adding the words ‘since it is contingent that all are good’. For the good may not hold of some animal unqualifiedly when it is also contingent that the good holds of all animals. 171 But since it is thought that not every animal is receptive of the good, he says that one should change the term and posit that either being awake or being asleep rather than good holds of no animal unqualifiedly. But animal holds of every horse by necessity. The result is that being awake or being asleep does not hold of some

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animal, but not by necessity, as he shows by adding the words ‘for these can apply to every animal’; for if they can apply to all, there will be no animal of which being awake or being asleep does not hold by necessity.

[31b11172 It has then been said when the conclusion will be necessary if the terms are related universally to the middle term. If one is related universally, the other particularly and both are affirmative, when the universal is necessary, the conclusion will also be necessary. The demonstration is the same as in the preceding; for the particular affirmative also converts. If, then, it is necessary for B to hold of all C and A is under C, it is necessary that B holds of some A; but if it is necessary that B holds of some A, it is also necessary that A holds of some B, since it converts.]

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Having spoken about mixtures of two universal premisses, he next speaks about those having one premiss particular. When one premiss is universal, the other particular, and both are affirmative, if the universal premiss is the necessary one, the conclusion will be necessary. For if the particular affirmative unqualified premiss is converted the result is again the first figure having the major premiss necessary. He first proves this, taking the minor premiss to be universal affirmative necessary, the major to be particular affirmative unqualified. For he takes it that A holds of some C and B holds of all C by necessity; keeping the latter fixed, he converts AC and takes ‘C holds of some A’; for this is ‘A is under C’;173 for this comes about by conversion. From these propositions in the third combination of the first figure174 it follows that B holds of some A by necessity. But if B holds of some A, A holds of some B. (It is necessary for the latter to be proved since A is the major extreme.) Thus the proposed conclusion is proved if the particular affirmative unqualified premiss, which is AC, is converted and so is the conclusion BA, which is particular necessary affirmative. [31b19175 Similarly, too, if AC is necessary and universal. For B is under C.] He says that the conclusion will be necessary similarly if the major premiss is taken to be affirmative universal necessary, the minor particular affirmative unqualified. For again, if the latter premiss is converted the result is a major universal affirmative necessary premiss in the first figure, so that again the conclusion is also particular

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affirmative necessary. The difference is that in the previous combination it was necessary to convert the conclusion as well, but in the present case in which the major is universal affirmative necessary, we only convert the minor premiss. He briefly indicates the reason why the conclusion is necessary when he says, ‘For B is under C’,176 thereby indicating also that the result of converting the minor premiss BC is affirmative particular.

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[31b20177 But if the particular premiss is necessary, the conclusion will not be necessary. For let BC be particular and necessary, and let A hold of all C, but not by necessity. If BC is converted, the first figure results, and the universal premiss is not necessary whereas the particular one is necessary. But when the premisses were related in this way the conclusion was not necessary, and so it isn’t necessary in the present case either. 31b27178 This is also evident from terms. For let A be being awake, B two-footed, C animal. Then it is necessary that B holds of some C, and it is contingent that A holds of C, and A holds of B but not necessarily. For it is not necessary that something two-footed not be asleep.179] If the particular premiss is taken to be necessary, the conclusion is no longer necessary, because then the minor premiss in the first figure becomes necessary; and when this was the case and the major was unqualified, the conclusion was unqualified. For let B hold of some C by necessity and A hold of all C. If we convert BC and keep the universal unqualified premiss AC fixed (for we cannot proceed in any other way),180 it results that A holds of all C, C of some B by necessity; but when the premisses were related in this way, the conclusion in the first figure was not necessary. (31b27) He also shows that the conclusion is not necessary by setting down terms. He takes A to be being awake, B two-footed, C animal. Two-footed holds of some animals (e.g., humans) by necessity, being awake holds of every animal unqualifiedly, and being awake holds of some two-footed things, but not by necessity. Again he says that ‘it is contingent that A holds of C’,181 to indicate that A holds of all C but not necessarily. It should be noted that here too he speaks of contingency instead of holding, as he also did in the case of the conversion of necessary propositions.182 183 The words ‘For it is not necessary that something two-footed not be asleep’ may indicate that it is contingent that being awake holds of some animal and also contingent that it holds of none. For, if some animal slept by necessity it would not be the case that being awake held of all animals contingently. (The words ‘For it is not necessary

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that something two-footed not be asleep’ may indicate that it is contingent that being awake holds of some animal and contingent that it does not hold of some.) [31b31184 It will be proved in the same way through the same terms if AC is particular and necessary.]

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He says that the conclusion will be proved not to be necessary through the same terms if the major premiss AC is taken to be particular necessary affirmative, and BC to be universal affirmative unqualified. However, it will be necessary to interchange the terms and change their order. For if the terms are assumed in the same way the premisses are not true. Thus it is necessary to assign two-footed to A, being awake to B. Then it will be the case again that two-footed holds of some animal by necessity, being awake holds of all animals unqualifiedly, and two-footed holds of something which is awake, but not by necessity. 185It is no less possible to reduce this to the first figure by conversion of the particular affirmative necessary premiss and prove that again the minor premiss in this figure is necessary (the conclusion also being converted ). 31b33186 But if one of the terms is affirmative and one privative, when the universal is privative [and necessary, the conclusion will also be necessary. For if it is not contingent that A holds of any C, and B holds of some C, it is necessary that A not hold of some B. 31b37 But when the affirmative is posited as necessary – whether it is universal or particular – or the privative is particular, the conclusion will not be necessary. The others are the same as we also said in the case of the previous ones. 31b40 Terms when the affirmative necessary premiss is universal: being awake, animal, human; the middle is human. 32a1 When the affirmative necessary premiss is particular: being awake, animal, white; for it is necessary that animal holds of something white, and it is contingent that being awake holds of nothing white, and it is not necessary that being awake does not hold of some animal. 32a4 When the privative particular premiss is necessary: two-footed, moving, animal; the middle is animal.]187

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He says that if one premiss is privative, the other affirmative, if the privative is universal and necessary, the conclusion is also necessary. For he is now discussing combinations having one particular premiss. But if the affirmative is necessary, the conclusion is unqualified and not necessary. The reason is the following: when the privative pre-

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miss is universal and the affirmative premiss is converted, the privative premiss becomes the major premiss in the first figure so that if it is necessary, the conclusion is also necessary (and he speaks briefly about this combination, since the mixture is already known188); but if the negative premiss is not necessary but the affirmative one, which becomes a necessary minor in the first figure, is, the conclusion will be unqualified. For if it is not contingent that A holds of any C, i.e., if it holds of none by necessity, and B holds of some C, BC will be converted and C will hold of some B; AC remains the major premiss and is universal necessary negative. Therefore, it will follow189 that A does not hold of some B by necessity. (31b37)190 Suppose the affirmative is necessary, either universal or particular, as when the major is either universal negative unqualified or particular negative unqualified, and the minor is either universal affirmative necessary or particular affirmative and likewise necessary. When the major is universal negative unqualified, the minor is particular affirmative necessary, but when the major is particular negative unqualified, the minor is universal affirmative necessary. In neither case will a necessary conclusion follow, because if the minor is converted, when it is particular,191 there results the first figure having the minor necessary. He indicates this reference to the argument from conversion when he says, ‘The others are the same as we also said in the case of the previous ones.’192 (32a1)193 However, he also shows that the conclusion is not necessary by setting down terms. He assigns being awake to A, animal to B, white to C. For it is contingent that being awake holds of nothing white, but animal does hold of something white by necessity, and being awake does not hold of some animal, but not by necessity. (This is what is meant by ‘it is not necessary that being awake does not hold of some animal’.) (31b40)194 He proves by setting down terms that the conclusion is not necessary if the universal premiss is affirmative necessary and the other premiss, the major, is particular negative unqualified; he assigns being awake to A, animal to B, and human to C. For animal holds of all humans by necessity, and being awake does not hold of some human. It follows that being awake does not hold of some animal simply, but not by necessity. (32a4) However, in his proof he has spoken first about the combination having a universal affirmative necessary premiss and a particular negative unqualified one – this is the sixth syllogism195 – and then about the one having a particular affirmative necessary premiss and a universal negative unqualified one.196 The latter is the only combination to which the proof by conversion also applies. Wishing to indicate this fact, he says, ‘The others are the same as we

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also said in the case of the previous ones.’197 For when the minor premiss is universal and affirmative, whether it is necessary or unqualified, the proof is not by conversion. However, if the particular premiss is negative necessary, the conclusion will not be necessary either. For as in the combination of the fourth syllogism in the second figure the conclusion was not necessary no matter which premiss was necessary because such a combination is not reduced to the first figure by conversion,198 so also in the combination of the sixth syllogism in the third figure the conclusion is not necessary no matter which premiss is necessary. It was proved a little earlier199 that the conclusion is not necessary when the affirmative premiss is universal and necessary by using terms.200 201When the particular negative premiss is necessary, terms are two-footed for A, moving for B, and animal for C. For two-footed does not hold of some animal by necessity, moving holds of every animal, and two-footed will not hold of some thing that moves, but not by necessity. For if the proposition ‘two-footed does not hold of some thing that moves’ is equivalent to ‘Some two-footed thing does not move’, then the proposition ‘Twofooted will not hold of some thing that moves by necessity’ will not be true because nothing two-footed does not move by necessity.202 The words ‘the middle is two-footed’ are added to the text in accordance with the blunder of the person who transcribed the book originally. For two-footed cannot be the middle term; animal must be. And this is also shown in the setting down of terms. For he mentions two-footed first, but his custom in the case of this figure is always to posit the middle term last. Thus, it is necessary to suppose that he does not say203 ‘the middle is two-footed’ but ‘the middle is animal’. For, if two-footed were the middle, neither of the terms mentioned, moving and animal, would not hold of some of it by necessity.204 205 Alternatively, it is possible by saying the same things which we have also said in the case of the fourth combination of the second figure which has the negative necessary premiss particular to prove that to the degree that it was necessary that the conclusion be necessary in cases discussed previously, it must also be necessary in this combination. For, using ekthesis we will get the particular necessary negative premiss to be universal negative necessary. Since this premiss is necessary, when the minor premiss is converted, it becomes a universal negative necessary major in the first figure. But the setting down of terms, which he also uses, shows that the conclusion in the case of this sort of combination is not necessary.

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1.12 Summary remarks on the necessary and the unqualified206 32a6207 It is then evident that there is no syllogism of holding unless both premisses are of holding, but there is a syllogism of necessity if only one of the premisses is necessary. If this assertion were made universally and without qualification, it would be false and conflict with what has already been said. For he has proved that the conclusion is also unqualified when one of the premisses, e.g., the minor premiss in the first figure, is necessary. And it has also been proved that in the second and third figure the conclusion is unqualified when one of the premisses is necessary. 208 But if one were to understand the assertion as made of the third figure, this would be true in a rather restricted sense. For in the third figure when both premisses are universal affirmative, the conclusion is not unqualified unless both premisses are unqualified, because if either of them is taken to be necessary, the conclusion is necessary. This was not true in the first figure when both premisses were universal affirmative; for when just the minor premiss was necessary, the conclusion was unqualified.209 And in the second figure nothing at all followed from two affirmative premisses. With these words he might then indicate the feature peculiar to the combination of two universal affirmative premisses in the third figure. For he was talking about this figure. For in this figure when there are two universal affirmative premisses, an unqualified conclusion will not follow unless both premisses are unqualified, and the conclusion will be necessary even if only one (no matter which) is necessary. 210 But since some combinations in which, conversely, unless both premisses are necessary, the conclusion is not necessary either (as in the case of the sixth) have already been proved, it is also possible to understand ‘holding’ as standing for ‘affirmative’; in this case he would be saying that the conclusion is not affirmative unless both premisses are affirmative, but it is necessary if only one of the premisses is necessary. But perhaps such an understanding of what is said will seem forced, since it would seem that the word ‘holding’ has not been taken to stand for ‘affirmative’, but as opposed to necessity. This is evident from what is added. For he says, ‘but there is a syllogism of necessity if only one of the premisses is necessary’. 211 If what is said was not just about the third figure but universal and general, it might mean that there is no conclusion which is simply unqualified . For the words ‘there is no syllogism’212 mean that no conclusion follows unless both premisses are unqualified. For even in those combinations in which just one of the premisses is necessary

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and the conclusion is unqualified, the conclusion results from two unqualified premisses, since the necessary is also unqualified. For the necessary is also unqualified, but the unqualified is not necessary. Therefore, when one of the premisses is necessary, it is possible to say that both are unqualified, but if one is unqualified, it is not possible to say that both are necessary. Therefore, an unqualified conclusion follows from two unqualified premisses and a necessary one follows ‘if only one of the premisses is necessary’. 213 It is perhaps better to understand the words ‘It is then evident that there is no syllogism of holding unless both premisses are of holding’ as if he were saying that it has become evident from what has been said that sometimes in some syllogistic combination an unqualified conclusion is impossible unless both premisses are unqualified; for one should not understand what is said universally, because said universally it is not true, but what is said is true of the combination of two universal affirmative premisses in the third figure, as he thinks has been proved. 214On the other hand, in every syllogistic combination it is possible for there to be a necessary conclusion, even if only one of the premisses is necessary, as he himself also says. Or is it possible to say the same thing also in the case of the necessary? For there are some combinations in which necessity does not follow unless both premisses are necessary, as has been proved of the fourth combination in the second figure and of the sixth in the third. 32a8 But there is a syllogism of necessity if only one of the premisses is necessary.

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He has proved in the case of the third figure that when there are two universal affirmative premisses215 the conclusion is necessary whichever of the premisses is taken to be necessary; thus also if only one of the premisses is necessary, the conclusion is necessary. It has also been proved in the case of the other figures that if one of the premisses is necessary the conclusion is also necessary.216 For it is not possible to say that both premisses are necessary if one is unqualified; for the unqualified is not necessary in the way that the necessary is also unqualified. 32a8-9217 In both, when the syllogisms are affirmative or privative, [it is necessary that one premiss be similar to the conclusion. By similar I mean that if the conclusion is unqualified a premiss is, and if necessary, necessary.] By saying ‘in both’ he indicates what he means by the words ‘when the syllogisms are affirmative or negative’.218 Or rather he says ‘in both’ with reference to what has just been said, that is, to the

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necessary and the unqualified, as he indicates through the things he adds. For in the case of necessary and unqualified conclusions, if the conclusion is affirmative or if it is negative, in either case ‘it is necessary that one premiss be similar to the conclusion’ in all the figures. He says in what respect it is necessary for one of the assumed premisses to be similar to the conclusion, and he adds the word ‘one’,219 since both premisses might be similar to the conclusion. (Or rather this is not possible in the case of mixtures, which he is now discussing.220) He says how they are similar in the previously discussed mixtures; for he says, ‘if the conclusion is unqualified a premiss is, and if necessary, necessary’. For it is not the case that if the conclusion is particular it is necessary that one of the premisses always be particular because in the third figure when the two premisses are universal,221 the conclusion is also particular. (Or is it the case that too a premiss has also been taken as potentially particular? That this is so is made clear by the reduction of combination to the first figure through a conversion by which one of the universal premisses becomes particular.) 222 The words ‘in both’ could also indicate the mixture. For when in combinations there is both an unqualified and a necessary premiss, he says that it is necessary for one of the premisses to be similar to the conclusion in modality. It is also clear that if the conclusion is negative it is necessary that a negative premiss be assumed. This is not added because he is now only talking about the necessary and the unqualified, but it has been noted for the future since it will be useful for him.223 224 He now indicates that he meant what was said a moment ago – namely that there will not be a syllogism with an unqualified conclusion unless both premisses are unqualified – to apply to the combination of two universal affirmative premisses in the third figure . For speaking universally he now says, ‘it is necessary that one premiss be similar to the conclusion  if the conclusion is unqualified a premiss is’, it being also possible for an unqualified conclusion to result from one unqualified premiss.

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32a12225 So this is also clear: the conclusion will not be either necessary or unqualified unless a necessary or an unqualified premiss is taken.

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the conclusion be either necessary or unqualified. He indicates that the conclusion of a combination cannot be necessary or unqualified unless a premiss has been taken to be necessary or unqualified because this will also be useful for him in the future.227 228 Against him one might ask how this can be sound in the case of a necessary premiss. (It is apparently sound in the case of an unqualified premiss.) For the conclusion seems to be necessary even if neither of the premisses is necessary. For consider, if you will, the following case: moving holds of all humans and human of all that walks unqualifiedly, and moving holds of all that walks necessarily. (a) Or is it the case that he says that the conclusion is not necessary unless one of the premisses is necessary since the situation is not always this way,229 but it is always true that when a necessary or an unqualified premiss is taken, the conclusion is similar, as he has said? (b) Or is it rather because necessity is signified by the addition of the modality, and this will not be added unless it has also been added to one of the premisses? (c) Or, if someone were to judge necessity not by the addition of the modality but by the nature of the thing, would it also be necessary that moving holds of everything human, even if he customarily uses this as if it were unqualified? For there is never a time when some human or, in general, animal is not moving. (d) Or is it also necessary to specify further how ‘moving holds of human’ is being taken? For, if this is done, ‘moving holds of human’ will be unqualified, and the modality of the conclusion will be found . For, moving in the sense of changing does not hold of all that walks by necessity – and this is the conclusion – if moving in the sense of changing holds of every human unqualifiedly. (e) Or is it the case that even if it is taken that all that walks is human and all humans move, still the conclusion ‘all that walks moves’ is not necessary without qualification but with the additional condition ‘as long as it is walking’? For all that walks does not move by necessity, if, indeed, it is true that what walks does not even walk necessarily except, as I said, on the condition ‘as long as it is walking’. 32a15 [We have said pretty much enough] about necessity, how it comes about [and how it differs from the unqualified.] In other words, how the conclusion comes to be necessary (and sometimes it comes about from one premiss, when the major premiss is necessary), and that necessary combinations are syllogistic in a way similar to unqualified ones, and that the necessary differs from the

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unqualified because it holds eternally.230 He is speaking about necessity as the necessity of a conclusion.

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32a16 Next we speak232 about the contingent, about when and how and through what things there will be a syllogism. He turns to the third specification of propositions, which involves the contingent. And he first establishes when there is a syllogism from two contingent premisses, and how they are combined with one another, and what is combined. For arbitrary combinations will not produce a syllogism, nor will they do so in an arbitrary way, just as was the case with unqualified and necessary propositions. 32a18 I call P contingent or say it is contingent that P if P is not necessary and if, when P is posited to hold, nothing impossible will be because of it.233 For we call what is necessary contingent homonymously. Since he is going to discuss syllogisms from contingent premisses, he first defines the contingent. He does not define it in its homonymous use since it is not possible to define something as it is used homonymously. Rather he isolates contingency as said of the necessary and the unqualified from the contingent. For he showed234 that the contingent is also predicated of these things. By saying ‘when P is posited to hold’ he indicates that, in addition to not being necessary, the contingent is not unqualified either.235 For what is contingent according to the third adjunct236 is of this kind and it differs from what is necessary and what is unqualified because if P is said to be possible,237 P is not yet the case. So, P would be contingent in the strict sense238 if P is not the case and if when P is posited to be the case it has nothing impossible as a consequent.239 And he would have spoken more strictly about the contingent if he said ‘P is not the case and when P is posited to hold’. For although what is not the case is not necessary, what is not necessary is not ipso facto not the case. Or does he deny that what is contingent is either necessary or unqualified, necessary by saying ‘is not necessary’, unqualified by saying ‘posited to hold’, since the word ‘posited’ also denies holding of the contingent? Or does he deny that what is contingent is unqualified by saying ‘if P is not necessary’; for, according to him, necessity is also predicated of the unqualified; for what holds of something holds of it with necessity, as long as it holds. At any rate Theophras-

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tus in the first book of his Prior Analytics, discussing the meanings of necessity, writes the following: ‘Third, what holds; for when it holds it cannot not hold.’240 Hypothesizing that what is not the case is the case is a peculiar feature of the contingent, since if someone takes it that either the unqualified or the necessary is the case, nothing impossible will follow either, and the account of contingency would also apply to the necessary and the unqualified. Therefore, it is a peculiar feature of the contingent that when what itself does not hold is hypothesized to hold it has nothing impossible as a consequent; for if when P is hypothesized something impossible follows, P is impossible; but, if something possible is hypothesized, nothing impossible follows (as he will show241) since the peculiar feature of the possible is not that it does not hold, but that when it does not hold and is assumed to hold, nothing impossible results. 32a21242 It is evident from oppositions of affirmations and negations243 that this is what the contingent is. [For ‘It is not contingent that X holds’, ‘It is impossible that X holds’, and ‘It is necessary that X does not hold’ are either the same or follow from each other, so that their opposites, ‘It is contingent that X holds’, ‘It is not impossible that X holds’, and ‘It is not necessary that X does not hold’ will also be either the same or follow from each other; for either the affirmation or the negation is said of each thing. Therefore, what is contingent will not be necessary, and what is not necessary will be contingent.]

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He shows that the contingent is as he defined it. What is said will be more understandable if we assume in advance what he asserts and shows shortly hereafter;244 for this will make what is said more evident to us. What he will show is this: if A follows from B and vice versa, then the contradictory of B follows from the contradictory of A and vice versa, since a thing or its contradictory is said of each thing245 (as he also adds here). For the sake of discussion let there be two affirmations, A and B; and let C be the contradictory negation of A, D the contradictory negation of B; and let the affirmation B follow from the affirmation A and be true at the same time as it. Then the negation D will also follow from the negation C, and will be true at the same time as it. For let something of which the negation C is true be taken, and let it be E. I say that the negation D is also true of this. For if it is not, the affirmation B will be true of it. And if it is, so will the affirmation A be. For they were assumed to follow one another and to be true at the same time. Therefore, the affirmation A will be true of that of which the negation C is true. Therefore, the contradictory is true of the same thing at the same time – which is impossible.

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Therefore the negation D will apply to246 E. But the negation C also applied to it. Therefore, it is universally true that if A and B are true together with one another, their contradictories are also true together with one another and follow from one another. Now that this is proved, what he says will be understandable. For if ‘It is not contingent that X holds’, ‘It is impossible that X holds’, and ‘It is necessary that X does not hold’ are either the same or follow from each other’ – as anyone would say and as has been shown in On Interpretation247 (for even if these are not all the same, because one is a negation and the others are affirmations, if one of them is true of something, so are the others) –, it is clear that the contradictories of these will also follow from one another. The affirmation ‘It is contingent that X holds’ is the contradictory of ‘It is not contingent that X holds’, which is negative; and ‘It is not impossible that X holds’ and ‘It is not necessary that X does not hold’248 are the contradictories of the affirmations ‘It is impossible that X holds’ and ‘It is necessary that X does not hold’, which follow from the negation ‘It is not contingent that X holds’. Therefore these things will also follow from one another and ‘It is contingent that X is’ and ‘It is not impossible that X is’ and ‘It is not necessary that X is not’ will be true at the same time. Therefore if it is contingent that X is, it will be neither impossible that X is nor necessary that X is not. Therefore, it is not necessary, it is not unqualified, and, when it is hypothesized to hold nothing impossible will follow since it is not impossible. This is true since an impossibility follows from an impossibility, as he, in proceeding, will soon assert and show.249

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32a28 Therefore, what is contingent will not be necessary, [and what is not necessary will be contingent.] He indicates why he has considered the implication relation between the propositions: in order to show that in the definition of contingency it is reasonable to lay down that P is not necessary and when P is hypothesized to hold, nothing impossible follows. For what is contingent is neither necessary nor impossible. 250 32a29 It results that all contingent propositions convert with one another. [I do not mean that the affirmative converts with the negative, but rather that whatever has an affirmative form converts with respect to its antithesis, e.g., that ‘It is contingent that X holds’ converts with ‘It is contingent that X does not hold’,

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Translation 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 does not hold of all B’, and ‘It is contingent that A holds of some B’ converts with ‘It is contingent that A does not hold of some B’, and the same way in the other cases.]

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Conversion is a peculiar feature of contingency, that is, that affirmative propositions and negative propositions with respect to it are equivalent to one another. I mean by negations with respect to contingency ‘contingent negatives’,251 not negations of contingency. For a contingent negative and a negation of contingency are different, and so are a necessary negative and a negation of necessity. For the proposition which says that it is necessary that X does not hold is a necessary negative, but it is not the negation of the proposition ‘It is necessary that X holds’. The proposition ‘It is not necessary that X holds’ is the negative252 of the necessary proposition ‘It is necessary that X holds’ and is negative in the strict sense. The other proposition is in itself an affirmative, but the whole is a ‘necessary negative’. In the same way also, in the case of contingency the negation in the strict sense, ‘It is not contingent that X is’, is a negation of contingency, while ‘It is contingent that X is not’, which is contingent, is in itself an affirmation. It results from the meaning which contingency is now taken to have253 that contingent negatives and contingent affirmatives are equivalent to one another. For, if it is contingent (in this sense) that X is, it is always also contingent that X is not. Because he is going to use this kind of conversion of contingency in connection with syllogisms from contingent premisses he first establishes that conversion holds in this way. However, it is necessary to understand that this kind of conversion of propositions is not sound according to the associates of Theophrastus;254 nor do they use it. For there is the same reason for saying that the universal negative contingent proposition converts with itself in the way that the unqualified and necessary propositions do, and that contingent affirmatives do not convert with contingent negatives, as Aristotle maintains. We will explain this when he discusses the term-conversion of a contingent proposition.255 The words ‘I do not mean that the affirmative converts with the negative, but rather that whatever has an affirmative form converts with respect to its antithesis’ indicate what has just been said. For the proposition ‘It is contingent that X is’ seems to be the opposite of the one which says that ‘It is contingent that X is not’, because ‘is’ is co-ordinated with the one and ‘is not’ with the other, and these seem to be opposite to one another. But in fact, both have an affirmative

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form because they are affirmations in which what negates is not attached to the modality. Having discussed indeterminate propositions, what they are and how contingent negatives are taken to be equivalent to contingent affirmatives, he next takes the determinate ones. He says that ‘It is contingent that A holds of all B’ is equivalent to ‘It is contingent that A holds of no B’ and likewise also to ‘It is contingent that A does not hold of all B’; of these ‘It is contingent that A holds of no B’ is a universal contingent negative, and the second, in which it is taken that it is contingent that A does not hold of all B, is a particular one; he shows that all contingent negative propositions are true when the contingent universal affirmative is. For neither of these propositions, one of which is like a contrary, the other like a contradictory, is in truth either a contrary or a contradictory. For if they were, they would not be true together . And it is clear from induction that propositions taken in this way are equivalent. And in the case of contingent propositions ‘to hold of some’ and ‘to not hold of some’ convert even when ‘some’ is taken to apply to the same thing. For it is a peculiar feature of the contingent that things asserted in this way of the same thing are true at the same time. 256 For if the B’s are taken as different things not only will the contingent particular propositions be true together, but unqualified and necessary propositions can also be true together sometimes; and in these cases ‘It is contingent that A holds of some B’ and ‘It is contingent that A does not hold of some B’ are equivalent because one takes Socrates or Plato or some individual and makes an antithesis with respect to contingency in relation to it in the way discussed. He says ‘and the same way in the other cases’ in order to refer us also to the other things which are equivalent to the contingent. These are (i) the affirmation ‘It is possible257 that X is’ with which ‘It is possible that X is not’ converts, no matter what the quantitative determination258 of X, whether it is universal affirmative, universal negative, particular affirmative or particular negative, as he proved in the case of the contingent proposition; (ii) the negations, (iia) ‘It is not necessary that X is not’ – for this follows from ‘It is contingent that X is’ – from which ‘It is not necessary that X is’ follows and with which it converts, and (iib) ‘It is not impossible that X is’ – for this follows from ‘It is contingent that X is’ – with respect to which the proposition which says that it is not impossible that X is not converts.

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Translation 32a36 For since the contingent is not necessary, [and what is not necessary may 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 that it holds of all, it is also contingent that it does not hold of all. And similarly in the case of particular affirmations. For the demonstration is the same. Propositions of this kind are affirmative, not privative; for ‘It is contingent that X’ is ordered in the same way as ‘X is’, as was said earlier.259]

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Having said that contingent negative propositions are equivalent to contingent affirmative ones and what these propositions are, he now sets down and gives the reason why they are equivalent and convert with one another. For since ‘It is contingent that X holds’ posits that X does not hold by necessity – since, as was indicated in his definition,260 P is contingent ‘if P is not necessary’–, and what is not necessary may also not hold, it is reasonable that ‘It is contingent that X does not hold’ is true together with ‘It is contingent that X holds’. The situation is the same in the case of all the quantitative determinations. He says that indeterminate propositions and determinate universal affirmative and negative propositions are equivalent to each other, and that particular affirmations and negations and individual ones said of the same thing are equivalent. And they are equivalent for the same reason: the affirmation under consideration indicates that what is predicated cannot be by necessity. If the contingent is of this sort, those people261 who do not think that affirmative propositions convert with contingent negative ones are wrong. He reminds us that contingent negatives are affirmations and not negations. For in On Interpretation262 he showed that propositions having the modality predicated but not negated are all affirmations. This is why he adds here the words ‘as was said earlier’. From these very words it is also possible to show, contrary to Andronicus, that On Interpretation is Aristotle’s. However, it is also true that he has already spoken about this in this work when he said that the universal negative does not convert.263 264 I inquired how, if the contingent is such as it was defined to be and is neither necessary nor unqualified, nevertheless negative contingent propositions will convert with affirmative contingent ones with contingency in the way specified265 being preserved in both. For suppose if it is contingent that X is then X is not yet – for this is thought to be the peculiar feature of the contingent, that what is said to be contingent is not yet; then what is contingent will not hold. And if it is contingent that X is not, X does not yet not hold; therefore it holds; thus a affirmative is true when an unqualified

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negative holds, and a negative one is true when an unqualified affirmative holds. But it is impossible that both hold at the same time. Therefore, contingency in the way specified cannot be being taken in the negative proposition.266 267 Or did he for this reason define the contingent268 by saying that P is contingent ‘if P is not necessary’ without adding that P absolutely does not hold on the grounds that not holding is not the peculiar feature of the contingent; rather, the peculiar feature is that when it does not hold and is hypothesized to hold, nothing impossible follows from it. He also indicated that he means this by saying269 ‘For we call what is necessary contingent homonymously’ without adding ‘and also the unqualified’ – which should have been a necessary implication ; for when we call the unqualified contingent we do not speak homonymously. It is not the case that, since if ‘It is contingent that X is’ is true of something, ‘It is contingent that X is not’ is also true of it, it is thereby also true that when one is true the other is; but at alternating times. For it is when X holds, not when it does not hold, that it is contingent that it does not hold. For if X does not hold and it is contingent that it does hold, then it is also contingent that X does not hold when it does hold. Alternatively, if ‘It is contingent’ is said of things which are not yet, then ‘It is contingent that X is’ might be said instead of ‘It is contingent that X comes about’. But if X is not and it is contingent that X comes about, it is always also contingent that X does not come about, so that both would apply to what is not yet.

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270 32b4 Having made these distinctions let us say next that contingency is spoken of in two ways.271 [In one way what comes about for the most part and falls short of necessity is contingent, for example, for a person to get grey hair or to grow or decline, or in general what holds naturally – for this does not have necessity continuously since a human does not always exist, but when a human does exist, this is the case either by necessity or for the most part. In another way, the indefinite, which can be one way or the other, for example, that an animal walks, or that there is an earthquake when it is walking, or, in general, what comes about by chance – for it is not by nature any more one way than the other.] Having said what the contingent is and that contingent affirmative and negative propositions convert with one another, he says that the contingent is spoken of in two ways, meaning it is spoken of homony-

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mously; for he does not specify of which contingency he makes this division; for the account which has been given applies to both. He makes the division either as the division of the genus of contingency into species which he sets out, or as the division of a whole into parts. He says that one kind of contingency signifies that which applies to things said for the most part. Since these things do not always come about in this way, but happening always and by necessity are interrupted because some of them do not turn out the same way sometimes, these things are also contingent. Things which come about by nature are of this sort; these things do not come about by necessity because in some cases things sometimes also happen in another way; but they come about the same way for the most part. For human beings naturally turn grey when they age, and they turn out this way for the most part – for some people are already old and they have not turned grey. And people grow for a certain length of time, and they do so naturally and usually. 32b5 [In one way what comes about for the most part and falls short of necessity is contingent, for example, for a person to get grey hair or to grow or decline, or in general what holds naturally –] for this does not have necessity continuously since a human does not always exist, but when a human does exist, this is the case either by necessity or for the most part. [In another way, the indefinite, which can be one way or the other, for example, that an animal walks, or that there is an earthquake when it is walking, or, in general, what comes about by chance – for it is not by nature any more one way than the other. Each of the kinds of contingency converts with respect to the opposite propositions also, but not in the same way. If P is by nature, it converts with ‘P does not hold by necessity’ – it is contingent for a person not to turn grey in this sense. But if P is indefinite, P converts with ‘no more P than not P’.]272 He introduces two reasons why these things which come about usually are not necessary. One is that the things of which they hold do not always exist. For what comes to be by nature is not eternal. What is natural is not by necessity because the things of which we say it is contingent that what is natural holds do not always exist – for what is natural holds of individuals. For what is necessary is eternal and always in the same way in the case of things which are in the same condition. So it could be said that what holds by necessity of what is not eternal is also contingent, even if it doesn’t cease to exist before because it is contingent that what comes about by necessity (if a human being existed forever) does not come about because does

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cease to exist before . For example, if by necessity every human went grey on reaching the age of sixty, nevertheless it would still be contingent that a given human being will become grey, because it is contingent that he also not reach such an age. He indicates what is contingent for this reason when he says ‘for this does not have necessity continuously since a human does not always exist’, meaning that such things are not necessary because they are not continuous. A second reason why things which come about naturally are not by necessity is that even if a person of which what is natural holds were to have reached the age of sixty, he will only turn grey usually, but not by necessity. He indicates this reason when he says, ‘but when a human does exist, this is the case either by necessity or for the most part’. For, if something is for the most part, it is clear that it is not by necessity. One of the things which contingency signifies, then, is this. Also ordered under this would be things that come about as a result of choice.273 For being usual also applies to these things. He says that the other thing which contingency signifies is ‘the indefinite’. He means by indefinite both what is equally balanced and will be no more one way than the other (for example, that Socrates will take a walk in the afternoon or converse with a certain person), and in addition what is opposite to the kind of contingency which comes about usually, namely what comes about infrequently and because of which, when it intervenes, what comes about usually is prevented from coming about always and being necessary. A sixtyyear-old not becoming grey would be this sort of thing. What is by chance is also included in the kind of contingency signified by infrequency. He indicates the kind of contingency which is equally balanced with respect to opposites with the words ‘ walks’; and he indicates the infrequent which is opposite to what happens usually with the words ‘that there is an earthquake when it is walking, or, in general, what comes about by chance’. Both of these are indefinite, one because it could equally go either way; for what is ‘not any more one way than the other’ is indefinite. And what is infrequent is indefinite because it comes about more or less without a cause. For what is by chance is of this kind and comes about in this way. For chance is a cause accidentally, not per se, and the future itself is in general indefinite and unclear. For what is most of all determined is what is necessary. In second place is what is close to the necessary; this is what happens usually. But it is reasonable to call what stands furthest from the determined indefinite. What is equally balanced is of this kind, and what happens infrequently is even more so. For, the necessary is like a line which has been stretched from eternity into eternity, and the contingent comes into being from this line when it is cut. For if this line is cut into unequal

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segments, the result is the contingent as the natural and what is for the most part, and also the contingent as the infrequent, which includes chance and spontaneity. But if the line is cut into equal segments there results the ‘who can tell’.274 Of these, if the contingent is taken in the sense of what is for the most part, ‘It is contingent that P’ converts with ‘It is not necessary that P holds’. Consequently, if it is contingent (in the sense of usual) that P holds, it is also true that it is contingent that P does not hold, because if P is usual, it is true that it is not necessary that P holds; for ‘It is not necessary that P is’ follows from ‘It is contingent that P is’, just as ‘It is not necessary that P is not’ does. If the contingent is taken in the sense of ‘who can tell’, ‘It is contingent that P holds’ converts with ‘It is contingent that P does not hold’; for what is contingent in this sense is equally balanced and is no more one way than the other. He himself also makes these things clear, by adding the words ‘Each of the kinds of contingency converts with respect to the opposite propositions also, but not in the same way’. By ‘each of the kinds of contingency’ he means the usual and the indefinite. If the contingent is taken in the sense of what is usually, he says that the proposition ‘It is contingent that P does not come about’ is true, not because it is equally true with the proposition ‘It is contingent that P comes about’, but because in the case of the contingent in the sense of what is usually the proposition ‘It is not necessary that P is’ is true; for the contingent is not necessary. And the proposition which says that it is contingent that P does not come about is true because interrupts the continuity of necessity. The contingent as indefinite converts insofar as it is no more one way than the other. For in the case of what is contingent in this way the proposition ‘It is contingent that P is not’ is true together with the affirmation ‘It is contingent that P is’ and converts with it because one is no more true than the other. The words ‘each of the kinds of contingency converts with respect to the opposite propositions also’ are equivalent to ‘also each of the kinds of contingency converts with respect to the opposite propositions’.275 The conjunction ‘also’ seems to be oddly placed. Or perhaps he says ‘with respect to the opposite propositions also’ because they also convert in other ways. For they also convert by interchange of terms.276 32b18 There is no science or demonstrative syllogism of what is indefinite277 [because the middle is without order. However, there is science and demonstrative syllogism of what is natural, and almost all arguments and investigations concern what is

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contingent in this way. There might be a syllogism in that way,278 but it is not customary to inquire about it. These things will be made more precise in what follows.] Having said that one of the two meanings of contingency is the indefinite, he says now that there is no science of what is contingent in this way. For there cannot be a demonstration of anything from premisses which are contingent in this way because what is proved through the syllogism is no more than its opposite. The reason for this is the indefiniteness of the middle term and its being no more related to the extremes in the way it is taken to be than in the opposite way. Consequently he rejects proving the syllogisms in the figures in the case of this kind of contingency, not because such proof is impossible, but because it is useless; he thereby indicates to us that in the present subject it is necessary to take and work out only what is useful for things which will be proved and to reject what is useless, even if it has a certain formal validity.279 Therefore it is clear that it is because of their uselessness and not because of his ignorance that he leaves out those things which recent thinkers280 discuss but he has not talked about and which are useless for demonstration. Examples are duplicated arguments, duplicating arguments, what is called ‘infinite matter’, and in general the thema which is called the second by recent thinkers.281 For the usefulness of what is shown or comes about using it is the measure of any instrument.282 And what is not useful would not even be an instrument; for an adze which is useless for a carpenter is not an adze except homonymously. He indicates this most clearly by rejecting discussion of contingency in the sense of indefiniteness. He says, ‘There might be a syllogism in that way,283 but it is not customary to inquire about it’, on the grounds that it is syllogisms liable to have an application to things which are inquired into and need proof which one must discuss. Thus he rejects talking about this kind of contingency because it is useless for inquiries.284 He promises to speak about the other kind of contingency because there are many arts which are conjectural285 and deduce something under consideration on the basis of what is contingent in this way, for example, medicine, navigation, gymnastics. But also, in general, things based on deliberation are proved by means of this kind of contingency. For example, if someone were to inquire whether he ought to sail now and he argued as follows: when the winds have been judged favourable, those who sail get through safely for the most part; now the winds have been judged favourable; therefore those who sail now will get through safely for the most part. He says that a syllogism is demonstrative if a person who wished to prove something would use it.

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32b23 These things will be made more precise in what follows. [Let us now say when there will be a syllogism from contingent premisses, how it will be, and what it will be.]

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He says that in the sequel he will make a determination about the fact that in the case of contingent propositions an affirmative converts with a negative, and about how it converts, and that contingency is twofold, and that the indefinite is useless for present inquiries.286 He adds what he says now about combinations of contingent premisses in each figure which are syllogistic and non-syllogistic. 32b25287 It is possible to take ‘It is contingent that A holds of B’ in two ways, [namely, as either ‘It is contingent that A holds of that of which B holds’ or ‘It is contingent that A holds of that of which it is contingent that B holds’; for the expression ‘It is contingent that A holds of that of which B is said’ means either of these things: ‘It is contingent that A holds of that of which B is said’ or ‘It is contingent that A holds of that of which it is contingent that B is said’.]

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He says that it is possible to understand the proposition which says that it is contingent that A holds of all B in two ways. For sometimes one can understand the person who says this as saying that it is contingent that A holds of all of that of which B holds, and sometimes one can understand him as saying that it is contingent that A holds of all of that of which it is contingent that B holds. He shows that the proposition which says that it is contingent that A holds of all B can mean each of these things, but transforms it into the proposition which says that it is contingent that A holds of that of which B is said.288 Having shown that it is possible to understand this proposition as either ‘It is contingent that A holds of that of which B is in fact said’ or as ‘It is contingent that A holds of that of which it is contingent that B is said’, he then adds: 32b29 There is no difference between ‘It is contingent that A holds of that of which B is said’ and ‘It is contingent that289 A holds of all B’. [So it is evident that ‘It is contingent that A holds of all B’ can mean two different things. 32b32 Let us then first say what syllogism and what sort of syllogism there will be if it is contingent that B holds of that of which C is said and that A holds of that of which B is said. For

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in this way both premisses are taken as contingent. But when it is contingent that A holds of that of which B is said , then one premiss is unqualified, the other contingent. Consequently, we should begin with premisses similar in form, as in the other cases.] Consequently ‘It is contingent that A holds of all B’ can also mean two things. But when the proposition means the first one mentioned, the premisses are not both contingent; rather the minor is unqualified, the major contingent, and such a combination is mixed. When the proposition means the second, both premisses are contingent. Since, then, he always discusses combinations with premisses similar in form before mixed combinations, he says that it is first necessary to discuss combinations with both premisses contingent. In the proof that ‘It is contingent that A holds of all B’ has two meanings he used the formulation ‘It is contingent that A holds of that of which B is said’. He now indicates that he has not proved anything other than what he proposed to prove. For ‘It is contingent that A holds of that of which B is said’ means the same thing as ‘It is contingent that A holds of all B’, as we have said. For ‘X is said of Y’ indicates universality and that X is said of all Y. Thus if ‘It is contingent that A is said of all of that of which B is said’ has two meanings, ‘It is contingent that A holds of all B’ will have the same two meanings. It is through these things that he shows that the premiss which is expressed prosleptically290 has the same meaning as the affirmative premiss. But if ‘It is contingent that A holds of that of which B is said’ has two meanings, so will ‘By necessity A holds of that of which B is said’ have two meanings; for it will mean either ‘A holds by necessity of all of that of which B is said unqualifiedly’ or ‘A holds by necessity of all of that of which 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 all of that of which B is said’, as is said by some of those291 who show that it is true that the conclusion of a necessary major and an unqualified minor is necessary. 32b32 Let us then first say [what syllogism and what sort of syllogism there will be] if it is contingent that B holds of that of which C is said and that A holds of that of which B is said. [For in this way both premisses are taken as contingent. But when it is contingent that A holds of that of which B is said , then one premiss is unqualified, the other contingent. Consequently, it is necessary to begin with premisses similar in form, as in the other cases.]

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He says that it is first necessary to discuss the combination of two contingent premisses. Since he assumes that ‘X holds of that of which Y is said’ is equivalent to ‘X is said of all Y’, he takes the premiss ‘It is contingent that B holds of all C’ because it means the same thing as the following: it is contingent that B holds of that of which C is said. Similarly in the case of ‘A holds of that of which B is said’; for this is equivalent to ‘A is said of all B’. He says that it is necessary to begin first from similar in form, and adds ‘as in the other cases’.292 For in treating the unqualified and the necessary he first spoke about each on its own and about premisses similar in form in their case, and then about mixtures. He proposes to do the same thing and will do it in the case of the contingent. * 1.17, 36b35-37a31 Failure of EE-conversion for contingent propositions293 36b35 It should first be shown that a privative contingent294 proposition does not convert; [that is, if it is contingent that A holds of no B, it is not necessary that it is also contingent that B holds of no A. 36b37295 For let this be assumed and let it be contingent that B holds of no A. Then, since contingent affirmations convert with negations – both contraries and opposites – and it is contingent that B holds of no A, it is evident that it will also be contingent that B holds of all A. But this is false. For it is not the case that if it is contingent that X holds of all Y, it is necessary that it be contingent that Y holds of all X. So the privative does not convert. 37a4 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 is contingent that white does not hold of any human being – for it is also contingent that it holds of every human being –, but it is not true to say that it is contingent that human holds of nothing white. For it does not hold of many white things by necessity, but what is necessary is not contingent.]

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When he discussed propositional conversions and showed which propositions convert with which,296 he said that a universal negative contingent proposition does not convert with itself, but he postponed giving the reason until later. He shows this now, as the situation demands because syllogistic combinations in the second and third figure require conversions. Since he is going to show that there is no syllogism from contingent premisses in the second figure, and since

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he is also going to make use of the fact that a universal negative contingent proposition does not convert with itself, he proves this first. By setting down terms he indicates what kind of conversion he is talking about – viz., the interchange of terms and not the transformation of a negative proposition to an affirmative one; for it is assumed to convert to that.297 However, as we mentioned at the beginning,298 Theophrastus and Eudemus say that the universal negative also converts with itself just as both the unqualified and the necessary universal negative do. They show that it converts in the following way. If it is contingent that A holds of no B, it is also contingent that B holds of no A. For since it is contingent that A holds of no B, when it is contingent that it holds of none, it is then contingent that A is disjoined from all the things of B.299 But if this is so, B will then also have been disjoined from A, and, if this is so, it is also contingent that B holds of no A. It seems that Aristotle expresses a better view than they do when he says that a universal negative which is contingent in the way specified does not convert with itself. For if X is disjoined from Y it is not thereby contingently disjoined from it. Consequently it is not sufficient to show that when it is contingent that A is disjoined from B, then B is also disjoined from A; in addition that B is contingently disjoined from A. But if this is not shown, then it has not been shown that a contingent proposition converts, since what is separated from something by necessity is also disjoined from it, but not contingently. (36b37) Aristotle shows that there is no conversion using reductio ad impossibile.300 For, if possible, let it be assumed that there is conversion, and if it is contingent that A holds of no B, let it also be contingent that B holds of no A. However, we are assuming that negative contingent propositions also convert with respect to affirmative contingent ones. But it is assumed that it is contingent that B holds of no A. So it is clear that it is also contingent that it holds of all A. But this is false. For it is not the case that, if it is contingent that A holds of all B for the reason that it is assumed contingent that it holds of none, it is necessary that it is also contingent that B holds of all A. For if it is the case, it results that a universal affirmative contingent proposition converts with itself, which isn’t true even according to them.301 For notice that it is contingent that white holds of every human – since it is also contingent that it holds of none –, but it is not contingent that human holds of everything white; for it does not hold of some white things, e.g., swan, snow, and many other things, by necessity. But if it is false that it is contingent that human holds of everything white, it is also false that it is contingent that it holds of nothing white. Consequently, it is not the case that if it is

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contingent that A holds of no B, it will also be contingent that B holds of no A. For what holds of nothing does not thereby contingently not hold. But they maintain consistency by saying that the universal negative converts with respect to terms and also denying that an affirmative contingent proposition converts with a contingent negative one. The latter conversion is not possible because according to them contingency in the way specified is not the only contingency.302 303 He has said ‘Furthermore nothing prevents’ instead of ‘For nothing prevents’. For, as is apparent from what is said, there is no other proof that things are this way than the one from the construction just described.304 Or perhaps he first showed this by contingent negative propositions being transformed into affirmative ones, affirmative contingent universal propositions being assumed not to convert with themselves; and now he gives a proof with respect to negative contingent universal propositions themselves, setting down terms and showing through them that the propositions do not convert. If this were so this proof would be different from the one before it. But he says: 36b38 ... since contingent affirmations convert with negations – both contraries and opposites – [and it is contingent that B holds of no A, it is evident that it will also be contingent that B holds of all A. But this is false. For it is not the case that if it is contingent that X holds of all Y, it is necessary that it be contingent that Y holds of all X. So the privative does not convert. 37a4 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 is contingent that white does not hold of any human being – for it is also contingent that it holds of every human being –, but it is not true to say that it is contingent that human holds of nothing white. For it does not hold of many white things by necessity, but what is necessary is not contingent.]

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He calls the universal propositions ‘It is contingent that X holds of all Y’ and ‘It is contingent that X holds of no Y’ contraries; and he calls the universal propositions ‘It is contingent that X holds of all Y’ and ‘It is contingent that X holds of no Y’ opposites of the particular propositions ‘It is contingent that X holds of not all Y’ and ‘It is contingent that X holds of some Y’. But he does not do so because these are genuine contraries or opposites of each other. How could they be if they are true together? Rather, he does so because these propositions are related verbally to one another in the same way as contraries are related to one another in the case of necessary and

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unqualified propositions. For the proposition which says that X holds of all Y by necessity is contrary to ‘X holds of no Y by necessity’, and ‘X holds of all Y’ is contrary to ‘X holds of no Y’. And as far as what is implied by the words, ‘It is contingent that X holds of all Y’ is also contrary to ‘It is contingent that X holds of no Y’; and again ‘X holds of all Y’ is the opposite of the proposition which says that X holds of not all Y, and ‘It is not the case that X holds of all Y by necessity’306 is the opposite of ‘X holds of all Y by necessity’. And it seems that ‘It is contingent that X holds of all Y’ is similarly related to ‘It is contingent that X does not hold of all Y’. This is why he also calls these propositions contraries and opposites. 307 However, he would not say that the particular propositions are true together with the universal ones to which they seem to be opposite. For it is not the case that if ‘It is contingent that X holds of some Y’ is true, ‘It is contingent that X holds of no Y’ is thereby also true. However, he would say that the propositions which seem to be opposite to the universal ones – i.e., the particular ones, whether affirmative or negative, – do convert with one another: the universal propositions convert with one another and again the particular propositions which appear to be opposite to the universal ones convert with one another. He also says this in De Interpretatione; for in speaking about contraries he says ‘Therefore these cannot both be true at the same time’, and adds ‘But it is possible308 for their opposites to be true with respect to the same thing’.309 And it is also possible that he has said that particular propositions are opposite to one another when they are taken with respect to the same subject as having their subject determinate. Or is this not a peculiar feature of opposites? 310 Perhaps he is saying that particular contingent propositions convert from those universal contingent ones which seem to be opposite to them, but not saying that the universal propositions convert from the particular contingent ones. (37a4) He shows in a clear way using material terms that universal negative propositions which are contingent in the way specified do not convert with respect to terms. For it is contingent that white (and similarly walking and also being asleep) holds of no human, but it is not contingent that human holds of nothing white (or walking or asleep), because it is not also contingent that it holds of all; for human necessarily does not hold of some things which are asleep or white. It is even more evident that it is contingent that moving holds of no human because it is contingent that it holds of every human, but it is not contingent that human holds of nothing that moves because it is not also contingent that it holds of all that moves; for it is not contingent that human holds of the rotating body,311 since it does not hold of that by necessity.

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Someone might ask about the conversion of contingent affirmative propositions with respect to negative ones whether perhaps the contingent propositions do not convert with one another, but do convert with unqualified ones. For if propositions about the future are contingent in the strict sense, then it is clear that, if a contingent affirmative is true, it is true that what was assumed to be contingent does not yet hold. Consequently ‘It is contingent that X holds of no Y’ – said of what does not hold now – would convert with respect to ‘It is contingent that X holds of all Y’. (The latter proposition is true because what it says will hold.) The same thing can be said of the contingent negative proposition since the affirmation of what holds converts with ‘It is contingent that X holds of no Y’, which is true. For it is not the case that what is going to hold is also going not to hold, since it already doesn’t hold. But perhaps even if the thing which the affirmation says is contingent most definitely does not now hold, nevertheless it is contingent that it later does not hold; for even if P is said to be contingent and P does not come about, it remains the case that it is contingent that P not hold again later. And if it is said that it is contingent that P holds and P does come about again, it would remain the case that it was contingent that P not hold at the time when it was also contingent that P would hold. For if it is true to say of a person that it is contingent that he walk tomorrow, it is true to say of him that it is contingent that he not walk tomorrow. Thus, since a proposition about the future is contingent, it is necessary to take both propositions in relation to the future. For even if it is true that the unqualified is the opposite of the contingent, it is not assumed to convert with respect to it. 312

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37a9313 Moreover, it will not be proved from impossibility that there is conversion either, [for example, if someone were to maintain that since it is false that it is contingent that B holds of no A, it is true that it is not contingent that it holds of none – this is a case of affirmation and negation –, and if this is so, then it is true that B holds of some A by necessity; consequently A also holds of some B , but this is impossible. 37a14 For it is not the case that if it is not contingent that B holds of no A, it is necessary that B holds of some A. For ‘It is not contingent that B holds of no A’ is said in two ways; it is said if B holds of some A by necessity and if it does not hold of some by necessity. For if B does not hold of some A by necessity, it is not true to say that it is contingent that it does not hold of all, just as if B does hold of some A by necessity, it is not true to say

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that it is contingent that it holds of all. So, if someone were to maintain that, since it is not contingent that C holds of all D, it does not hold of some by necessity, he would take things falsely. For it holds of all, but we say that it is not contingent that it holds of all because it holds of certain of them by necessity. Consequently both ‘X holds of some Y by necessity’ and ‘X does not hold of some Y by necessity’ are opposite to ‘It is contingent that X holds of all Y’. And similarly in the case of ‘It is contingent that X holds of no Y’. 37a26 It is clear then that with respect to things which are contingent and not contingent in the way which we have specified initially it is necessary to take ‘B does not hold of some A by necessity’ and not ‘B holds of some A by necessity’. But if this is taken, nothing impossible results, so there is no syllogism. Thus it is evident from what has been said that a privative proposition does not convert.] Someone might think that it is at least possible for it to be proved that a universal negative contingent proposition converts by reductio ad impossibile. And his associates314 have used this same proof. For, if it is contingent that A holds of no B, it is also contingent that B holds of no A. For if this is false, the opposite is true, but the opposite of ‘It is contingent that B holds of no A’ is ‘It is not contingent that B holds of no A’, which is thought to be equivalent to ‘B holds of some A by necessity’. Therefore, B holds of some A by necessity. But since a particular necessary affirmative proposition converts, A also holds of some B by necessity, which is impossible, since it was hypothesized that it is contingent (in the way specified) that A holds of no B. Accordingly, if this is impossible, so is the hypothesis from which it followed, namely ‘B holds of some A by necessity’, which was obtained by transforming ‘It is not contingent that B holds of no A’. Therefore, the opposite, ‘It is contingent that B holds of no A’ is true. (37a14) Aristotle rejects this proof as not being sound. Having set out the proof and being about to refute it, he does not first say ‘This is false’ or something of that kind; rather he turns directly to showing that such a proof has not proceeded correctly. Consequently what is said seems in a way rather obscure. For he says ‘for it is not the case that if it is not contingent that B holds of no A, it is necessary that B holds of some A’. With these words he censures the transformation of ‘It is not contingent that B holds of no A’ (which is the opposite of ‘It is contingent that B holds of no A’) into ‘B holds of some A by necessity’ as unsound. For it is not at all the case that if ‘It is not contingent that B holds of no A’ is true, thereby and as a result it is true that B holds of some A by necessity. For the proposition which says ‘It is not contingent that B holds of no A’ is also true if B does

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not hold of some A by necessity. And the reason is that ‘It is contingent that B holds of all A’ converts with ‘It is contingent that B holds of no A’; and the following are uniquely opposite to them:315 to

(i) B does not hold of some A by necessity (ii) It is contingent that B holds of all A; and

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(iii) B holds of some A by necessity (iv) It is contingent that B holds of no A.

Either (i) or (iii) will do away with both (ii) and (iv); at least if (ii) and (iv) are equivalent to one another and convert with one another, each of (i) and (iii) does away with both (ii) and (iv); and when one of (ii) or (iv) is done away with the other is. Consequently (iii) and (i) do away with (iv), and (iii) does so per se, (i) accidentally (since it does away with (ii) and thereby also does away with (iv)). But, if this is so, the negation of (iv), ‘It is not contingent that B holds of no A’, will be true not only because (iii) is true but because (i) is. For both do away with the opposite of this, (iv), since (iv) cannot be true when (i) is. Consequently the person who hypothesizes ‘It is not contingent that B holds of no A’ does not always hypothesize it because (iii) holds, but also because (i) does. So, if, given the hypothesis that it is not contingent that B holds of no A, someone transforms it into (i) – which is no less a consequence of the hypothesis than (iii) –,316 nothing impossible will follow.317 For it is not the case that if B does not hold of some A by necessity, thereby A will not also hold of some B by necessity. For a particular negative necessary proposition does not convert. This being so, nothing is proved by reductio ad impossibile. For if animal is divided into rational and irrational and there are rational and irrational animals and someone were to assume the existence of an animal and say absolutely that it is irrational, he would say what is absurd and not true, since it is contingent that it is rational when rational is posited to be a consequence of animal no less than irrational is; so too, if someone were to assume that ‘It is not contingent that B holds of no A’ and say that it signifies (iii) only, he would say what is absurd, since it is also possible318 that (i) is true. 319 And also it seems that only when (i) holds does the contingent negative proposition not convert. For although it is contingent that white holds of no human, it is not true that it is contingent that human holds of nothing white. However, ‘It is not contingent that human holds of nothing white’ is true not because

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human holds of something white by necessity (since it wouldn’t be contingent that white holds of every human if it held of some human by necessity) but because human does not hold of something white by necessity. Therefore in the case of conversions from ‘It is not contingent that B holds of no A’ to (iii) the transformation would not be proper when the negation is not true because of (iii) but because of (i).

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37a17 For if B does not hold of some A320 by necessity, it is not true to say that it is contingent that it does not hold of all, [just as if B does hold of some A by necessity, it is not true to say that it is contingent that it holds of all. So, if someone were to maintain that, since it is not contingent that C holds of all D, it does not hold of some by necessity, he would take things falsely. For it holds of all, but we say that it is not contingent that it holds of all because it holds of certain of them by necessity. Consequently both ‘X holds of some Y by necessity’ and ‘X does not hold of some Y by necessity’ are opposite to ‘It is contingent that X holds of all Y’. And similarly in the case of ‘It is contingent that X holds of no Y’.] He says ‘It is contingent that it does not hold of all’ instead of ‘It is contingent that it holds of none’. Taking it that (i) follows from ‘It is not contingent that B holds of no A’,321 he shows how it follows. For ‘It is not contingent that B holds of no A’ is true when (i) holds and when (iii) does. For example, if (i), it is not then true that it is contingent that B holds of no A. For, as I said, he takes ‘It is contingent that it does not hold of all’ instead of ‘It is contingent that it holds of none’, which also makes what he says less clear. And if (iv) is not true when (i) is true, it is clear that the negation of (iv) which says that it is not contingent that B holds of no A is true then. He also shows that this is how things are because of the fact that again the affirmation (ii) is false not only if (i) is true but also if (iii) is. For if (iii) holds, (ii) is false, since it is not contingent that B holds of that of which it holds by necessity. And (iii) is related to (ii) in the same way as (i) is to (iv). So (iv) will not be true when (i) is,322 since it is not true that it is contingent that B does not hold of that of which it does not hold by necessity. Therefore, the negation of (iv), ‘It is not contingent that B holds of no A’, will be true when (i) is. So ‘It is not contingent that B holds of no A’ is true not just when (iii) is, but also when (i) is, since both (iii) and (i) do away with each of the universal contingent propositions (ii) and (iv). So the negation of either (ii) or (iv) is true no matter which of (i) and (iii) is.

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37a20 So, if someone were to maintain that since it is not contingent that C holds of all D, it does not hold of some by necessity, he would take things falsely. [For it holds of all,323 but we say that it is not contingent that it holds of all because it holds of certain of them by necessity. Consequently both ‘X holds of some Y by necessity’ and ‘X does not hold of some Y by necessity’ are opposite to ‘It is contingent that X holds of all Y’. And similarly in the case of ‘It is contingent that X holds of no Y’.] Taking it that (ii) does not follow from (i) but that it is clear that its negation ‘It is not contingent that B holds of all A’ does, he uses this fact to make it evident that ‘It is not contingent that B holds of all A’ is not always true because (i) is. For it has been shown that it is also true because (iii) is. Consequently the two particular necessary affirmative and negative propositions will be opposites of (iv). So, if it is hypothesized that C holds of all D and of some of D by necessity, the proposition which says that it is contingent that C holds of all D is not then true. The reason is not that C does not hold of some D by necessity – for that isn’t true – but that it holds of some by necessity. There is a reason why he takes it that C holds of all D and of some D by necessity and that consequently the proposition which says that it is contingent that C holds of all D is false; for by means of this he says that the contingent negative proposition which is taken in the conversion, namely (iv), is false because B holds of no A and does not hold of some by necessity. For just as (ii) is false if B holds of all A and of some A by necessity, so too (iv) is false if B holds of no A and does not hold of some by necessity. Not just (i) but also (iii) does away with (ii), as has been shown. But if both (i) and (iii) do away with (ii), both of them – not just (iii) but also (i) – will do away with (iv), which is equivalent to (ii) and converts with it. But since (iv) is equivalent to (ii) and it has been shown that both (i) and (iii) do away with (ii), it is clear that the same two will do away with (iv). 37a26 It is clear then that with respect to things which are contingent and not contingent in the way which we have specified324 initially [it is necessary to take ‘B does not hold of some A by necessity’ and not ‘B holds of some A by necessity’. But if this is taken, nothing impossible results, so there is no syllogism. Thus it is evident from what has been said that a privative proposition does not convert.] Having shown that the two particular necessary and affirmative or negative propositions do away with each of the universal

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affirmative and negative propositions which are contingent in the way specified , he sets down the purpose for which he proved these things. He says that in the case of the reductio ad impossibile involving the conversion of a proposition which is contingent in the way specified, it is necessary, having hypothesized ‘It is not contingent that B holds of no A’, to transform this into (i). For (iv) was not true325 because (i) was, but its negation, which says ‘It is not contingent that B holds of no A’, was true because (i) was. For if ‘It is contingent that A holds of no B’ is true then the proposition which says ‘It is contingent that B holds of no A’ can be false only because B does not hold of some A by necessity . For if it were false because B holds of some A by necessity , ‘It is contingent that A holds of no B’ could not be true. For if B holds of some A by necessity , A holds of some B by necessity because a necessary particular affirmative proposition converts. But since the negation is not transformed into this but into the particular negative necessary proposition which makes it true , nothing impossible follows because a negative particular necessary proposition does not convert. Therefore, a universal negative contingent proposition is not proved or inferred to convert by reductio ad impossibile. And at the same time from the fact that they are sometimes false together it is also clear that (iii) is not the opposite of (iv) nor is it equivalent to the negation of (iv), ‘It is not contingent that B holds of no A’; but the person who wishes to show, using (iii), that a negative contingent proposition converts transforms ‘It is not contingent that B holds of no A’ into (iii), as if they were equivalent. For, (iv) is also false when B holds of no A by necessity (because then it is not contingent that B holds of all A), and so is (iii). For it is false that it is contingent that irrationality holds of no human and also false that it holds of some by necessity. For it is not the case that if the reductio ad impossibile goes through in some cases in which the negation holds (and it does go through when (iii) holds) and does not go through in some (e.g., when (i) holds), then it is any more proved than not proved that a universal negative contingent proposition converts; rather it is not proved because it is not this way in all cases in which the negation is true. For it is necessary that what is syllogistic be the same in all cases, and a counter-example to it is sufficient, even if it is shown to hold in some case.326

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Notes 1. On this chapter see section II.B of the introduction. The most interesting part of Alexander’s commentary concerns the ekthesis arguments, and starts at 121,15. 2. We quote Aristotle’s text wherever we think it is helpful for understanding the commentary. Material added in square brackets represents part of the lemmas not in Alexander’s text. 3. A clear expression of the idea that what is contingent does not hold; see section III.A of the introduction. 4. Aristotle next argues that the same NNN and UUU combinations will be syllogistic. For an account of the argument see section II.B of the introduction. 5. Baroco2(NNN) NEC(AaB) NEC(AoC) NEC(BoC). 6. Bocardo3(NNN) NEC(AoC) NEC(BaC) NEC(AoB). 7. See sections III.A and D of the introduction. 8. Baroco2(UUU) was justified at 1.5, 27a36-b3, Bocardo3(UUU) at 1.6, 28b17-21. 9. At 29b36. 10. Reading gnôrimon tí for the gnôrimón ti of Wallies. 11. That is, the combinations of premisses with the same modality. 12. Baroco2(NNN) and Bocardo3(NNN). 13. We suspect corruption in this long sentence (121,18-26), but the sense is clear enough, and what Alexander has in mind is made perfectly clear by the proofs of Baroco2(NNN) and Bocardo3(NNN) which follow. 14. i.e. the part D of C of which NEC(AeD). 15. We give a representation of Alexander’s account of Aristotle’s proof of: Baroco2(NNN) NEC(AaB) NEC(AoC) NEC(BoC) Let D be a part of C such that NEC(AeD). Then (EE-conversionn) NEC(DeA). Hence (Celarent1(NNN)) NEC(DeB) and (EE-conversionn) NEC(BeD). But D is part of C, so that NEC(BoC). 16. Alexander gives the argument for: Bocardo3(NNN) NEC(AoC) NEC(BaC) NEC(AoB). Again he takes a part D of C such that NEC(AeD). But since D is a part of C and NEC(BaC), NEC(BaD) and (AI-conversionn) NEC(DiB). Hence (Ferio1(NNN)) NEC(AoB). 17. Wallies prints: eilêphthô palin ti tou G, hôi merei autou mê huparkhon to A ex anankês ekeito tini tôi G ex anankês mê huparkhein, kai estô to D. We have translated eilêphthô palin ti tou G, hôi merei autou mê huparkhon to A ex anankês – ekeito tini tôi G ex anankês mê huparkhein – kai estô to D 18. In chapter 6, 28a24-6 and 28b14-15, 20-1. What Alexander goes on to say can be explained by reference to the first of these passages in which Aristotle offers a proof by what he calls ekthesis for: Darapti3(UUU) PaS RaS PiR In it he takes an S’ (some of S) and asserts that, since both P and R hold of S’,

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P holds of some R. Alexander does not want this to be understood as a matter of reducing one case of Darapti3 to another, viz., PaS’ RaS’ PiR and so he claims that in this case S’ is not a subclass of S but an individual, consideration of which makes evident the coincidence of P and R with respect to something. He insists that Aristotle is not arguing about S’, i.e., performing logical operations on propositions involving S’. (If one takes S’ as an individual S and formalizes syllogistic using quantification theory, one will be arguing that S’ is P and S’ is R, so it is both P and R [conjunction introduction], so something is both P and R [existential quantifier introduction].) See also in An. Pr. 99,16-100,26 and 32,32-34,2 with the notes in Barnes et al. 19. ‘Just what some of the original term is’ is our very loose paraphrase of Aristotle’s terse hoper ekeino ti. Alexander explains this as a part or species of the original term. Alexander’s prose is also terse, and we have inserted letters from the proof of Bocardo3(NNN) to indicate his meaning. 20. See the Greek-English index under rhêthêsetai. 21. Alexander’s dense explanation of his (minor) point applies most directly to Bocardo3(NNN). For the ekthesis, he says, Aristotle must take a part D of C such that NEC(AeD). For then, given NEC(BaC), he can be sure that NEC(BaD). However, if he had taken a part D’ of C of which NEC(BaD’), he would have no way of knowing whether or not NEC(AoD’), even though NEC(AoC). The extension of the point to Baroco2(NNN) presupposes only the equivalence of NEC(AeD) and NEC(DeA). 22. At 1.6, 28b20-1 in connection with Bocardo3(UUU). 23. Alexander’s point is that the ekthesis arguments involve transforming the second-figure Baroco2(NNN) into the second-figure Camestres2(NNN) and the third-figure Bocardo3(NNN) into the third-figure Felapton3(NNN). 24. This paragraph is Theophrastus 104 FHSG. See section II.B of the introduction. 25. On Alexander’s commentary on this chapter see section II.C of the introduction. The most interesting part is the beginning discussion of whether a necessary conclusion can follow from a necessary and an unqualified premiss. 26. Aristotle asserts the validity (completeness) of Barbara1(NUN) and Celarent1(NUN). 27. Alexander may have read ‘conclusion’ here; see the note on 128,7. 28. For a textual point see the note on 127,25. 29. i.e. 1. On the term ‘indemonstrable’ – which only occurs here in Alexander’s discussion of modal syllogistic, see Barnes et al., p. 21. 30. hoi de ge hetairoi autou hoi peri Eudêmon te kai Theophraston. What follows is Theophrastus 106A FHSG and Eudemus fr. 11a Wehrli. The rest of the section is concerned with this dispute about the validity of Barbara1(NUN) and Celarent1(NUN). The same dispute arises for Darii1(NUN) and Ferio1(NUN), but they are not given an interesting separate consideration. For discussion of the topic see section II.C of the introduction. Aristotle accepts the combinations as syllogistic. Theophrastus and Eudemus reject them. Alexander focuses on Barbara1. From his account of the discussion, Theophrastus and Eudemus would seem to triumph. There is additional material on this subject at 129,26-130,24, 132,23-34, and 140,14-141,6. See also Philoponus, in An. Pr. 123,12-126,29. 31. This law, according to which the conclusion is as weak as the weakest premiss in terms of its modality (where necessity is stronger than unqualifiedness, which in turn is stronger than contingency), quality (where affirmative is

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stronger than negative), and quantity (where universality is stronger than particularity), is sometimes called the peiorem rule. Note that it may be taken negatively as a way of ruling out claims to validity or positively as a way of deciding validity for modal versions of unqualified syllogisms. For an application of the rule by Theophrastus and Eudemus see 173,32-174,3. 32. Alexander uses the terminology of being disjoined (apozeugnunai) only in passages in which he is describing or discussing the views of Theophrastus (and Eudemus). (At 132,23-34 he uses the terminology of separation (khôrizein) in a similar context.) Two of the passages concern their explanation of EE-conversionu (31,4-10; 34,14-16), and a third their assertion that EE-conversion holds for contingent propositions (220,9-23). On the basis of these passages it seems clear that ‘A is disjoined from B’ is tantamount to ‘AeB’. We have not found a fully satisfactory reconstruction of the argument sketched by Alexander in the present passage, but it looks to us to start by assuming the premisses of Barbara1(NU_), NEC(AaB) and BaC, and adding the assumption (legitimate for the purpose of argument) that  NEC(BaC); (i) it then somehow moves from that to ‘It is contingent that BeC’, and then (legitimately) to the claim that B and C will be totally disjoint at some time, from which it infers (ii) that A and C will be disjoint at that time, so that  NEC(AaC). There seems to be no way to make (ii) a sound inference; from the fact that AaB and BeC it does not follow that AeC (or even that AoC). Perhaps we should understand the future ‘A will also be disjoined’ as ‘A might also be disjoined’, but justifying the claim that A might be disjoined from C would seem to require some kind of argument, the simplest of which is the specification of terms. The best we can do with (i) is to supply the interpretation CON(BaC) for ‘B holds of all C but not by necessity’ and then use Aristotelian AE-transformationc to get CON(BeC). Philoponus (in An Pr. 124,9-24) gives a more long-winded but not more clear version of this argument. 33. Philoponus’ examples (in An Pr. 124,24-8) are moving, walking, human, and virtue, practical wisdom, human. 34. Other passages which suggest support for Theophrastus and Eudemus on this issue are 129,26-130,24, 132,23-34. But note that Alexander describes Aristotle’s position as reasonable at 129,18-20. 35. Wallies marks a lacuna in the text and suggests supplying ‘It is strange’ (atopon). In a parallel passage [Ammonius] (in An. Pr. 39,31-40,2) ascribes to Herminus the view which Alexander describes in this paragraph, according to which Aristotle means to say that some instances of a single mood are valid and some are not. Alexander is right to reject this line of interpretation, but unfortunately he continues to be unclear about the difference between the false claim that ‘Every human is a thing that laughs’ follows from ‘Every human is an animal’ and ‘Everything that laughs is an animal’ and the true one that one gets true sentences when one takes A as animal, B as human, and C as thing that laughs in the schema: AaC AaB CaB. Alexander will say that the schema is not syllogistic even though in certain instances the premisses imply the conclusion. We might say that the premisses don’t imply the conclusion because the schema is not syllogistic. See part I of the introduction. On Herminus, who is said to have been a teacher of Alexander, see Moraux (1984), pp. 361-98. 36. Wallies’ text reads eti de ei touto ebouleto dêloun, hôs edeixen, edeixen an, eph’ hês hulês touto houtôs ekhei: ‘if Aristotle wanted to make this point, as he

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proved, he would have proved ’. Since we do not know what to make of ‘as he proved’ we suggest that edeixen was written twice and the dittography rectified by the addition of hôs. 37. On this lost work see Sharples (1987), p. 1196. Alexander refers to it again at 127,15-16, 207,35-6, 213,25-7, 238,37-8, 250,1-2; cf. 188,16-17, and 191,17-18. These passages suggest that Alexander deliberately refrains from expressing views on certain controversial subjects, possibly on the grounds that this is not the task of a commentator, i.e, a person who is expounding a text to students. 38. Alexander now considers Barbara1(NUN). 39. Alexander here quotes his paraphrase of 1.1, 24b29-30. See 24,27-30. 40. 1.8, 30a2-3. 41. The issue raised here concerns the proper interpretation of an unqualified formula. The counter interpretation to Barbara1(NUN) offered by Theophrastus at 124,24-5 takes ‘Everything moving is a human’ to be true, but it is presumably false; the defenders of Aristotle call this kind of proposition a postulate (aitêma) because its use depends on a concession that it might be true at some time that, e.g. the only moving things are humans. However, Aristotle himself sometimes takes ‘Every animal is moving’ to be true (e.g. at 30a28-32). At 126,17-22 Alexander points out that this line of defense will not work for Aristotle’s claim at 30a28-30b2 that Darii1(NUN) and Ferio1(NUN) are valid, since the non-necessary premiss is particular. At 130,24-5 he says that Aristotle (at least at 30a28-32) treats an unqualified universal proposition as a hypothesis. 42. en tôi huparkhein. 43. Alexander finds this view to be expressed or at least to underlie what Aristotle says in 1.41, 49b14-32. According to Alexander, although ‘AaB’ is equivalent to ‘A holds of all of that of all of which B holds’, it is not equivalent to ‘A holds of all of that of which B holds’, which Alexander (mistakenly) thinks is equivalent to ‘AiB’. At 379,9-11 Alexander reports Theophrastus as saying in On Affirmation that ‘A holds of what B does’ is equivalent to ‘A holds of all of that of all of which B holds’. On this question see also 1.13, 32b29-37 with Alexander’s discussion at 165,25-166,25. (We thank Robin Smith for help with this note.) 44. The argument here seems to concern Barbara1(NUN) primarily. It takes AaB to mean ‘If BaC then AaC’, and apparently infers that NEC(AaB) ought to mean ‘If BaC, then NEC(AaC)’. If the inference were correct Barbara1(NUN) would be justified, but one needs an explanation why NEC(AaB) should not be rendered ‘NEC(if BaC then AaC)’. 45. The remainder of this section is Theophrastus 106C FHSG. In the first paragraph Alexander considers an alleged reductio argument for: Barbara1(NUN) NEC(AaB) BaC NEC(AaC) Assume  NEC(AaC), ‘i.e.’, CON  (AaC), i.e., CON(AoC). But NEC(AaB); so (Baroco2(N‘C’‘C’)) ‘CON’(BoC), but this, it is alleged, conflicts with BaC, so that NEC(AaC). This reductio is not valid since ‘CON’(BoC) and BaC are compatible. Alexander says nothing about the quality of this argument. Perhaps his words ‘try to show’ are a sign that he is sceptical. In any case he follows it with a devastating argument against Theophrastus. That argument is repeated by Philoponus (in An. Pr. 123,26-124,1) At 1.19, 38b27-9 Aristotle rejects Baroco2(NC_). Alexander gives an indirect reduction of Baroco2(NCCt) to Barbara1(NNN) at 240,32-241,1. Alexander’s invocation of Baroco2(NC_) is the first of many examples of what

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we will call a circle argument, an argument which uses a syllogism not yet established to argue for a syllogism under consideration. Although this procedure does not establish the syllogism under consideration, it can be used to confirm the coherence of a version of syllogistic. 46. Alexander gives a reductio argument for Barbara1(NUN). Assume  NEC(AaC), ‘i.e.’, CON (AaC), i.e., CON(AoC). But BaC. Therefore (Bocardo3(CUC)), CON(AoB), contradicting NEC(AaB). Aristotle justifies Bocardo3(CU‘C’) by a problematic reductio to Barbara1(NUN) in chapter 21 at 39b31-9. See Alexander’s commentary on that passage 247,9-248,30. [Ammonius] (in An. Pr. 39,10-15) ascribes this argument to Alexander. See Appendix 3 on conditional necessity. Philoponus (in An. Pr. 124,30-125,18) reports as a response to this defence of Aristotle the point that the same kind of argument could be used to establish Barbara1(NUU) and Barbara1(NUCt). If Aristotle were clear about the different kinds of contingency, neither derivation need have bothered him, since NEC(P) implies both P and  NEC( P). 47. Theophrastus and Eudemus. The claim that Theophrastus accepted some version of Bocardo3(CUC) (presumably Bocardo3(CtUCt)) is problematic precisely because it is equivalent to Barbara1(NUN), which Theophrastus rejected. But the claim is repeated at 248,19-30. In a parallel passage (in An. Pr. 124,1-4) Philoponus speaks of ‘Aristotle and everyone else’ accepting Bocardo3(CUC). 48. At 125,30-1. 49. Barbara1(NUN) and Celarent1(NUN). 50. The lemma reads huparkhein ê mê huparkhein to A keitai where Ross prints huparkhei ê oukh huparkhei to A. 51. Aristotle accepts Barbara1(UNU), but insists that this pair of premisses does not yield a necessary conclusion, referring to arguments using the first or third figure. He offers what we have called an incompatibility rejection argument. See section II.C of the introduction. Alexander’s fullest discussion of the general form of these arguments starts at 131,8. Alexander takes the third-figure argument referred to by Aristotle to use Darapti3(NNN), the first-figure argument to involve the step of reduction of Darapti3(NNN) to Darii1(NNN). We combine the two arguments as follows: Assume that AaB,  NEC(AiB), and NEC(BaC) – these three propositions are compatible. If Barbara1(UNN) were valid, we could infer NEC(AaC), which with NEC(BaC) would imply (Darapti3(NNN)) NEC(AiB), which contradicts  NEC(AiB). (Alexander places stress on the consistency of AaB and  NEC(AiB).) We might then interpret Aristotle as arguing that there are interpretations which make AaB and NEC(BaC) true and something implied by NEC(AaC) and NEC(BaC) false, hence an interpretation which makes AaB and NEC(BaC) true and NEC(AaC) false. The terms Aristotle uses will provide such an interpretation if we assume that it is not necessary that some animal be in motion although in fact all are. For then all of: Motion a Animal  NEC(Motion i Animal ) NEC(Animal a Human) would be true. But NEC(Motion a Human) cannot be true in this circumstance because, if it were, so would NEC(Motion i Animal) be true, contradicting:

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 NEC(Motion i Animal) At 128,31ff. Alexander points out that no similar refutation can be given for Aristotle’s Barbara1(NUN). However, he does not point out that once the complete Darii1(NUN) is available this same method could be used to show that Barbara1(UNU) is not valid. Indeed, at 133,19-29 Alexander implies that this method couldn’t be used against Barbara1(UNU). Alexander discusses the method used here again in connection with the next chapter at 138,5-139,11. 52. endekhetai. For Alexander’s text of this line see 128,29 with note and the next unit of the commentary. We take it that endekhetai is here used informally rather than as a term of logic and so translate informally in the next stretch of text. 53. At the beginning of the chapter, 30a15-16. For the reason Alexander, who substitutes the word ‘conclusion’ for the ‘syllogism’ of our text of Aristotle, makes this point see 125,3-29. 54. Alexander’s text of this sentence reads endekhetai de toiouton einai to B, hôi enkhôrei to A mêdeni huparkhein whereas our manuscripts read endekhetai gar toiouton einai to B, hôi enkhôrei to A mêdeni huparkhein. See also the lemma at 129,8 (which reads endekhetai de toiouton ti einai) with Alexander’s discussion. 55. Alexander points out that Aristotle is not claiming that  NEC(AiB) is implied by AaB, only that the two are compatible. 56. Wallies prints all’ epei mê anankaion esti to panti huparkhon euthus en tôi panti huparkhein kai to ex anankês tini autôi (autôi a; au  M) huparkhein periekhein (periekhei aM). We tentatively propose all’ epei mê anankaion esti to panti huparkhon euthus en tôi panti huparkhein kai to ex anankês tini autou huparkhein periekhetai. 57. See the note on 128,29-30. 58. cf. Denniston (1954), p. 169: ‘de is not infrequently used where the context admits, or even appears to demand, gar.  The Scholia often observe: ho de anti tou gar.’ 59. Alexander considers the possibility that the kind of argument Aristotle has given against Barbara1(UNN) might be applied against the validity of: Barbara1(NUN) NEC(AaB) BaC NEC(AaC) He points out that NEC(AaC) and BaC will yield either NEC(AiB) or AiB, depending on whether or not one accepts Darii1(NUN), neither of which is incompatible with NEC(AaB). He might also have mentioned that NEC(AaB) and NEC(AaC) yield nothing at all. Such an argument, which we have called an incompatibility acceptance argument, does not, of course, establish the validity of Barbara1(NUN); at most it establishes that its conclusion is compatible with its premisses. 60. Contrast 124,31-2, where the Theophrastean arguments are called ‘reasonable’ – or at least apparently reasonable. 61. Aristotle points out that Barbara1(UNN) is refuted because all humans are animals by necessity and it might be true that all animals were in motion, so that all humans were in motion; nevertheless it would not be true that all humans are in motion by necessity. Alexander recurs to the Theophrastean counterexample from 124,24 to Barbara1(NUN): even if humans were the only things in motion at some time, it would not be true that animals are the only things in motion by necessity. He then gives what looks to be a compelling justification of the Theophrastean position, but he makes no attempt to corre-

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late what he says with Aristotle’s acceptance of the first-figure NUN cases. This next section is discussed by Flannery (1995), pp. 86-92. 62. But if Barbara1(NUN) were valid and ‘animal holds of all human’ is taken as necessary, it would follow that animal holds of all that moves by necessity. Perhaps something has dropped out of the text at this point. 63. cf. 126,4. 64. Alexander here substitutes Y being under (hupo) X for what was expressed as X being said of Y in the previous sentence. He goes on to distinguish between being under Y in the sense of being a part of the essence of X and being under X in the general sense of being an X. If we take being under (or something of) X in the general way – as we should – Barbara1(NUN) will not be valid. 65. Following the suggestion of an anonymous reader, we retain the manuscript hê where Wallies emends to ên and take the words ‘is the following’ to be understood. 66. In setting down terms for Barbara1(UN_) in the passage under discussion. Alexander presumably means that a proposition such as ‘All animals are moving’ is not really true. On the notion that a universal unqualified proposition is a hypothesis see 126,9-22. 67. Aristotle rejects: Celarent1(UNN) AeB NEC(BaC) NEC(AeC) Alexander gives his apparent reasons. Again he offers a first- and a third-figure argument, which we combine into the latter and expand. Suppose NEC(AeC). But NEC(BaC). Therefore (Felapton3(NNN)), NEC(AoB). Alexander asserts that NEC(AoB) is false. Again, what he means is that AeB and  NEC(AoB) can be true together, but the latter is incompatible with NEC(AoB). See 128,1ff. with the note on 128,1. We observe that a similar argument cannot be given against Celarent1(UNU). 68. We adopt Wallies’ conjecture hupothêsometha for the ekthêsometha which he prints. 69. Ferio1(NNN). 70. Felapton3(NNN). 71. Alexander points out that the proofs that the conclusions of Barbara1(UN_) and Celarent1(UN_) are not necessary are not the standard kind of reductio used to establish that a conclusion follows from a pair of premisses by showing that the contradictory of the conclusion and one of the premisses are incompatible with the other premiss. 72. Alexander makes the correct point that, e.g. in the rejection of Barbara1(UNN) NEC(AaC) is not incompatible with the premisses, but incompatible with them plus something which is compatible with them, namely CON(AeB). 73. At 30a27: ‘But this is false.’ Unfortunately this is not a good way of putting the matter, and Alexander takes it over. Aristotle should say something like ‘But there are concrete terms which make this false and the other propositions true.’ 74. What lies behind this remark is the idea that an impossibility cannot follow from a possibility, which Aristotle argues for at 1.15, 34a5-33 and uses in his justification of Barbara1(UC‘C’) at 34a34-35b17; see Alexander’s lengthy discussion starting at 175,22. 75. Another version of the point made at the beginning of the paragraph. 76. i.e. AaB in the case of Barbara1(UNU). 77. This remark is rather surprising. It perhaps reflects Alexander’s correct

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sense that Aristotle’s method is very problematic. See section II.C of the introduction. 78. of Barbara1(UN_) and Celarent1(UN_). 79. Alexander here introduces another way of showing that Barbara1(UNU) and Celarent1(UNU) are syllogistic, but Barbara1(UNN) and Celarent1(UNN) are not. Attempt to do a reductio on the premisses and the denial of the conclusion for each alternative, and show that a contradiction follows when the unqualified conclusion is denied, but not when the necessary one is. The method presupposes the validity of modal syllogisms which Aristotle has not yet discussed; cf. 134,17-20. It is possible that Alexander introduces the method as one way of dealing with the obscure 30b2-4; see 133,29-134,20. Alexander uses the method again in his discussion of the next chapter at 139,12-26. 80. We propose khrômenou for the khrômenois printed by Wallies. The Aldine has khrômenôn. 81. Alexander applies his method to justify: Barbara1(UNU) AaB NEC(BaC) AaC He assumes  (AaC), i.e. AoC, and then applies Bocardo3(UNU) (briefly treated by Aristotle at 1.11, 31b40-32a1) to get AoB, contradicting AaB. He then argues that no similar contradiction can be produced for: Barbara1(UNN) AaB NEC(BaC) NEC(AaC) For if one assumes  NEC(AaC), ‘i.e.’, CON(AoC), Bocardo3(CN‘C’) yields only ‘CON’(AoB), which is quite compatible with AaB. Aristotle’s treatment of Bocardo3(CN_) at 1.22, 40a40-b3 is not crystal clear. See 252,3ff. with the note on 252,3. One might think of trying to apply Baroco2(UC_), but Aristotle rejects this combination at 1.18, 37b39-38a2. We note that this argument may have been unproblematic for Theophrastus, who – we think – accepted Bocardo3(CtNCt) and should have rejected Baroco2(UCtCt). 82. Alexander asserts that his method works for accepting: Celarent1(UNU) AeB NEC(BaC) AeC and rejecting: Celarent1(UNN) AeB NEC(BaC) NEC(AeC) The argument for the first involves assuming  (AeC), i.e., AiC, and applying Disamis3(UNU) to get AiB, contradicting AeB. For the second, if one assumes  NEC(AeC), ‘i.e.’, CON(AiC) and applies Disamis3(CN‘C’) there is no contradiction. But if one applies Festino2(UC‘C’), one gets ‘CON’(BoC), which does contradict NEC(BaC). So Alexander is incorrect here, and Aristotle’s modal logic once again shows its incoherence. Again, this argument would seem to be all right for Theophrastus, who – we believe – would have accepted Disamis3(CtCtCt) and should have rejected Festino2(UCtCt). 83. This last paragraph is Theophrastus 106B FHSG. 84. i.e. Barbara1(UN_). 85. Viz., that Barbara1(NUN) is syllogistic. The text, as it stands, is opaque here, but that may reflect Alexander’s excerpting. 86. Aristotle asserts the completeness of Darii1(NUN) and Ferio1(NUN). 87. Aristotle asserts that an incompatibility rejection argument will work against Darii1(UNN) and Ferio1(UNN). He then gives terms for rejecting them. Those terms take as true: Motion a Animal Motion e Animal NEC(Animal i White)  NEC(Motion i White)  NEC(Motion o White)

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Alexander understands NEC(Animal i White) to be true because, e.g., swans are white by necessity, a clear de re reading of NEC. 88. The punctuation adopted here corresponds to Alexander’s understanding of the text; see 133,17-19. Ross puts ‘for nothing impossible results’ in parentheses to make ‘just as in the universal syllogisms’ attach to ‘the conclusion will not be necessary’. 89. hêgeitai. Presumably an expression of Alexander’s misgivings about Aristotle’s position; cf. 42,29, 127,29, 235,21 and 239,12. For similar uses of dokei see 37,15, 153,10, 236,33, and 242,12. 90. Alexander offers three interpretations of Aristotle’s brief remarks at 30b2-5, the first at 133,19-29, the second at 133,29-134,20, the third at 134,2131. 91. Alexander refers to Aristotle’s rejections of Barbara1(UNN) at 30a23-8 and Celarent1(UNN) at 30a32-3. Here he implies falsely that Aristotle (or Alexander himself) showed that a similar argument couldn’t be given against Barbara1(UNU) AaB NEC(BaC) AaC He sketches an argument, assuming all three propositions and pointing out that the last two imply (Darapti3(UNU)) AiB, which is implied by AaB. Unfortunately, Aristotle accepts Darapti3(UNN), which yields NEC(AiB), and  NEC(AiB) ought to be compatible with AaB. Again we have a situation in which Alexander gives an argument which works for Theophrastus but not for Aristotle. At 134,32-135,6 and 135,12-19 Alexander shows that on this (probably correct) understanding of Aristotle’s words what he says about Darii1 and Ferio1 is false. 92. cf. 128,31-129,7. 93. Alexander’s second interpretation involves the method he himself introduced at 132,5. He offers an indirect circle justification for: Darii1(UNU) AaB NEC(BiC) AiC He assumes  (AiC), i.e., AeC, and uses Ferison3(UNU) to infer AoB, contradicting AaB. He argues that no such justification can be given for: Darii1(UNN) AaB NEC(BiC) NEC(AiC) because if one assumes  NEC(AiC), ‘i.e.’, CON(AeC), Ferison3(CNC) would yield CON(AoB), which is compatible with AaB. (There is no possibility of applying Camestres2(UC_) because Aristotle rejects it at 1.18, 37b19-23 and Theophrastus presumably rejected it as well.) Unfortunately an argument of the same kind can be given to justify: Ferio1(UNN) AeB NEC(BiC) NEC(AoC) For if  NEC(AoC), ‘i.e.’, CON(AaC), then (Cesare2(UC‘C’)) ‘CON’(BeC), contradicting NEC(BiC). The method also confirms: Ferio1(NUN) NEC(AeB) BiC NEC(AoC) For if  NEC(AoC), ‘i.e.’, CON(AaC), then (Datisi3(CUC)) CON(AiB), contradicting NEC(AeB). Finally we mention that the method also confirms: Darii1(NUN) NEC(AaB) BiC NEC(AiC) For if  NEC(AiC), ‘i.e.’, CON(AeC), then (Ferison3(CUC)) CON(AoB), contradicting NEC(AaB). None of these arguments for necessary conclusions will work in what we take to be the Theophrastean system. 94. The words in parentheses may be an incorrect gloss. Alexander is now talking about his method of assuming the contradictory of a purported conclusion and trying to derive a contradiction. Nothing was said in his application of that method about whether something ‘false’ was generated. That issue is relevant to the method employed by Aristotle at 30a25-8.

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95. At 132,5-23. 96. Alexander’s third suggestion is that the words ‘nothing impossible results’ somehow apply to the terms which Aristotle supplies to show the invalidity of Darii1(UNN) and Ferio1(UNN). This makes the most logical sense of what Aristotle says (cf. Smith (1989) ad 30b4), but Aristotle’s text suggests that he sees the specification of terms as an alternative way of showing invalidity. Alexander seems quite uncertain here. 97. Alexander points out that the method applied by Aristotle at 30a23-8 to reject Barbara1(UNN) will not work for: Darii1(UNN) AaB NEC(BiC) NEC(AiC) since nothing follows from NEC(AiC) and either of the other two premisses. He indicates that the same thing is true of Ferio1(UNN) at 135,12-19. 98. Alexander now argues that if no ‘difficulty’ can be produced from the assertions making up Barbara1(UNU) (or Celarent1(UNU)) none can be produced from the weaker assertions making up Darii1(UNU) or Ferio1(UNU). The argument is correct, but the assumption that no difficulty can be derived for Barbara1(UNU) is wrong. See the note on 133,20. Moreover, showing that no difficulty follows from, e.g. Darii1(UNU) does not show that Darii1(UNN) is not valid. 99. Alexander seems to point out that an incompatibility rejection argument produces no difficulty when applied to any of Darii1(UNU), Darii1(UNN), Ferio1(UNU) AeB NEC(BiC) AoC and: Ferio1(UNN) AeB NEC(BiC) NEC(AoC) He has given the argument for Darii1(UNN) at 134,32-135,6, and now sketches the argument for the Ferio1 cases: the conclusion and the minor premiss (or the minor premiss converted) are both particular, and the conclusion and the major premiss are both negative. 100. Reading touto kai for Wallies’ touto, kai. 101. i.e. Ferio1(UNN). 102. Taking off from the valid UUU cases with universal premisses, Aristotle asserts that Cesare2(NU_) and Camestres2(UN_) yield a necessary conclusion, whereas Cesare2(UN_) and Camestres2(NU_) yield only an unqualified one. He takes up the combinations with one particular premiss at 31a1. What he says in the present passage will carry over to Festino2(NU_) and Festino2(UN_), but not to Baroco2(UN_) and Baroco2(NU_), which Aristotle will argue at 31a10-17 both yield only unqualified conclusions. For a minor divergence between the text of the lemma and our text of Aristotle see Appendix 6 (the textual note on 30b7). 103. Baroco2. 104. 31a1. For a minor divergence between this citation and our text of Aristotle see Appendix 6 (the textual note on this line). 105. i.e. all but Baroco2. What Alexander says here can be understood by looking at the summary of assertoric syllogistic and seeing that there are no alternative procedures for directly reducing a second-figure combination to a first-figure one, and remembering that Aristotle accepts the first-figure NUN and UNU combinations. 106. Aristotle reduces Cesare2(NUN) to Celarent1(NUN). 107. Aristotle says to A tôi B mêdeni endekhesthô. Alexander points out that this means the same thing as, e.g. to A tôi B mêdeni ex anankês huparkhetô. Here and elsewhere we render such phrases as ‘It is not contingent that A holds of any B’.

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108. Alexander means the dictum de omni et nullo. 109. Aristotle reduces Camestres2(UNN) to Celarent1(NUN). 110. Cesare2 and Camestres2. 111. We would have thought that Aristotle means only that the conclusion converts in the same way as the minor premiss. 112. Aristotle now asserts that Camestres2(NU_) and Cesare2(UN_) yield only unqualified conclusions. He discusses only Camestres2. Alexander takes up Cesare2 at 141,17. Aristotle first points out that the straightforward reduction of Camestres2(NU_) is to Celarent1(UNU); he then argues that the assumption that Camestres2(NUN) holds leads to absurdity, and finally gives problematic terms for rejecting it. For minor deviations between the lemma and our text of Aristotle see Appendix 6 (the textual notes on 30b18 and 30b19). 113. Aristotle applies the incompatibility rejection method he introduced in the previous chapter at 30a23-8. See the note on 128,1, and section II.C of the introduction. We may express his argument as follows. Assume, as is possible, that NEC(AaB), AeC and  NEC(CoA) (i.e.,  NEC (CaA), ‘i.e.’, CON(CaA)). Then, if Camestres2(NUN) held NEC(BeC), so that NEC(CeB); but also, since NEC(AaB), (AI-conversion) NEC(BiA), and (Ferio1(NNN)) NEC(CoA), contradicting  NEC(CoA). At 138,30 Alexander gives as an example of  NEC(CoA) and AeC, ‘ NEC(Moving o Animal)’ and ‘Animal e Moving’, i.e., ‘It is not necessary that some animal is not moving’ and ‘Nothing moving is, in fact, an animal’. For a minor deviation between the lemma and our text of Aristotle see Appendix 6 (the textual note on 30b24-5). 114. At 1.9, 30a23-8. 115. The first of these two conditions is irrelevant to the argument. 116. Alexander takes these terms from 1.9, 30a29-30. Alexander goes on to argue that if the minor premiss AeC of Camestres2(NU_) is interpreted as ‘No moving things are animals’, it could be true even though ‘It is contingent that all moving things are animals’ and ‘It is contingent that all animals are moving things’ might both be true. 117. Alexander’s use of various terms for expressing possibility is somewhat loose in this paragraph. We have used ‘contingent’ whenever he uses a form of endekhesthai, but all he really means in the present sentence is that it is possible for AeC and CON(AaC) to both be true. 118. Alexander is here worried about someone saying that the outer heavens move by necessity, so that there is no possibility that everything moving be an animal. See 199,1-4. 119. On Alexander’s method here see 132,5-23 and section II.C of the introduction. Alexander gives a circle argument that Camestres2(NUU) NEC(AaB) AeC BeC is valid because the combination of NEC(AaB), and the denial, BiC, of its conclusion implies (Darii1(NUN)) NEC(AiC), which is incompatible with AeC. On the other hand, he says, Camestres2(NUN) NEC(AaB) AeC NEC(BeC) is not valid. He assumes  NEC(BeC), ‘i.e.’, CON(BiC) and uses Darii1(NC‘C’) to infer ‘CON’(AiC), which is perfectly compatible with AeC. However, he does not consider using Ferison3(UC‘C’) to infer ‘CON’(AoB), which is incompatible with NEC(AaB). 120. This brief passage causes some difficulty. Aristotle wishes to give terms for rejecting Camestres2(NUN). For his terms to work the following have to be true: All humans are animals by necessity,

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No white things are animals, It is not necessary that no white things are human. The problem with Aristotle’s choice of terms is that they make the second premiss false if all swans are necessarily white, as Alexander suggests at 141,8-9. Alexander goes on to suggest substituting moving for white and also supplies terms which yield: It is necessary that what walks moves, No humans are moving, It is not necessary that no humans are walking. The third proposition is true if necessity is taken ‘simply’, but the proposition ‘No humans are walking’ must be true if the premisses are; in Alexander’s phrase (or Theophrastus’; see 36,25-8) the ‘No humans are walking’ is necessary ‘on a condition’ (meta diorismou). Alexander takes Aristotle to express this point when he says ‘the conclusion will be necessary if certain things hold, but it will not be necessary simply’. He takes the opportunity to point out that Aristotle cannot be talking about necessity on a condition when he takes a conclusion to be a necessary proposition. 121. As Alexander has done at 138,30ff. and will do again at 141,8ff. 122. On the next two paragraphs see Appendix 3 on conditional necessity. 123. The text of this sentence is corrupt and heavily emended by Wallies. Alexander is apparently saying that Aristotle’s denial that the conclusions of Cesare2(UN_) and Camestres2(NU_) are necessary shows that he does not mean ‘necessary on a condition’ when he says a conclusion is necessary. 124. Barbara1(NUN). 125. We read to mê haplôs auto anankaion ginesthai alla [to] meta diorismou, which we take to be a paraphrase of toutôn men ontôn anankaion estai to sumperasma, haplôs d’ ouk anankaion (30b38-40; cf. 32-3) 126. i.e. Theophrastus and Eudemus. 140,14-18 and 141,1-6 constitute Theophrastus 100D FHSG, 141,1-6 is Eudemus fragment 12 Wehrli. 127. 9.19a23-4. 128. In the remainder of this section Alexander considers: Cesare2(UN_) AeB NEC(AaC) in a quite confusing way. He first suggests that there is no incompatibility rejection argument against the conclusion being NEC(BeC) (141,17-27), and then gives two (141,27-142,8 and 142,8-14). He concludes by providing terms for the rejection (142,14-17). In this first paragraph Alexander says that no incompatibility rejection argument would ‘seem’ possible because either we will get two negative premisses or the non-syllogistic combination AE_3(NN_). What he doesn’t point out is that Felapton3(NNN) would give NEC(BoA), but  NEC(BoA) is compatible with AeB. This is the point he makes in the next paragraph, although he carries out the reduction to Ferio1(NNN). 129. Here and in the next parenthetical remark Alexander refers back to the rejection of Camestres2(NUN) at 30b24-31. 130. Alexander’s formulation is again rather unfortunate. His point is that ‘It is contingent that CaA’ is compatible with the minor premiss AeC of Camestres2(NUN), but NEC(CoA), which could be false when AeC is true, was derived from the major premiss and conclusion of Camestres2(NUN) 131. At 138,30-139,11. 132. This sentence expresses the idea that BeA and ‘CON’(BaA) are compatible. 133. Alexander gives an alternative incompatibility rejection argument for

Notes to pp. 77-79

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Cesare2(UNN), converting the purported conclusion to NEC(CeB) and the minor premiss to NEC(CiA) and using Festino2(NNN) to infer again NEC(BoA). 134. Aristotle asserts the validity of Festino2(NUN) and denies that of Baroco2(NUN) and Baroco2(UNN). Alexander appears to take him not to be considering Baroco2(UNN) and also to be denying the validity of Festino2(UNN). 135. Aristotle establishes Festino2(NUN) by reduction to Ferio1(NUN). 136. Aristotle asserts the invalidity of Baroco2(NUN) and then of Baroco2(UNN). For a minor deviation between the lemma and our text of Aristotle see Appendix 6 (the textual note on 31a10). 137. Alexander discusses a textual issue here starting at 144,4. 138. Alexander first points out that Aristotle does not explain why Festino2(UNN) is invalid. He points out (in the obscurely formulated second sentence) that if one tries to follow Aristotle’s procedure for Festino2(NU_) and convert the major premiss of Festino2(UN_), one gets a reduction to Ferio1(UN_) and hence no necessary conclusion. He then gives terms to show the invalidity of Festino2(UNN), taking as true: Nothing moving is an animal, Some white thing is an animal by necessity, It is not necessary that something white is not moving. He does not mention that an incompatibility rejection argument will not work in this case or that his own method could be used to justify: Festino2(UNN) AeB NEC(AiC) NEC(BoC) For if  NEC(BoC), then (Celarent1(UC N ))  NEC (AeC), i.e.,  NEC(AiC), contradicting NEC(AiC). 139. 1.5, 27a36-b1. 140. 1.8, 30a3-14. 141. At 30b33-4. In the passage now under consideration Aristotle apparently takes the following propositions to be true: (i) All humans are animals by necessity; (ii) Some white things are not animals; (iii) It is not necessary that some white thing is not human. Alexander thinks that (ii) is a necessary rather than an unqualified truth, since, for example, snow is necessarily not an animal, and that (iii) is false since snow is necessarily not human. He therefore substitutes his own terms for Aristotle’s: (i) All that walk move by necessity; (ii) Some human is not moving; (iii) It is not necessary that some human is not walking. 142. See 141,6-16. 143. Before discussing Aristotle’s obscure remark about Baroco2(UNN) Alexander gives his own terms to show its invalidity. They would make the following sentences true: A (i) Everything literate is awake; (ii) Some human (a sleeping one) is necessarily not awake; (iii) It is not necessary that some human is not awake. B (i) All moving things are animals; (ii) Something white (snow) is necessarily not an animal; (iii) It is not necessary that some white thing is not moving. C (i) Everything which is awake is two-footed. (ii) Some animals are necessarily not two-footed. (iii) It is not necessary that some animal is not awake.

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Notes to pp. 79-80

At 144,15 he opts for B because it involves changing only one of Aristotle’s terms. 144. We read [horoi] to . We suspect that horoi is an intrusive gloss. Or perhaps one should read horoi to egrêgorenai  . 145. i.e. some animal is not awake. 146. The words ‘if just one of them is changed’ (henos monou metalambanomenou) do not occur in our texts of Aristotle. We shall interpret what Alexander says step by step in the following notes. 147. At 30b33-4. Alexander rules out these terms as an interpretation for rejecting Baroco2(UNN) because they make the first premiss the necessary ‘All humans are animals’. Alexander apparently says that they also make the second premiss the (allegedly) false ‘It is necessary that some white things are not animals’. This is presumably a careless formulation since at 143,23-5 he indicated that this proposition is true. (A reader has suggested that this problem would be avoided if the words we have translated ‘but not’ (all’ oukh) were translated ‘rather than’ and understood as ‘rather than just’.) 148. We do not know whether Alexander just means that it is false that all white things are animals, or that it is impossible, since, e.g., snow is necessarily not an animal. We suspect the latter, since he is aware that Aristotle sometimes interprets unqualified sentences by false but possible propositions. See 130,234. On the other hand, Alexander proceeds to say that ‘All humans are white’ will not do as an interpretation for AaB. 149. Note that here Alexander takes it that NEC(Human o Animal). 150. At 141,6-16. 151. That is, with moving substituted for white. 152. We take the remark in parenthesis to be a response to an attempt to defend Aristotle either by altering the text of 31a17 from dia gar tôn autôn horôn hê apodeixis to dia gar tôn autôn hê apodeixis or interpreting the text to mean something like ‘The same kind of proof will work.’ Alexander insists that horôn would have to be understood even if it weren’t there. 153. Alexander here raises the question why the procedure of ekthesis which Aristotle invoked at 1.8, 30a6-14 to validate Baroco2(NNN) cannot be applied to: Baroco2(UNN) AaB NEC(AoC) NEC(BoC) He first goes through the proof for Baroco2(NNN) and then adapts it to Baroco2(UNN). Let D be a part of C such that NEC(AeD). Then (Camestres2(UNN)) NEC(BeD); but D is some of C; so NEC(BoC). Alexander’s argument is longer because he carries out the reduction of Camestres2(UNN) to Celarent1(NUN). 154. i.e. by necessity. 155. This seems to be a minor slip on Alexander’s part since only one of the premisses is converted. We move from AaB and NEC(AoC), to AaB and NEC(AeD), to (converting NEC(AeD)) AaB and NEC(DeA), premisses for Celarent1(NUN). Alexander is presumably looking ahead to the conversion of the conclusion. 156. Again the questioning of Aristotle’s treatment of first-figure NUN cases. 157. In the manuscripts ‘D’ and ‘A’ are interchanged; apparently the Aldine reads ‘D holds of no D by necessity’. 158. Alexander confronts the embarrassing fact that Baroco2(NUN) can apparently be disproved by terms and proved by the same method which Aristotle used for Baroco2(NNN). For similar difficulties with Bocardo3(NU_)

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see the end of the commentary on the next chapter, 150,25-151,30, and for a related difficulty 238,22-38. 159. The terms given (against Baroco2(NUN)) are not Aristotle’s, but the ones Alexander gave at 143,28-144,2. 160. i.e., doing an ekthesis. 161. Alexander defends Aristotle’s rejection of Baroco2(NUN), claiming that if the ekthesis proof were correct it would not be possible to give terms to reject it. 162. In the previous paragraph. 163. Alexander appears to claim that it would not be possible to produce terms to reject Baroco2(NNN), but he lacks the resources to argue for this claim; he contents himself with giving three terms which do not invalidate it. 164. Aristotle announces that Darapti3(NU_), Darapti3(UN_), and Felapton3(NU_) have necessary conclusions, but Felapton3(UN_) an unqualified one. 165. Aristotle reduces Darapti3(NUN) to Darii1(NUN). 166. Aristotle reduces Darapti3(UNN) to Darii1(NUN), albeit in a somewhat sketchy way; Alexander fills in the details and insists that they are necessary, showing his commitment to the idea that the major term must be the predicate of the conclusion. [Ammonius] (in An. Pr. 40,9-13) mentions – apparently as an argument of ‘those around Theophrastus’ – that the modality of the conclusion of Darapti3(UN_) and Darapti3(NU_) would seem to depend on which of the premisses one chooses to convert for the reduction. 167. Darii1(NUN). 168. Aristotle reduces Felapton3(NUN) to Ferio1(NUN). This is the last lemma in the Greek text until 149,23. 169. Aristotle now argues that the standard reduction of Felapton3(UN_) would be to Ferio1(UNU). 170. Aristotle gives terms to show that Felapton3(NU_) yields an unqualified conclusion. Alexander understands him to be affirming the following: (i) No horse is good; (ii) It is necessary that all horses are animals; (iii) Some animal is not good and it is not necessary that some animal is not good. Alexander has some difficulty with Aristotle’s justifying (i) by saying that ‘it is contingent that good holds of no horse’ and (iii) by saying ‘it is contingent that all are good’. 171. Alexander explains Aristotle’s substitution of being awake (or asleep) for good by reference to the doxa that some animals are incapable of goodness. 172. Aristotle announces the validity of Disamis3(UNN) and Datisi3(NUN), and reduces Disamis3(UNN) to Darii1(NUN). 173. Alexander writes esti where our texts of Aristotle have estin. 174. Darii1(NUN) 175. Aristotle remarks that Datisi3(NUN) can also be reduced to Darii1(NUN). Alexander fills in the details and explains the formal difference from the previous reduction of Disamis3(UNN). 176. Again Alexander writes esti where our texts of Aristotle have estin. 177. Aristotle argues for the invalidity of Datisi3(UNN), using the same method which he applied to Felapton3(UNN) at 31a37. 178. Aristotle now rejects Datisi3(UNN) on the grounds that the following are true: (i) Awake a Animal (ii) NEC(Two-footed i Animal)

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Notes to pp. 85-86

(iii)  NEC(Awake i Two-footed) Aristotle expresses (i) by saying that it is contingent that A holds of C. Alexander explains that Aristotle means only to make clear that ‘All animals are awake’ holds, but not by necessity. Cf. 147,17-25. 179. For Alexander’s text here see 149,7-12. 180. Alexander’s point is that if we try converting the universal premiss AaC of Datisi3(UN_), we will end up with two particular premisses which yield no conclusion at all. 181. For a minor divergence from our text of Aristotle see Appendix 6 (the textual note on 31b29). 182. This remark presupposes Alexander’s interpretation of 1.3, 25a31-2. See section III.C of the introduction. 183. Aristotle’s words at 31b30-1 involve a textual difficulty. One would expect him to say, ‘It is not necessary that something two-footed be awake.’ In Ross’s text he says ‘it is not necessary that something two-footed be asleep or awake’ (ou gar anankê dipoun ti katheudein ê egrêgorenai). As this paragraph shows, Alexander’s text has him saying that ‘it is not necessary that something two-footed not be asleep’ (ou gar anankê dipoun ti mê katheudein). It seems to us likely that Aristotle inadvertently wrote ou gar anankê dipoun ti katheudein instead of ou gar anankê dipoun ti egrêgorenai and that this underwent two different corrections, the one printed by Ross and the one read by Alexander. However, we are not sure how to construe what is said in this paragraph about the words ‘It is not necessary that something two-footed not be asleep’. Two suggestions are made, but it is not clear whether each says that the words may indicate a conjunction of two things or each offers two alternatives for what the words mean. Moreover, both substitute ‘some animal’ for ‘something twofooted’ and ‘awake’ for ‘not asleep’. The second substitution is a perfectly reasonable way to bring the words in line with Aristotle’s terms. We take the first to be irrelevant. We take each suggestion to be assigning a conjunction to ‘It is not necessary that some animal be awake’. The first offers ‘It is contingent that some animal be awake’ and ‘It is contingent that no animal be awake’. The second offers ‘It is contingent that some animal be awake’ and ‘It is contingent that some animal not be awake’. However, we say all this without much confidence. 184. Aristotle says that Disamis3(NUN) can be rejected using the same terms as were just used for Datisi3(NUN). Alexander points out that the order has to be changed. Instead of: A: being awake: B; two-footed; C: animal, one will have to assign being awake to B and two-footed to A, so that the propositions become: (i) NEC(Two-footed i Animal) (ii) Awake a Animal (iii)  NEC(Two-footed i Awake) 185. Alexander points out that what Aristotle said about Datisi3(UN_) at 31b20 could be applied to Disamis3(NU_): the standard reduction reduces it to Darii1(UN_), which has an unqualified, not a necessary conclusion. 186. The brevity of this last section of chapter 11 causes Alexander some difficulty. Aristotle is considering the four cases Ferison3(NU_), Ferison3(UN_), Bocardo3(NU_), and Bocardo3(UN_). At 31b33-7 he says that Ferison3(NUN) is valid; Alexander supplies the argument. At 31b37-40 Aristotle apparently asserts the non-validity of Ferison3(UNN) and Bocardo3(UNN), adding the obscure remark that ‘The others are the same as we also said in the case of the previous ones’ (ta men gar alla tauta ha kai epi tôn proteron eroumen). Alexan-

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der interprets these words to mean that only Ferison3(UN_) can be handled by showing that the standard reduction reduces it to a first-figure UNU case. Aristotle goes on to give terms purporting to show the invalidity of Bocardo3(UNN) at 31b40-32a1, of Ferison3(UNN) at 32a1-4, and of Bocardo3(NUN) at 32a4-5. Alexander does the terms for Ferison3(UNN) and then Bocardo3(UNN) before turning to the difficult case of Bocardo3(NUN). 187. On the text here see 151,14-22. 188. Presumably because of the similarity to Felapton3(NUN). 189. Ferio1(NUN). 190. Alexander gives elaborate abstract descriptions of Ferison3(UN_) and Bocardo3(UN_). 191. This qualification is important: only Ferison3(UNN) can be ‘shown’ invalid by conversion, since Bocardo3 is reduced to the first figure by reductio. Alexander believes Aristotle is referring only to this case when he says ‘The others are the same as we also said in the case of the previous ones.’ See further 150,25ff. 192. See Appendix 6 (the textual note on 31b39) for a slight difference between Alexander’s citation and our text of Aristotle. 193. Alexander treats Aristotle’s second rejection, that of Ferison3(UNN) first. In rejecting Ferison3(UNN) Aristotle takes these sentences to be true: (i) Nothing white is awake; (ii) Something white is an animal by necessity; (iii) It is not necessary that some animal is not awake. 194. Alexander now takes up Aristotle’s first rejection, that of Bocardo3(UNN). Aristotle takes the following sentences to be true: (i) Some humans are not awake; (ii) All humans are animals by necessity; (iii) It is not necessary that some animal is not awake. 195. Bocardo3(UN_). 196. Ferison3(UN_). 197. See Appendix 6 (the textual note on 31b39). 198. See Aristotle’s treatment of Bocardo2 at 1.10, 31a10-17 with Alexander’s discussion at 143,4-145,20. 199. At 31b40. 200. We follow the Aldine in omitting autôn, which would give ‘using the same terms’. 201. Aristotle’s assignment of terms for rejecting Bocardo3(NUN) presupposes the truth of the following propositions: (i) Some animal is not two-footed by necessity; (ii) All animals are moving; (iii) It is not necessary that some moving thing not be two-footed. Here animal is taken to be the middle term, but in Alexander’s texts Aristotle says that two-footed is the middle term. Such an assignment would work only if either ‘Some two-footed thing is not an animal’ or ‘Some moving thing is not an animal’ is a necessary truth. Alexander ascribes the mistake to a scribe rather than to Aristotle. Philoponus (145,5-6) says that the mistake is found in ‘some of the older manuscripts’. Both readings occur in our manuscripts. 202. i.e.  NEC(Moving o Two-footed). Alexander’s inference is correct, but the assumed equivalence is not. 203. Accepting Wallies’ conjecture hup’ autou for the ep’ autou which he prints.

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204. i.e. Both NEC(Moving o Two-footed) and NEC(Animal o Two-footed) are false. 205. Alexander now confronts the fact that Bocardo3(NUN), like Baroco2(UNN) (see 144,23-145,20), could be validated using the same kind of ekthesis argument which Aristotle used for Baroco2(NNN) and Bocardo3(NNN) at 1.8, 30a6-14. Suppose NEC(AoC) and BaC; take some of C, D, such that NEC(AeD); then, since D is some of C, BaD and (AI-conversionu) DiB; hence (Ferio1(NUN)), NEC(AoB). Alexander again insists that the setting down of terms shows the invalidity of Bocardo3(NUN). See also 238,22-38. 206. This brief chapter is notable mainly for its obscurity. We would like to thank an anonymous reader for greatly improving our understanding of Alexander’s commentary (which includes an interesting discussion of the notion of necessity at 155,3-25). 207. Taken as a generalization about N+U syllogistic combinations, this remark is false since it seems to say (a) that one must have two unqualified premisses to get an unqualified conclusion and (b) that if just one premiss is necessary the conclusion will be necessary. If one assumes that Aristotle is talking about pairs of combinations (e.g. Barbara1(NU_) and Barbara1(UN_)), then the only exceptions to (b) are Baroco2 and Bocardo3, but the only pair satisfying (a) is Darapti3. Alexander has already expressed qualms about Aristotle’s rejection of Baroco2(UNN) (144,23-145,20) and Bocardo3(NUN) (151,22-30). His position here on those two cases is not clear to us, but it appears that he decides that Aristotle’s remarks are only intended to apply to Darapti3; see 154,17-22. (For two minor questions about Alexander’s text of this passage see Appendix 6 (the textual notes on 32a7 and 32a8)). 208. Alexander considers taking Aristotle to be referring to the third figure, which he has most recently been discussing. Alexander first points out that of third-figure syllogisms with no contingent premisses only Darapti3 yields a necessary conclusion whenever either premiss is necessary. 209. i.e. although Barbara1(NUN) is valid, Barbara1(UNN) is not. 210. Alexander now points out that Bocardo3 violates Aristotle’s claim that one necessary premiss will suffice to yield a necessary conclusion and considers reading 32a6-7 as if it said ‘It is then evident that there is no syllogism of an affirmative conclusion unless both premisses are affirmative’. As Alexander indicates, the context makes the interpretation ‘forced’. 211. Alexander now considers the suggestion that cases, such as Barbara1(UNU), in which Aristotle holds that an unqualified and a necessary premiss yield an unqualified conclusion can be made to cohere with his problematic remark by pointing out that a necessary proposition also holds simply (or, as we might put it, if NEC(P), then P) and Barbara1(UUU) is valid. Aristotle’s remark would then be read as saying something like, whenever there is an unqualified conclusion, it can be reached with two unqualified premisses, but a necessary conclusion requires at least one necessary premiss. Alexander does not comment on this suggestion, which is implausible in itself and obviously does not help with Aristotle’s apparent claim that a necessary premiss and an unqualified one always yields a necessary conclusion. 212. Alexander paraphrases Aristotle’s ouk esti sullogismos by sullogismos ou ginetai. 213. Alexander expresses a preference for restricting Aristotle’s first remark to Darapti3, a view he confirms at 154,17-22. 214. Alexander appears to affirm Aristotle’s remark that ‘there is a syllogism of necessity if only one of the premisses is necessary’, but proceeds to

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question it because of Aristotle’s belief (which Alexander has questioned in the case of Baroco2(UN_) and Bocardo3(NU_)) that Baroco2 and Bocardo3 only yield a necessary conclusion if both premisses are necessary. 215. Darapti3. 216. Here either Aristotle’s problematic rejection of Baroco2(UNN) is being overlooked or dismissed, or what is said is only that in each figure there are some moods in which one necessary premiss suffices to get a necessary conclusion. 217. The principal content of Aristotle’s statement is clear: in a syllogism if the conclusion is unqualified, at least one premiss is unqualified, and if it is necessary, at least one premiss is necessary. Alexander worries mainly about the meaning of ‘in both’, which we take to mean ‘whether the syllogism is of a necessary or an unqualified conclusion’. Before considering this interpretation Alexander mentions the possibility that ‘in both, when the syllogisms are affirmative or negative’ means ‘if the syllogism is of an affirmative or a negative conclusion’. 218. Here Alexander substitutes apophatikos for Aristotle’s sterêtikos. 219. Reading kai prostithêsin . 220. i.e. if Aristotle is only talking about mixed syllogisms at most one premiss can be of the same modality as the conclusion. 221. Darapti3 and Felapton3. Alexander goes on to point out that the justification of these moods involves a weakening step of AI-conversion. 222. It is now apparently suggested that the words ‘in both’ means something like ‘in cases in which there are both necessary and unqualified premisses’. 223. Alexander is thinking specifically of 1.19, 38b14-17. 224. Alexander adds further ‘evidence’ for restricting the scope of 32a6-8 to Darapti3. 225. It seems reasonably clear that Aristotle is repeating what he has just said: a necessary (unqualified) conclusion requires at least one necessary (unqualified) premiss. Alexander reads into what Aristotle says the idea that N+U combinations corresponding to syllogistic UU combinations must yield either a necessary or an unqualified conclusion. 226. e.g. at 1.9, 30a23-8. 227. cf. 172,15-19. 228. Alexander raises the question whether the following example doesn’t show that we can have a necessary conclusion without necessary premisses: what walks is human; what is human moves; therefore, what walks moves. His discussion is of some interest although it seems to involve a confusion of two questions: (i) is it possible for a proposition NEC(P) to follow from two modally unqualified propositions? (ii) is it possible for a necessary proposition P to follow from two propositions which are not necessary? The answer to (i) is negative; Alexander’s example shows that the answer to (ii) is positive. Alexander offers a number of alternative responses without espousing any: (a) The first (155,8-10) is the correct one if it is interpreted as saying that, e.g. Barbara1(UUN) is not a valid combination despite the example of true contingent propositions AaB, BaC, where AaC is necessary. (b) The second (155,10-12) develops the same point in another way by suggesting that the issue is not whether particular propositions AaB, BaC might imply a necessary proposition AaC, but whether they imply NEC(AaC). (c) Alexander’s third suggestion (155,12-15) is that ‘What is human moves’ is – despite Aristotle’s use of it – a necessary truth, because humans are always

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in motion (in the sense of changing in some way). Hence, in an informal sense, the example is a case of Barbara1(NUN) and not a demonstration that Barbara1(UUN) is valid. (d) Alexander’s fourth suggestion (155,16-20) insists on making precise the sense of motion involved, pointing out that it is not true that what walks changes in quality by necessity even if it is assumed that all humans are changing and all walking things are humans. (e) The final suggestion (155,20-5) invokes the distinction between simple necessity and necessity on a condition. ‘What walks moves by necessity’ is not true simply, but only on the condition ‘as long as it is walking’, i.e., what is walking might not be walking and so might not be moving, but what walks moves by necessity as long as it is walking. See Appendix 3 on conditional necessity. 229. That is, as indicated by the preceding example. 230. This remark does not seem to apply to what Aristotle says in the Prior Analytics; for the connection between necessity and eternity see Appendix 3 on conditional necessity. The next sentence may be an insertion designed to correct this apparently irrelevant remark. 231. Chapter 13 divides into four parts: (i) 32a16-29: The diorismos of contingency, (ii) 32a29-32b3: Conversion rules, (iii) 32b4-25: What is true for the most part and the indefinite, (iv) 32b25-37: Two readings of ‘It is contingent’; transition to the treatment of pairs of premisses. We have discussed the first two parts in section III.A of the introduction. 232. legomen. Ross prints legômen, which is also in the Aldine. 233. legô de (d’ Ross) endekhesthai kai to endekhomenon, hou mê ontos anankaiou tethentos de huparkhein ouden estai dia tout’ adunaton. 234. See 1.3, 25a38-9 with Alexander’s comments at 37,28-38,9. 235. Alexander wants to insist that what is contingent in the strict sense is neither necessary nor unqualified. See section III.A of the introduction. 236. The third adjunct (prosrhêsis) is ‘It is contingent that’. See 1.2, 25a2-3 with Alexander’s explanation at 26,29-27,1. 237. dunasthai. 238. cf. Ammonius, in Int. 245,1-32. Ammonius distinguishes between what is contingent in the strict sense (kuriôs) and what is contingent without qualification (haplôs), the latter being what we represent by  NEC . 239. Alexander consistently uses some word for ‘follows’ or ‘results’ (here hepomenon ekhei) in citing the diorismos in which Aristotle says ouden estai dia tout’ adunaton. See also 157,6-7, 157,9-10, 158,13-14, 158,19-20, 169,30-2, 174,5-6. 240. This is ‘necessity on a condition’; see Appendix 3 on conditional necessity. 156,26-157,2 constitute Theophrastus 100B FHSG. 241. At 1.15, 34a5-33. 242. Since Becker (1933), pp. 11-14, this passage has frequently been treated as an interpolation because it seems to presuppose Theophrastean contingency rather than the contingency which Aristotle has just introduced. Alexander does not show any sense of difficulty with this passage. He understands its purpose to be a confirmation of the correctness of the clause ‘P is not necessary’ in the definition of contingency. According to him, Aristotle confirms the second clause at 1.15, 34a5-33 by showing that if an impossibility follows from something it is impossible. Alexander points out that what is said here agrees with the doctrine of Int. 13 (see Appendix 4 on On Interpretation, chapters 12 and 13), but it is clear from other passages that he recognizes that what is said there

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is not compatible with the notion of contingency which is central to Aristotle’s modal syllogistic. 243. Most mss. of Aristotle read ‘negations and affirmations’. 244. Wallies takes the reference to be to chapter 46. The tautology: (i) (P  Q)  ( P   Q) gives a reasonable approximation of what Alexander is talking about, although he discusses predicational rather than propositional relations. At 1.46, 52a39 ff. Aristotle proves what might be formulated as: (ii) (P  Q) &  (Q  P)  ( Q   P) &  ( P   Q). His reasoning could obviously be used to establish (i), since in the course of proving (ii) Aristotle proves what amounts to: (iii) (P  Q)  (  Q   P). Aristotle proves what amounts to (iii) again at 2.2, 53b12-16. 245. hôn antiphaseôn ta hetera moria allêlois hepetai. toutôn tôn antiphaseôn kai ta loipa moria anankê hepesthai allêlois  Alexander gives another general formulation at 157,30-2, but the letters in the immediately following text are his. 246. Alexander here uses the verb akolouthein to express the predication relation. For such uses in Aristotle, see Bonitz (1870), 26b1-9. 247. Chapter 13. 248. We omit Wallies’ insertion of apophaseis which indicates that these two propositions are negations. 249. At 1.15, 34a5-33. 250. See section III.A of the introduction. 251. A contingent negative (endekhomenê apophatikê) is a proposition of the form CON(P), where P is a negative proposition; in this case NEC(P) is a necessary negative (anankaia apophatikê). These are not ‘simple negations’, since they are, in fact, affirmations, as opposed to the negations of contingency (endekhomenês apophatikê) or necessity (anankaias apophasis),  CON(P) and  NEC(P). If P is an affirmation, CON(P) and NEC(P) are called contingent and necessary affirmations by Alexander. Philoponus, in An. Pr. 53,15-24 rejects the distinctions made by Alexander here. His discussion makes clear that the distinctions were introduced to explain a perceived discrepancy between Aristotle’s treatment of, e.g.  CON(P) as a negation of CON(P) in On Interpretation, chapter 12 and his tendency in the Prior Analytics to speak as if, e.g. CON(AeB) were a negation. See Appendix 4 on On Interpretation, chapters 12 and 13. 252. i.e. the negation, and similarly for the next occurrence of ‘negative’. 253. Wallies prints kata to sêmainomenon to keimenon nun lambanomenou tou endekhomenou. We would prefer kata to sêmainomenon to [keimenon] nun lambanomenon tou endekhomenou. 254. tous peri Theophraston. 158,24-5 and 159,8-15 constitute Theophrastus 103A FHSG. 255. See 1.17, 36b35-37a31 with Alexander’s discussion at 219,35-227,9. 256. Alexander’s point in the next tortured sentence is that when ‘Some B is A’ and ‘Some B is not A’ or ‘Some B is A by necessity’ and ‘Some B is not A by necessity’ are true together, the B’s in question will be different. But ‘It is contingent that some B is A’ and ‘It is contingent that some B is not A’ are true of the same B’s. 257. Alexander uses dunaton rather than endekhesthai in expressing these two affirmations. We understand Alexander to be making points about ordinary expressions not the logic of syllogisms.

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258. diorismos. For this use of the term diorismos see 44,26-7 with the note of Barnes et al. ad loc. 259. On this phrase see 160,28-161,2. 260. logos. The reference is, of course, to 32a18-21. 261. i.e. Theophrastus and Eudemus. See 159,8-15 with notes. 262. Alexander refers to chapter 12 of On Interpretation. 263. Here Alexander refers to 1.3, 25b15-21. Immediately after that passage Aristotle says ‘for “It is contingent that X” is ordered in the same way as “X is” ’ (25b21-2 to gar endekhetai tôi estin homoiôs tattetai). There is no evidence from the commentary that Alexander had the words ‘for “It is contingent that X” is ordered in the same way as “X is” ’ at 25b21-2, and it seems unlikely that he would refer the words ‘as was said earlier’ to On Interpretation if he had them. On Andronicus’ rejection of On Interpretation see Moraux (1973), pp. 117-19. 264. Alexander raises a difficulty about the convertibility of CON(P) to CON(  P) on the assumption that CON(P) entails ‘P is not yet the case’, that is,  P. Obviously, we cannot assume that CON(  P) entails   P unless we are willing to say that contingent propositions are contradictory. Alexander first suggests that the solution to this problem is to deny that CON(  P) is genuinely contingent. 265. kata ton diorismon, the first of 60-odd occurrences of this way of referring to the contingency which excludes necessity and simple holding. The phrase is taken from Aristotle; see 1.14, 33b23, 1.15, 33b28-31 and 34b27. 266. i.e. if CON(P) is understood in the way specified, CON(  P) cannot be so understood. 267. The remainder of this section is difficult. We take Alexander to canvass three ways out of the problem without opting for any of them. The first is to deny that Aristotle held that CON(P) excludes P. The second is to say that CON(P) and CON(  P) can be true at different times. This will hardly suffice for converting one of the propositions into the other. The third alternative is stated very briefly, and seems to be this. Take CON(P) to mean ‘It is contingent that P comes about’; both ‘It is contingent that P comes about’ and ‘It is contingent that P does not come about’ can be true when P is not true; so CON(P) and CON(  P) can be converted into one another. With this passage one should compare 222,16-35. Alexander appears to subscribe to the view that CON(P) is incompatible with P at 174,15-16. 268. At 32a18-20. 269. At 32a20-1. 270. In the next section Aristotle and, following him, Alexander discuss what we might call a pragmatic issue without clearly separating it from what for us are more strictly logical questions. Aristotle says that contingency is spoken of in two ways. If P is contingent, then either: (i) P for the most part (epi to polu) or usually (epi to pleiston or epi pleiston or epi pleon), or: (ii) P as often as not (ep’ isês). Alexander adds a third: (iii) P infrequently (ep’ elatton), but lumps (ii) and (iii) together with Aristotle’s term ‘indefinite’. It is clear that this distinction of kinds of contingency would be difficult to incorporate into Aristotle’s formalism. For, although if P as often as not, then  P as often as not, if P for the most part, then  P infrequently and if P infrequently,  P for the most part. Aristotle announces that ‘P for the most part’ ‘converts’ to 

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NEC(P) and ‘P indefinitely’ to ‘No more P than  P’. Alexander interprets him as saying that these convert to CON(  P). At 32b18 Aristotle says that science is concerned with (i). Alexander’s comment on this remark is of some interest; it includes an attack on the Stoics. 271. For a minor difference between this lemma and our text of Aristotle see Appendix 6 (the textual note on 32b4). 272. We have translated these last two sentences in conformity with Alexander’s understanding of them; see 163,23-164,8. The standard modern understanding of them is conveyed by the following translation. ‘If P is by nature it converts because it does not hold by necessity – it is contingent for a person not to turn grey in this sense. But if P is indefinite it converts because P is no more one way than the other.’ 273. cf. 165,10-14. 274. to hopoter’ etukhen. 275. Alexander changes the position of the word kai in Aristotle’s formulation. 276. e.g. CON(AaB) converts to both CON(AeB) and CON(BiA). 277. For a minor divergence between the lemma and our text of Aristotle see Appendix 6 (the textual note on 32b19). 278. On the text here see the note on 165,4. 279. sumplôkê; cf. 169,10. We take the point to be that there are formally valid combinations which are useless. 280. hoi neôteroi, the Stoics; see the note on 3,3 in Barnes et al. Lines 164,27-31 constitute fragment 1169 Hülser (1987-1988) and SVF II.259. 281. Alexander gives the following explanation of duplicated (diphoroumenoi) and duplicating (adiaphorôs perainontes) arguments: ‘According to the Stoics, the following sort of thing is a duplicated argument: if it is day, it is day; but it is day; therefore, it is day. A duplicating argument is one in which the conclusion is the same as one of the premisses, as in the following case: either it is day or it light; but it is day; therefore, it is day.’ (Alexander, in Top. 10,7-12; the distinguishing feature of a duplicated argument is the occurrence of a premiss in which the same sentence (in the example ‘It is day’) occurs twice.) The second thema is one of the rules which the Stoics used to reduce valid arguments to the unprovable arguments. Its exact content is uncertain. See Mates (1961), pp. 77-82 or Frede (1974), pp. 181-90. Nothing is known about ‘what is called infinite matter’ (hê apeiros hulê legomenê); see Hülser (19871988), pp. 1622-3. 282. organon. 283. Ross prints ekeinôn, but reports two readings of ekeinôs. Wallies prints ekeinôs, noting that Aristotle and the Aldine have ekeinôn. 284. At 168,31-169,14 Alexander points out that Aristotle sometimes gets involved with contingencies which are infrequently true. Indeed, he has to, if he infers, e.g. CON(AeB) from CON(AaB) and AaB is true for the most part. 285. See 39,30-40,5 with the notes in Barnes et al. 286. Aristotle does not, in fact, speak again about the kinds of contingency or the uselessness of the indefinite. He takes up AE-transformationc again in chapter 17. The next sentence appears to be a general characterization of the remainder of this chapter. 287. This difficult passage has sometimes been taken as Aristotle’s way of justifying the claim that the standard first-figure CC combinations are syllogistic, introducing a reading of CON(AaB) as something like ‘What is contingently

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B is contingently A’, so that he can justify Barbara1(CCC) by taking it to infer ‘What is contingently C is contingently A’ from ‘What is contingently B is contingently A’ and ‘What is contingently C is contingently B’. See, e.g. Patterson (1995), pp. 141-9. Alexander, for whom the justification of complete syllogisms always turns on the dictum de omni et nullo, shows no trace of this idea; see 167,10-168,4 and 169,17-32. He takes Aristotle to be showing that CON(AaB) has two meanings: (C*) CON(A holds of that of which B in fact holds) (C1) CON(A holds of that of which it is contingent that B holds) or alternatively (see 165,30 with note): (C*’) CON(A holds of that of which B is said) (C1’) CON(A holds of that of which it is contingent that B is said) At 32b32 Aristotle distinguishes between the mixed syllogistic premisses: (1) Barbara1(CU_) CON(AaB) BaC and the unmixed pair: (2) Barbara1(CC_) CON(AaB) CON(BaC) and says that as before he will be discussing the unmixed pairs before the mixed. For Alexander (probably following Theophrastus) the C propositions are ‘prosleptic’, that is, they potentially contain the middle term and so express a pair of premisses. Alexander assumes (see 166,5-10) – naturally and probably correctly (see 1.14, 32b38-33a1) – that (1) is to be correlated with (C*) and (C*’), (2) with (C1) and (C1’). At 166,19-25 Alexander argues that NEC(AaB) admits two analogous interpretations, viz., (N*) It is necessary that A holds of that of which B is said (N1) It is necessary that A holds of that of which B is said by necessity and says that defenders of Aristotle’s position on Barbara1(NU_) presuppose that only (N*) is a correct interpretation. Cf. 126,23-8. 288. Alexander remarks that Aristotle changes from to endekhesthai tode tôide huparkhein at 32b25-6 to kath’ hou to B, to A endekhesthai at 27-8. 289. Alexander’s lemma has endekhesthai whereas our mss. of Aristotle have enkhôrein. 290. On prosleptic assertions see first 378,12-379,11, and other material gathered as Theophrastus 110A-D FHSG. The crucial point in connection with what Alexander says here is that the prosleptically formulated major premiss ‘potentially’ contains the minor premiss. 291. Defenders of Aristotle. 292. Alexander writes kathaper epi tôn allôn where our text of Aristotle has kathaper kai en tois allois. 293. For discussion of this passage see section III.D.2 of the introduction. 294. For a minor divergence from our text of Aristotle see Appendix 6 (the textual note on 36b35). 295. Aristotle takes for granted the equivalence of CON(AeB) and CON(AaB) and assumes their compatibility with  CON(BaA). But if CON(AeB) converted with CON(BeA), then since CON(BeA) is equivalent to CON(BaA), CON(AeB) would imply CON(BaA). So EE-conversionc must fail. Aristotle’s example to show that CON(AeB) (and CON(AaB)) is compatible with  CON(BaA) (and  CON(BeA)) takes it to be true that it is contingent that no human is white but necessary that some white things are not human. 296. In chapters 2 and 3. The discussion of universal negatives is at 25b3-25. 297. i.e. Aristotle is taking for granted EA-transformationc and arguing

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against EE-conversionc. Theophrastus accepts EE-conversionc and rejects EAtransformationc. 298. At 41,21-4. Note that only Theophrastus was mentioned before, and that in the present passage the verbs translated ‘say’ and ‘show’ are in the singular in the ms B. (In line 10 Wallies prints phasi with the Aldine instead of B’s phêsi, and in 12 deiknûsin against the deiknusin of both the Aldine and B. 220,9-221,5 are Theophrastus 102A FHSG. 220,9-16 are Eudemus fragment 16 Wehrli.) We discuss this argument and Alexander’s rejection of it in section III.D.1 of the introduction. In Theophrastus 102C FHSG this argument is briefly formulated as what is called an ekthetic argument: ‘If it is contingent that white is in no man, it is contingent that white is disjoined from all man, and man will be disjoined from all white.’ Alexander gives another Theophrastean argument for EE-conversion‘c’ at 223,4-14. 299. pantôn tôn tou B. On the terminology of disjointedness see the note on 124,20. 300. Aristotle’s argument is not a reductio, but what we have called an incompatibility rejection argument (introduction, p. 17). 301. Theophrastus and Eudemus. 302. That is to say,  NEC (AaB) is not equivalent to  NEC (AeB). 303. Alexander shows uncertainty about whether the specification of terms is a new argument against EE-conversionc or a way of showing that the compatibility assumption underlying it is correct. For a minor divergence between the citation and our text of Aristotle see Appendix 6 (the textual note on 37a4). 304. We read ek tês proeirêmenês kataskeuês with B where Wallies prints [ek] tês proeirêmenês kataskeuê, following the Aldine. 305. From here to 222,7 Alexander worries about the meaning of contraries and opposites, first about the fact that Aristotle appears to refer to, e.g., CON(AaB) and CON(AeB) as contraries and CON(AaB) and CON(AoB) as opposites when in fact the pairs can be true together, and then about the question of which ‘contraries and opposites’ convert with which. His solution to the first problem – presumably the correct one – is that such pairs are verbally similar to pairs of unqualified or necessary propositions which are genuinely contraries or opposites. Alexander may raise this issue because in On Interpretation, chapter 12 Aristotle says that, e.g., the opposite of NEC(P) is  NEC(P); cf. Philoponus, in An. Pr. 53,15-56,5 and Ammonius, in Int. 221,11-229,11. See also Appendix 4 on On Interpretation, chapters 12 and 13. Alexander’s treatment of the second question is more tentative, but in the course of it all the relevant transformations countenanced by Aristotle (AE-, EA-, IO-, OI-, AO-, and EI-transformationc) are mentioned positively and the two not countenanced by him (OA- and IE-transformationc) are mentioned negatively. 306. ouk ex anankês panti. 307. Alexander now worries about the fact that Aristotle’s words might be taken as implying that, e.g., a particular contingent affirmative is transformable into a universal contingent negative. He insists that Aristotle only means to assert AE-, EA-, IO-, and OI-transformationc. 308. endekhetai. 309. 7.17b22-3. Alexander now apparently suggests that Aristotle might be treating XiY and XoY as opposites only in the case where the Y in question is the same thing.

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310. Alexander now suggests that Aristotle is accepting AO-and EI-transformationc, but not OA-and IE-transformationc. 311. i.e. the heaven. 312. The remainder of this section is very difficult, and we are doubtful that we have grasped Alexander’s meaning entirely. What he says obviously involves his idea, that ‘It is contingent that P’ means that P does not hold but can hold in the future. Here Alexander considers a situation in which, say, ‘No X is Y’ is true. He says that if someone says, e.g., ‘It is contingent that no X is Y’ in this situation, the proposition will ‘convert’ into ‘It is contingent that all X are Y’, that is to say (we take it), the latter proposition becomes true. But we cannot transform ‘It is contingent that all X are Y’ back into ‘It is contingent that no X is Y’; for we cannot say ‘No X is Y’ will hold since it already does hold. In the second paragraph Alexander rejects the latter claim apparently with a question-begging argument which has an intuitive appeal: roughly, if it is contingent that P at one time, it is always contingent that P. 313. On this Aristotelian passage see section III.D.2 of the introduction. 314. hoi hetairoi autou, Theophrastus and Eudemus. 223,3-15 are Theophrastus 102B FHSG. Alexander gives a perfectly legitimate argument for EE-conversion  n or even for CON(AeB)   NEC (BeA). A somewhat garbled version of this same argument is ascribed to Theophrastus and Eudemus in Theophrastus 102C FHSG. 315. We here make use of numerical indices to avoid Alexander’s use of pronouns and repeated longer explicit formulations of these four propositions. Most of what Alexander has to say amounts to pointing out that each of (ii) CON(BaA) and (iv) CON(BeA) imply the negations of each of (iii) NEC(BiA) and (i) NEC(BoA), so that each of (iii) and (i) imply the negations of (ii) and (iv); however, the negations of (ii) and (iv) do not imply either (i) or (iii). He is particularly concerned to point out that the negation of (iv),  CON(BeA), does not imply (iii), NEC(BiA). He develops the point by giving cases in which all of  CON(BeA),  NEC(BiA), and NEC(BoA) are true. 316. A better formulation would be that neither (i) nor (iii) is a consequence of the denial of (iv). 317. In the alleged reductio justification of EE-conversionc. 318. endekhetai. 319. Alexander gives a case where CON(AeB) but  CON(BeA) because, even though  NEC(BiA), NEC(BoA). He takes as true CON(White e Human),  NEC(Human i White) (because if NEC(Human i White), NEC(White i Human), contradicting CON(White e Human)). But also, NEC(Human o White) (presumably because, e.g., NEC(Human o Swan)). This example illustrates the difficulty of reading Alexander in terms of the de re/de dicto distinction: NEC(Human o White) would seem to be true de re, but then, since there are white humans, so would NEC(Human i White), but Alexander takes this to be false. 320. For a minor difference between the lemma and our text of Aristotle see Appendix 6 (the textual note on 37a18). 321. The correct implication is in the opposite direction, and what Alexander says establishes the correct implication. 322. We have not translated the words ep’ ekeinôn, which would seem to mean ‘when either (i) or (iii) is true’ and hence to render the words ‘when (i) is true’ (alêthous ousês tês ex anankês tini mê) redundant. 323. Ross ad loc. takes the sense to be not that C does hold of all D but that C might hold of all D even though  CON(CaD) since  CON(CaD) is true just

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because NEC(CiD). Alexander takes it that Aristotle is assuming CaD and also  CON(CaD) because NEC(CiD). 324. For a minor difference between the lemma and our text of Aristotle see Appendix 6 (the textual note on 37a27). 325. See 224,18-27. 326. Perhaps Alexander’s clearest statement of the point that a logical rule must be true in every instance.

Appendix 1

The expression ‘by necessity’ (ex anankês) For the most part Alexander’s formulations of propositions involving contingency are not difficult to construe because he uses forms of the verb ‘be contingent’ (endekhesthai). His formulations of propositions involving necessity are often more problematic because, like Aristotle, he usually uses the phrase ‘by necessity’ (ex anankês). At 1.15, 33b29-31 Aristotle says of the conclusions  NEC(AiC) and  NEC(AaC) of Celarent1(UC N ) and Ferio1(UC N ), ‘ their conclusions will be that something holds of none by necessity or does not hold of all by necessity (mêdeni ê mê panti ex anankês huparkhein).’ At 1.15, 34b27-8 (cf. 35a1-2) he says of the conclusion of Celarent1(UC N ), ‘Thus the conclusion of this syllogism is not a proposition which is contingent in the way specified, but is “of none by necessity” (mêdeni ex anankês) .’ Commenting on this passage at 194,14ff. Alexander insists on the distinction between ‘by necessity of none’ (ex anankês mêdeni) and ‘of none by necessity’ (mêdeni ex anankês), where the first corresponds to NEC(AeC) and the second to  NEC(AiC), the conclusion of Celarent1(UC N ), apparently understood as something like ‘There’s no C of which A holds necessarily’. Alexander does not adhere to this distinction uniformly (see, e.g., 131,11-12, 136,25, 202,22), and at 196,28-33 he indicates that ‘of none by necessity’ (in this case oudeni ex anankês) is ambiguous. We have not thought it worthwhile to distinguish these two phrases and others except in cases where it seemed clear that Alexander wanted to stress a difference. We have instead adopted a uniform English translation of the four necessary propositions as follows to render what we take to be Alexander’s intentions: ‘A holds of all B by necessity’ when we take Alexander to mean NEC(AaB); ‘A holds of no B by necessity’ when we take Alexander to mean NEC(AeB); ‘A holds of some B by necessity’ when we take Alexander to mean NEC(AiB); ‘A does not hold of some B by necessity’ when we take Alexander to mean NEC(AoB). As is indicated by the discussion at the beginning of this appendix, negations of necessary propositions cause Alexander – and consequently us – more difficulty. When Alexander uses something like mêdeni ex anankês (which he says at 197,26 is equivalent to ouk ex anankês tini) to express  NEC(AiB) (=  NEC (AeB)) we write ‘A holds by necessity of no B’. At 198,5 Alexander takes up the question whether the conclusion of Barbara1(UC_) is CON(AaC) or  NEC  (AaC), i.e.,  NEC(AoC). He expresses this second alternative as oudeni ex anankês ou. We find this expression rather baffling, and we translate it and its analogues with the equally baffling ‘A does not hold by necessity of no B’. At 174,13 Alexander indicates that the conclusion of Ferio1(CU N ) is ou

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panti ex anankês, for which we use the translation ‘A does not hold by necessity of all B’. Thus we have the correspondences: ‘A holds by necessity of no B’ for  NEC(AiB) (=  NEC (AeB)); ‘A does not hold by necessity of no B’ for  NEC(AoB) (=  NEC (AaB)); ‘A does not hold by necessity of all B’ for  NEC(AaB) (=  NEC (AoB)). We have not noticed any passage in which  NEC(AeB) (=  NEC (AiB)) requires special attention.

Appendix 2

Affirmation and negation To talk about the quality of propositions Alexander mainly uses the adjectives ‘affirmative’ (kataphatikos) and ‘negative’ (apophatikos) and the nouns ‘affirmation’ (kataphasis) and ‘negation’ (apophasis). No problems arise in non-modal syllogistic where a- and i-propositions are affirmations and affirmative and eand o-propositions are negations and negative. The fact that a- and i-propositions might also be construed as negations of o- and e-propositions is avoided by talking about contradictories or opposites rather than negations. At 31,5-6 Alexander informs us that Aristotle calls a universal apophatikos a universal sterêtikos. We follow Barnes et al. in translating sterêtikos ‘privative’. Sterêtikos is Aristotle’s word of choice, but almost every occurrence of the word in this section of the commentary is in a citation or paraphrase of Aristotle. For reasons which are not clear to us Alexander has a strong preference for apophatikos. Alexander also eschews one of Aristotle’s words for ‘affirmative’, katêgorikos (see the Greek-English index.) What Aristotle says in the course of our text clearly implies that propositions such as NEC(AeB) and CON(AoB) are both negative (1.15, 35a30-40; 1.18, 38a10-11) and negations (1.17, 36b38-9). This leaves Alexander with a problem about how to talk about, e.g.,  NEC(AeB) and  CON(AoB). There is no question that Alexander understands the relevant difference between these kinds of proposition. At 158,24-159,23 he introduces a distinction between a contingent or necessary negative (e.g., CON(AoB) or NEC(AeB)) and a negation of contingency or necessity (e.g.,  CON(AoB) or  NEC(AeB)). Unfortunately he sometimes calls a proposition like NEC(AeB) an apophasis by contrast with a kataphasis such as NEC(AaB). In these cases, which are listed in the Greek-English index, we have opted for the translations ‘negative proposition’ and ‘affirmative proposition’.

Appendix 3

Conditional necessity Toward the end of his discussion in On Interpretation of contingent statements about the future Aristotle writes: It is necessary that what is is when it is and that what is not is not when it is not. But it is not necessary that everything which is be nor that what is not not be. For these are not the same: (a) everything that is is by necessity when it is; (b) everything that is is without qualification (haplôs) by necessity. (On Interpretation 9.19a23-6) It appears that Theophrastus and, following him, the ancient commentators took Aristotle to be marking here a distinction between (b) necessity without qualification and (a) a necessity which they typically labelled either ‘on a hypothesis’ (ex hupotheseôs) or ‘on a condition’ (meta diorismou). Alexander typically uses the latter expression.1 Ammonius (in Int. 153,13-154,2) explicates the distinction in terms of affirmative subject-predicate propositions. It is necessary without qualification that S is P if S cannot exist without being P; it is necessary on a condition that S is P as long as P holds of S. Ammonius makes a further distinction between two kinds of necessity without qualification on the basis of whether or not the subject is eternal. In his commentary on the Prior Analytics Philoponus invokes the distinction to defend Aristotle’s claim that Barbara1(NUN) is valid: Aristotle says in On Interpretation that necessity is said in two ways: in the strict sense (kuriôs) and on a hypothesis. And necessity on a hypothesis is said in two ways: something is said to be necessary as long as the subject exists (huparkhein); and something is necessary as long as what is predicated holds (huparkhein). For example, ‘The sun moves’ is said to be necessary in the strict sense; ‘Socrates is an animal’ is said to be necessary on a hypothesis, , since as long as Socrates exists, it is necessary that he is an animal – this type is closer to necessity in the strict sense; and the third sense occurs when we say that it is necessary that what is seated is seated; for as long as what is predicated holds, I mean being seated, it necessarily holds of what is seated in the sense of necessity on a hypothesis.2 Accordingly, we say that the major premiss has been taken as necessary in the strict sense, but the conclusion has been taken as necessary on a hypothesis, namely ‘as long as what is predicated is the case’. For as long as A holds of C, it holds of it by necessity. And Alexander, explicator of Aristotle, says in a certain short work (en tini monobiblôi)3 that his teacher Sosigenes4 is of the same opinion, namely that here Aristotle draws a conclusion which is necessary on a hypothesis. For, he says, that he means this is clear because when the major premiss is unqualified, the

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minor premiss necessary, and the conclusion is inferred to be unqualified, he sets out terms , but when the major premiss is necessary, the minor unqualified, and the conclusion is inferred to be necessary, he does not manage to set out terms which imply necessity in the strict sense. Therefore, he says, it is clear that he also takes the necessity to be on a hypothesis. (in An. Pr. 126,8-29) The argument about terms is opaque partly because of Philoponus’ misunderstanding of the role interpretations play as counterexamples to alleged syllogisms, a misunderstanding which we have seen to be Alexander’s as well.5 There is absolutely no reason for Aristotle to set out terms in connection with a premiss combination he deems syllogistic. However, it seems likely that what underlies the argument is the existence of counter-interpretations for Barabara1(NUN) of the kind which Alexander gives at 124,21-30. The first of these takes the following proposition to be true: It is necessary that all humans are animals; Everything moving is human; It is not necessary that everything moving is an animal. It appears that Sosigenes tried to defend Aristotle against such counter-interpretations by pointing out that ‘Everything moving is an animal’ is necessary on a hypothesis or condition. On the basis of what is said by Philoponus and Ammonius one would expect the condition to be that everything moving is an animal, but because of the necessary first proposition it is also possible to take the condition to be that everything moving is human. Alexander takes this second option in a passage which makes it likely that he dissented from the position of Sosigenes. In the passage Alexander is commenting on Aristotle’s rejection at 1.10, 30b31-40 of Camestres2(NUN) on the basis of the following interpretation: It is necessary that all humans are animals; Nothing white is an animal; It is not necessary that nothing white is human. Aristotle says of the third proposition: Then, human will also hold of nothing white, but not by necessity; for it is contingent that a human be white, although not so long as animal holds of nothing white. So the conclusion will be necessary if certain things are the case, but it will not be necessary without qualification. (1.10, 30b36-40) Alexander says what he takes to be the force of these words: He indicates that when he says, in connection with mixtures, that the conclusion is necessary, he means ‘necessary without qualification’ and not ‘necessary on a condition’, as some of the interpreters of the subject of mixture of premisses say, thinking that they strengthen his position; they assert that he does not speak about inferring necessity without qualification, but about inferring necessity on a condition. For they say that when animal holds of every human by necessity and – as in the first figure – human of all that moves or walks, the conclusion is necessary on a

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condition; for animal holds of all that moves or walks as long as the middle, human, holds of it. For it is not the case that if the minor premiss is necessary, the conclusion is necessary; for it is not the case that if moving holds of every animal and animal of every human by necessity, moving holds of every human by necessity as long as animal holds of every human – for that is false – but for as long as moving holds of every animal. (140,16-28)6 Alexander goes on to cite Aristotelian passages which suggest very strongly that the interpretation offered by Sosigenes is untenable.7 Alexander might, of course, have taken a different position in the ‘certain short work’ to which Philoponus refers, but we have no way of knowing this. Alexander concludes his discussion of Sosigenes’ position by citing the passage from On Interpretation with which we began this appendix: At the same time he has also indicated by the addition that he is aware of the division of necessity which his associates have made, and which he has also already established in On Interpretation, where, discussing contradiction of propositions about the future and individual things, he says, ‘It is necessary that what is is when it is, and that what is not is not when it is not’. For the necessary on a hypothesis is of this kind. (141,1-6) There are two other passages in the commentary connecting Theophrastus with necessity on a condition. In the first Alexander offers a possible justification for the view that, according to the diorismos of contingency, if CON(P),  P: Or does he deny that what is contingent is unqualified by saying ‘if P is not necessary’; for, according to him, necessity is also predicated of the unqualified; for what holds of something holds of it with necessity, as long as it holds. At any rate Theophrastus in the first book of his Prior Analytics, discussing the meanings of necessity, writes the following: ‘Third, what holds; for when it holds it cannot not hold.’ (156, 26-157,2) It seems reasonable to assume that this third sense of necessity is the third sense of Ammonius and Philoponus, the one according to which S is P by necessity as long as P holds of S. In the other passage the connection with the account of Ammonius and Philoponus is even clearer: What is necessary is either necesssary without qualification or is called necessary on a condition, e.g., ‘Human holds of everything literate by necessity, as long as it is literate.’ This proposition is not necessary without qualification. Theophrastus showed the difference between them; for there are not always literate things, and a human is not always literate. Since they differ in this way, we must recognize that Aristotle is now discussing what is called necessity in the strict sense and without qualification. (36,25-31)8 Although this passage occurs just before the lemma on AI-conversionn, we

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believe that Alexander invokes this distinction here in defence of AI-conversionn. His general point seems clear: we cannot convert a proposition like ‘Human holds of everything literate by necessity’ to ‘Literate holds of some human by necessity’. But the distinction between the two types of necessity does not seem to explain the failure of conversion. It only puts a label on the kinds of a-proposition which are assumed not to convert. However, we can perhaps see why Philoponus classified propositions with non-eternal subjects as necessary on a hypothesis. For if he did not, he would have to say that human holds of everything literate by necessity without qualification.9 Pursuit of this line of reasoning would seem to lead to the conclusion that AI-conversionn holds only when the subject of the a-proposition is eternal. Alexander never pursues this point, but it may have been one of the ways in which people tried to make sense of AI-conversionn. And the point does come up tacitly at three other places in the commentary where Alexander invokes necessity on a condition. One such passage concerns one of Aristotle’s most striking specification of terms for rejecting a combination when he takes as true: It is necessary that every sleeping horse is asleep and it is necessary that every horse-that-is-awake is awake. Alexander is certain that these propositions are only necessary on a condition, that is, hold only as long as the predicate asleep or awake holds, and he is certain that such propositions are really unqualified. But he is uncertain what to make of the situation because he is uncertain about the status of the combination which Aristotle rejects.10 Elsewhere Alexander suggests that ‘What walks moves’ is only necessary on a condition: Or is it the case that even if it is taken that all that walks is human and all humans move, still the conclusion ‘all that walks moves’ is not necessary without qualification but with the additional condition ‘as long as it is walking’? For all that walks does not move by necessity, if, indeed, it is true that what walks does not even walk necessarily except, as I said, on the condition ‘as long as it is walking’. (155,20-5; cf. 201,21-4) Although the exact construal of these words is uncertain, one plausible reading would commit Alexander to the view that a necessary truth requires an eternal subject term. Of course, such a view is not compatible with Aristotle’s practice in the Prior Analytics.11

Notes 1. See the entry on meta diorismou (anankaios) in the Greek-English index. 2. We note that Philoponus divides the three kinds of necessity differently from Ammonius, producing two kinds of necessity on a hypothesis where Ammonius has two kinds of necessity without qualification. Stephanus (in Int. 38,14-31) agrees with Ammonius. At 162,13-26 Alexander suggests that necessity which is conditional on the existence of a non-eternal subject is not necessity at all. 3. This is generally thought to be the work on mixtures of premisses; see 125,31 with the note.

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4. On Sosigenes, see Moraux (1984), 335-60. 5. See section I of the introduction. We mention here a suggestion of an anonymous reader, according to which Sosigenes espoused a method of showing that a pair of premisses assumed to imply a conclusion of one kind does not imply a stronger one by producing a counter-example rejecting the stronger conclusion (the method used by Aristotle in connection with first-figure UN cases). According to this suggestion, when Sosigenes claimed that Aristotle took the conclusion of, e.g., Barbara1(NU_) to be only necessary on a hypothesis, he was asked why he didn’t produce a counter-example to a strictly necessary conclusion. Sosigenes’ answer: Aristotle was not able to produce such terms. This suggestion has the advantage of providing a reasonably unobjectionable sense to the notion of providing terms to establish an implication, but we have not succeeded in working it out fully. 6. We are not certain what to make of this last sentence. The anonymous reader mentioned in the previous note has argued persuasively that it is part of the view against which Alexander is arguing, according to which syllogistic NU cases yield a conclusion which is necessary as long as the minor premiss is true whereas the corresponding UN cases do not yield a conclusion which is necessary in any sense. 7. Further evidence that Alexander did not follow Sosigenes on this issue is provided by the fact that [Ammonius] (in An. Pr. 39,10-25) ascribes to Sosigenes alone the position that the conclusion of Barbara1(NU_) is ‘necessary on a condition’, while ascribing to Alexander an argument in support of Barbara1(NUN). (For the argument see Alexander’s commentary at 127,3-14.) 8. For another (less clear) passage of Alexander (citing Galen) which connects Theophrastus with a distinction between necessary truths with eternal subjects and those with perishable ones see Theophrastus 100C FHSG. 9. The fact that ‘Human holds of everything literate’ is not necessary without qualification shows that necessity without qualification is not so-called de re necessity, since it is presumably de re necessary that human hold of everything literate. 10. See 251,11-252,2. 11. Alexander mentions necessity on a condition one other time in the commentary (179,31-180,3) in connection with the problematic conditional ‘If Dion has died, he has died’, but he does so in a way which seems marginally related to the topic of this appendix. He also twice (181,13-17 and 189,2-3) uses in what seems to be an informal way the standard formula (est’ an) for introducing the condition on which something is necessary.

Appendix 4

On Interpretation, chapters 12 and 13 In chapter 12 of On Interpretation Aristotle proposes to investigate ‘how affirmations and negations of the possible to be and the not possible to be and of the contingent to be and the not contingent to be are related to one another and about the impossible and the necessary’. (21a34-37) In what follows Aristotle makes no distinction between the possible and the contingent, but since the way he treats the two notions differs from the way he treats contingency in the way specified in the Prior Analytics, we shall introduce the operator POS to represent what he says here. We shall also ignore difficulties in the details of what Aristotle says. Since ‘It is impossible that’ and ‘It is necessary that it is not the case that’ end up as equivalent, we can formulate what Aristotle says in terms of possibility and necessity. In chapter 12 the results are: The negation of POS(P) is  POS(P), and it is not POS( P) The negation of NEC(P) is  NEC(P), and it is not NEC( P) These statements cause Alexander and other commentators some difficulty because, as indicated in the appendix on affirmation and negation, in the Prior Analytics Aristotle sometimes speaks as if, e.g., CON(AeB) is a negation. However, Alexander quite rightly takes the view expressed here as the norm to which Aristotle’s apparently discordant statements have to be adjusted (see, for example, 158,24-159,3 on 32a29, and 221,16-222,4 on 36b38). In chapter 13 Aristotle seems to come out strongly for Theophrastean contingency, that is, he seems to hold that: (i) POS(P) if and only if  NEC( P) Aristotle’s argumentation is confused, but he clearly commits himself to a consequence of this equivalence, which is obviously incompatible with the diorismos of contingency, namely: (ii) if NEC(P) then POS(P) At 22b29 he raises the question whether this implication is correct. He uses the example of being cut to suggest that POS(P) implies POS( P), which, with (ii), would produce the impossibility that NEC(P) implies POS( P). Aristotle’s way out is to speak of different kinds of possibility, only some of which are two-sided; he also suggests that possibility is homonymous, and introduces a notion which is something like what we represent by CONu: For some possibilities are homonymous. For possible is not said in just one way. But one thing is said to be possible because it is true in the sense

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of actually being – for example, it is possible for something to walk because it does walk, and, in general, it is possible for something to be because it already is in actuality; another thing is said to be possible because it might be actual, e.g., it is possible for something to walk because it might walk. (23a6-11) In his commentary on On Interpretation Ammonius fixes on the homonymy of possibility. He recognizes that in the Prior Analytics Aristotle denies (i) and (ii) for contingency in the way specified and that this contingency is the fundamental notion of possibility in the Prior Analytics, although it rather clearly is not in On Interpretation. Ammonius (see in Int. 245,1-32) adopts what we think is a rather unusual expedient to harmonize the two texts. He says that the notion of contingency in On Interpretation is contingency without qualification (haplôs) whereas contingency in the Prior Analytics is contingency in the strict sense (kuriôs). We find no clear trace of this distinction in Alexander, but it is clear from 37,28-38,10 (quoted in section III.A of the introduction) that Aristotle’s discussion in On Interpretation was a primary source of Alexander’s view that contingency is homonymous. Of course, the way Aristotle handles the notion of contingency in the Prior Analytics could only encourage such a view.

Appendix 5

Weak two-sided Theophrastean contingency Aristotle and Alexander are committed to: CON(P)   NEC( P) Given: AE-transformationc: EA-transformationc: IO-transformationc: OI-transformationc:

CON(AaB)  CON(AeB)  CON(AiB)  CON(AoB) 

CON(AeB) CON(AaB) CON(AoB) CON(AiB)

they are, indeed, committed to: CON(AaB)  CON(AeB)  CON(AiB)  CON(AoB) 

 NEC (AaB) &  NEC (AeB)  NEC (AaB) &  NEC (AeB)  NEC (AiB) &  NEC (AoB)  NEC (AiB) &  NEC (AoB)

or equivalently to CON(AaB)  CON(AeB)  CON(AiB)  CON(AoB) 

 NEC(AoB) &  NEC(AiB)  NEC(AoB) &  NEC(AiB)  NEC(AeB) &  NEC(AaB)  NEC(AeB) &  NEC(AaB)

In section III.D.2 of the introduction we mentioned equivalents of the first two of these propositions labelled as NCa and NCe. We here introduce analogous equivalents of the other two as well: (NCa) NEC(AoB) v NEC(AiB)  (NCe) NEC(AoB) v NEC(AiB)  (NCi) NEC(AeB) v NEC(AaB)  (NCo) NEC(AeB) v NEC(AaB) 

 CON(AaB)  CON(AeB)  CON(AiB)  CON(AoB)

In section III.D.2 we have sketched some reasons for thinking that Aristotle and Alexander may have accepted the converses of NCa and NCe: (  CaN)  CON(AaB)  NEC(AoB) v NEC(AiB) (  CeN)  CON(AeB)  NEC(AoB) v NEC(AiB) to which we now add:

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( CiN)  CON(AiB)  NEC(AeB) v NEC(AaB) ( CoN)  CON(AoB)  NEC(AeB) v NEC(AaB) Combining these pairs of propositions into equivalences makes explicit what we shall call weak two-sided Theophrastean contingency: (2TCa) CONtc(AaB)  (2TCe) CONtc(AeB)  (2TCi) CONtc(AiB)  (2TCo) CONtc(AoB) 

 NEC(AiB) &  NEC(AoB)  NEC(AiB) &  NEC(AoB)  NEC(AaB) &  NEC(AeB)  NEC(AaB) &  NEC(AeB)

These equivalences enable one to prove a number of principles which Aristotle accepts and to block some that he doesn’t accept. This is trivially true for AE-transformationc, EA-transformationc, IO-transformationc, and OI-transformationc. We give a proof that: CONtc(AaB)  CONtc(AiB) Suppose CONtc(AaB) and  CONtc(AiB). Then  NEC(AiB) &  NEC(AoB), and either NEC(AaB) or NEC(AeB). But in either case there is a contradiction since NEC(AaB) implies NEC(AiB) and NEC(AeB) implies NEC(AoB). This same proof establishes that: CONtc(AeB)  CONtc(AoB) On the other hand we can block both EE-conversiontc and OO-conversiontc, that is, we can show: CONtc(AeB) does not imply CONtc(BeA) CONtc(AoB) does not imply CONtc(BoA) that is

 NEC(AiB) &  NEC(AoB) does not imply  NEC(BiA) &  NEC(BoA)  NEC(AaB) &  NEC(AeB) does not imply  NEC(BaA) &  NEC(BeA) Since  NEC(AiB) does imply  NEC(BiA) and  NEC(AeB) does imply  NEC(BeA), these two claims reduce to the obviously true:

 NEC(AoB) does not imply  NEC(BoA)  NEC(AaB) does not imply  NEC(BaA) that is NEC(BoA) does not imply NEC(AoB) NEC(BaA) does not imply NEC(AaB) These arguments are, of course, also arguments against AA-conversiontc and II-conversiontc, that is, they show: CONtc(AaB) does not imply CONtc(BaA) CONtc(AiB) does not imply CONtc(BiA)

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The failure of II-conversiontc means that  CiN would have to be rejected by Aristotle and presumably by Alexander. We have found no evidence that either one accepts it or  CoN. In addition one might expect Alexander to be suspicious of  CaN and  CeN, since they are incompatible with his apparent belief that if CON(P), P does not hold now. However, there are passages which suggest that Alexander did accept these principles. The strongest is perhaps his presentation at 211,2-17 of a proof of what he asserts to be: Celarent1(CNC) CON(AeB) NEC(BaC) CON(AeC) He assumes  CON(AeC) and transforms it into NEC(AiC). But NEC(BaC), so that (Disamis3(NNN)) NEC(AiB), contradicting CON(AeB). Alexander insists that in this case the conclusion really is CON(AeC) and not  NEC(AiC), since he can also rule out NEC(AoC); for NEC(AoC) and NEC(BaC) imply (Bocardo3(NNN)) NEC(AoB), which is incompatible with CON(AeB). This inference obviously only makes sense if Alexander is assuming ( CeN). This argument throws light on a difficult passage in which Alexander is discussing: Celarent1(UC N )

AeB CON(BaC)  NEC (AeC)

which Aristotle established indirectly by refuting NEC(AiC). In his comment Alexander invokes NCe: He himself indicates by what he says that it is necessary to transform ‘It is not contingent that A holds of no C’ in this combination into a particular affirmative necessary proposition . For ‘It is not contingent that A holds of no C’ is no less true when the particular negative necessary proposition is, but the proof goes through in the case of the former. (195,6-10) What Alexander says suggests that the only reason Aristotle didn’t make NEC(AoC) the hypothesis for reductio is that it would not enable him to derive a contradiction. He does not make clear what sense he would make of a justification of EAA1(UC N ). In the light of the passage we have just discussed it seems to us likely that underlying Alexander’s remark here is the idea that the conclusion of Celarent1(UC_) is not CON(AeC) because the premisses do not imply  NEC(AoC). This idea obviously presupposes ( CeN); cf. 197,12-22, 198,9-11, 205,29-30, 207,9-11. Other passages suggest the same thing but not so decisively. For example, Alexander writes:  ‘A holds of some C by necessity’ is not equivalent to ‘It is not contingent that A holds of no C’, which was transformed into it. For ‘It is not contingent that A holds of no C’ is also true when A does not hold of some C by necessity; for it is true that it is not contingent that walking holds of no animal, but not because it holds of some animal by necessity, but because it does not hold of some by necessity. (194,25-9) This suggests that  CON(AeC) means that either NEC(AiC) or NEC(AoC) ( CeN). But everything Alexander says here is compatible with his accepting only NCe. He can deny the equivalence of  CON(AeC) and NEC(AiC) simply

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on the grounds that NEC(AoC) alone implies  CON(AeC). Unfortunately Alexander says nothing to make explicit this idea (see also 205,16–206,11). However, the question whether Alexander recognizes  CeN or  CaN is made more difficult by the way in which he searches for terms to confirm the difference between, say,  CON(P) and NEC( P). He does this for  CON(AaC) and NEC(AoC) at 198,13-199,4. Obviously, he could simply verify NEC(AaC) and thus verify  CON(AaC) and falsify NEC(AoC), but he chooses to verify  CON(AaC),  NEC(AoC), and NEC(AiC). Again, however, we are not forced to conclude that he is presupposing  CaN, and, indeed, we suspect that he adopts his method because he is discussing Barbara1(UC‘C’), and he does not want terms which would falsify the possible conclusion which he is considering at the moment, viz.  NEC(AoC) (=  NEC (AaC)).

Appendix 6

Textual notes In this appendix we indicate places where Alexander’s text may have been different from that printed by Ross. Places where we have not followed the text printed by Wallies are listed on p. 52 above. Aristotle 26a17 30a15 30a21-2 30a27 30a27 30b7 30b18 30b19 30b24-5 31a1 31a10 31a17 31b17 31b20 31b29 31b31 31b39 32a5 32a7 32a8 32a17 32a18 32a22

Ei d’ ho men (Ross); hotan ho men (citation, 252,8 (Wallies, following the Aldine); the lemma at 58,24 has Ei d’ ho men) ton sullogismon (Ross); to sumperasma (citation, 128,6-7; the lemma at 123,26 has ton sullogismon) huparkhei ê oukh huparkhei to A (Ross); huparkhein ê mê huparkhein to A keitai (lemma, 127,25-6). gar (Ross); de (citation, 128,29;129,9; lemma, 129,8). toiouton (Ross); toiouton ti (lemma, 129,8, but toiouton in the citation at 128,29. de tou deuterou (Ross); de deuterou (lemma, 135,20). de hê (Ross); d’ hê (lemma, 137,22). ouk estai to sumperasma anankaion (Ross and the Aldine of Alexander); to sumperasma ouk estai anankaion (lemma, 137,22-3). eti d’ ei (Ross); eti ei (lemma, 138,3). d’ hexei (Ross); de hexei (citation, 135,29, but the lemma at 142,18 has d’ hexei). te kai (Ross); kai (lemma, 143,3). Alexander reports (144,4-5) that some copies have henos monou metalambanomenou after hê apodeixis. estin (Ross); esti (citation, 148,10). estin (Ross); esti (citation, 148,24). to de A (Ross); to A (citation, 149,4). dipoun ti katheudein ê egrêgorenai (Ross); dipoun ti mê katheudein (citations, 149,8 and 10-11). tauta (Ross); ta auta (citation 150,12; at 150,30 Wallies follows the Aldine in printing tauta where B and M have panta). zôion, meson zôion (Ross); zôion, dipoun meson (treated as a scribal blunder, 151,14-16). ean (Ross); an (lemma 151,32, but ean in the citation at 153,4). monon (Ross); monês citation, 153,2; the mss. divide on the citation at 152,26 and on the lemma at 151,33). legômen (Ross and the Aldine of Alexander); legomen (lemma, 156,1 and some mss. of Aristotle). d’ endekhesthai (Ross); de endekhesthai (lemma, 156,8). apophaseôn kai tôn kataphaseôn (Ross); kataphaseôn kai tôn apophaseôn (lemma, 157,11).

160 32b4 32b19 32b21 32b30 32b37 33a1 33a4 33a21 33b4 33b8 33b27 33b28-9 33b34 34a2 34a4 34a6-7 34a8-10

34a10-11

34a12 34a32 34a38 34b3 34b7 34b40-1 35a9-10 35b1 35b8-9 35b11

Appendix 6 de (Ross); dê (lemma, 161,27. Both Ross and Wallies print legômen in this line, although the mss. of Aristotle which Ross cites and the Aldine of Alexander have legomen). esti (Ross); estin (lemma, 164,16). ekeinôn (Ross); ekeinôs (citation, 165,4). enkhôrein (Ross); endekhesthai (lemma, 166,4 and following citations). kathaper kai en tois allois (Ross); kathaper epi tôn allôn (citation, 167,3-4). elegomen (Ross); legomen (citation, 167,27). mê endekhesthai (Ross); endekhesthai (citation 168,4). Ean (Ross); An (lemma, 169,15). houtô (Ross); houtôs (lemma, 171,14). touton ton tropon ekhontôn tôn horôn hoti oudeis (Ross); hoti touton ton tropon ekhontôn tôn horôn oudeis (lemma, 172,6-7). t’esontai pantes (Ross); te pantes esontai (citation, 245,23). to elatton (Ross); ton elattona (citation, 245,24). huparkhein (Ross); huparkhon (lemma, 174,32). d’ enantiôs (Ross); de enantiôs (lemma, 175,7). kai hoti ateleis (Ross); hoti kai ateleis (citation, 175,16 [aB (d); hoti ateleis M]). estai kai to B (Ross; kai omitted in three main manuscripts); kai to B estai (lemma, 175,20). to men dunaton, hote dunaton einai, genoit’ an, to d’ adunaton, hote adunaton, ouk an genoito (Ross); to men [A] dunaton, hot’ dunaton, genoit’ an, to de [B] adunaton, hote adunaton, ouk an genoito (citation, 177,4-6). hama d’ eiê to A dunaton kai to B adunaton, endekhoit’ an to A genesthai aneu tou B (Ross; the mss have ei for eiê); the lemma at 177,1-2 reads hama d’ ei to A dunaton kai to B adunaton, endekhoit’ an genesthai to A aneu tou B. At 177,8 Alexander cites the lines as hama de endekhoit’ an genesthai to A aneu tou B. to adunaton kai dunaton (Ross); to dunaton kai adunaton (lemma, 182,20-1). dunaton estai to auto (Ross); to auto estai dunaton (citation, 185,27). panti tôi G (Ross); tôi G (citation, 185,15 and most mss. of Aristotle). thentas (Ross); thenta (lemma, 187,10). huparkhon (Ross); huparkhein (lemma, 188,18 and one of the mss. of Aristotle cited by Ross). The words kai ouk estai to sumperasma anankaion were not known to Alexander (citation, 200,6-7). The word endekesthai was not read by Alexander (citations, 200,24-5, 27-8). kai dia tês antistrophês (Ross); di’ antistrophês (citation, 202,11). hotan de to mê huparkhein lambanêi hê kata meros tetheisa (Ross); Hotan de to mê huparkhein tini lambanêi (complete lemma 203,10). adioristou (Ross); aoristou (citation, 203,16 and some mss. of Aristotle).

Appendix 6 35b12 35b17 35b23 35b32-3 35b34 35b35 36a11 36a14 36a17 36a23

36a32 36b1 36b16 36b19 36b26 36b35 37a4 37a18 37a27 37a35-6

37a38 37b19 37b30 38a17 38a22 38a25 38b10 38b24 38b32

161

to elattona akron (Ross and the Aldine of Alexander); ton elattona akron (lemma, 204,1). hêper kapi (Ross); hê kai epi (citation, 204,21-2). Hotan d’ hê men ex anankês huparkhein hê d’ endekhesthai (Ross); Hotan d’ hê men ex anankês huparkhein ê mê huparkhein hê d’ endekhesthai (lemma, 204,30-1). to d’ endekhesthai (Ross); to de endekhesthai mê huparkhein (citation, 205,21-2). tou d’ ex (Ross); tou de ex (citation, 207,34). heteron gar to mê ex anankês huparkhein (Ross); heteron gar esti to mê huparkhein ex anankês (citation, 206,16-17). huparkhein (Ross); huparkhon (citation, 208,22 and one ms. of Aristotle reported by Ross). hôst’ oudeni ê ou panti tôi G to B endekhoit’ an huparkhein (Ross); hôst’ ou panti tôi G to B endekhetai huparkhein (citation, 209,22, on which see the note). kataphatikê (Ross); katêgorikê (lemma, 209,33, although Alexander uses kataphatikê in the commentary at 209,36). to A tôi G tini huparkhein (Ross). The mss read to A tôi G mêdeni huparkhein, and Alexander clearly did as well (210,21-34). But he indicates (30-1) that some manuscripts read to A tôi G tini [mê] huparkhein (mê bracketed by Wallies, following Waitz). kapi (Ross); kai epi (citation, 212,3). to en tôi katêgorikôi (Ross); en tôi katêgorikôi (citation, 213,30 and the majority of the Greek mss. used by Ross). apsukhôi tini (Ross); tini apsukhôi (lemma, 215,19). Phaneron (Ross); Dêlon (lemma, 215,29). lambanôsin (Ross and most mss.); lambanôntai (lemma, 217,29). endekhesthai (Ross and the Aldine of Alexander); endekhomenôi (lemma, 219,34). d’ ouden (Ross); de ouden (citation, 221,6). tini tôn A (Ross); tini tôi A (lemma, 224,8). diôrisamen (Ross); diôrikamen (lemma, 226,12). tethentos gar tou B panti tôi G endekhesthai huparkhein (Ross); Alexander (citations, 227,27-8; 228,20-1 and 25-6) does not have the inserted words, but he mentions (228,25-6) the possibility of reading tethentos gar tou B panti tôi G endekhesthai huparkhein endekhesthai (Ross); endekhomenou (lemma, 229,1). sêmainei (Ross with most manuscripts and the Aldine of Alexander); sêmainoi (lemma, 230,25). d’ hê (Ross); de hê (lemma, 233,13). gar (Ross): oun (lemma, 235,3). oud’ huparxei (Ross); oukh huparkhei (citation, 235,9 and some mss. of Aristotle). kai (Ross); kan (lemma, 235,31). oun (Ross); goun (citation, 239,13). kapi (Ross); kai epi (lemma, 240,12) anankaia hê (Ross); anankaia êi hê (lemma, 241,11; Alexander also cites (241,16) the text printed by Ross, although Wallies prints anankaia hê because of the lemma.

162 38b39 39a29 39a30 39b7 40a2 40a8-9 40a39

Appendix 6 tês katholou (Ross); katholou (lemma, 242,6 and one ms. of Aristotle reported by Ross). esti (Ross); estin (citation, 244,27, but the lemma at 39a28 has esti). te kai (Ross); kai (citation, 244,29). Ean (Ross); An (lemma, 245,1). proteron (Ross); katholou (citation, 249,3, and all mss. of Aristotle). endekhesthai kai (Ross); endekhesthai mê huparkhein kai (citation, 249,11 and the major manuscripts of Aristotle). ei ho (Ross); ei d’ ho (citation, 252,4)

Bibliography Alexander, in Top. (Wallies, Maximilian (ed.), Alexandri Aphrodisiensis in Aristotelis Topicorum Libros Octo Commentaria (CAG II.2, 1891)). Alexander, Quaest. (Quaestiones in Bruns, Ivo (ed.), Alexandri Aphrodisiensis Praeter Commentaria Scripta Minora (Supplementum Aristotelicum II.2), Berlin: Georg Reimer, 1892). Ammonius, in Int. (Busse, Adolf (ed.), Ammonius in Aristotelis De Interpretatione Commentarius (CAG IV.5, 1897)). [Ammonius], in An. Pr. (Wallies, Maximilian (ed.), Ammonii in Aristotelis Analyticorum Priorum Librum I Commentarium (CAG IV.6, 1899)). Barnes, Jonathan et al. (=Bobzien, Susanne, Flannery, Kevin S.J., and Ierodiakonou, Katerina) (1991), Alexander of Aphrodisias On Aristotle Prior Analytics 1.1-7, London: Gerald Duckworth. Becker, Albrecht (1933), Die Aristotelische Theorie der Moglichkeitsschlusse; eine logisch-philologische Untersuchung der Kapitel 13-22 von Aristoteles’ Analytica priora I, Berlin: Junker und Dünnhaupt. Bochenski, Joseph M. (1947), La Logique de Théophraste (Collectanea Friburgensia, nouvelle série, fascicule 32), Fribourg en Suisse: Librairie de l’Université. Bonitz, Hermann (1870), Index Aristotelicus, included in vol. 5 of Aristotelis Opera, Berlin: Georg Reimer, 1831-1870. CAG (Commentaria in Aristotelem Graeca, 23 vols., Berlin: Georg Reimer, 1882-1909). Denniston, J.D. (1954), The Greek Particles, 2nd ed., Oxford: Clarendon Press. Döring, Klaus (1972), Die Megariker. Kommentierte Sammlung der Testimonien (Studien zur antiken Philosophie, 2), Amsterdam: B.R. Grüner N.V. FHSG (Fortenbaugh, William W., Huby, Pamela M., Sharples, Robert W., and Gutas, Dimitri, Theophrastus of Eresus: Sources for his Life, Writings, Thought, and Influence (Philosophia Antiqua 54), 2 vols., Leiden and New York: E.J. Brill, 1992). Flannery, Kevin (1995), Ways into the Logic of Alexander of Aphrodisias (Philosophia Antiqua 62), Leiden and New York: E.J. Brill. Frede, Michael (1974), Die stoische Logik (Abhandlungen der Akademie der Wissenschaften in Göttingen, Philologisch-Historische Klasse, 3. Folge, 88), Göttingen: Vandenhoeck und Ruprecht. Graeser, Andreas (ed.) (1973), Die logischen Fragmente des Theophrast (Kleine Texte für Vorlesungen und Übungen, 191), Berlin and New York: Walter de Gruyter. Hintikka, Jaakko (1973), Time & Necessity; Studies in Aristotle’s Theory of Modality, Oxford: Clarendon Press. Hülser, Karlheinz (ed. and trans.) (1987-1988), Die Fragmente zur Dialektik der Stoiker, 4 vols., Stuttgart: Frommann-Holzboog. Lee, Tae-Soo (1984), Die griechische Tradition der aristotelischen Syllogistik in der Spätantike (Hypomnemata 79), Gottingen: Vandenhoeck und Ruprecht.

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Long, A.A. and Sedley, D.N. (1987), The Hellenistic Philosophers, 2 vols., Cambridge, England and New York: Cambridge University Press. Mates, Benson (1961), Stoic Logic, Berkeley and Los Angeles: University of California Press. Mignucci, Mario (1972), On a Controversial Demonstration of Aristotle’s Modal Syllogistic; an Enquiry on Prior Analytics A 15 (Universita di Padova. Pubblicazioni dell’Istituto di storia della filosofia e del Centro per ricerche di filosofia medioevale. Nuova serie 11) Padova: Editrice Antenore. Moraux, Paul (1973), Der Aristotelismus bei den Griechen, vol. 1 (Peripatoi 5), Berlin and New York: Walter de Gruyter. Moraux, Paul (1984), Der Aristotelismus bei den Griechen, vol. 2 (Peripatoi 6), Berlin and New York: Walter de Gruyter. Muller, Robert (1985), Les Megariques: fragments et temoignages, Paris: J. Vrin. Nemesius. (Morani, Moreno (ed.), Nemesii Emesini De Natura Hominis, Leipzig: B.G. Teubner, 1987). Patterson, Richard (1995), Aristotle’s Modal Logic: Essence and Entailment in the Organon, Cambridge, England and New York: Cambridge University Press. Patzig, Gunther (1968), Aristotle’s Theory of the Syllogism. A Logico-philological Study of Book A of the Prior Analytics, translated by Jonathan Barnes, Dordrecht: D. Reidel. Philoponus, in An. Pr. (Wallies, Maximilian (ed.), Ioannis Philoponi in Aristotelis Analytica Priora Commentaria (CAG XIII.2, 1905)). Repici, Luciana (1977), La logica di Teofrasto: studio critico e raccolta dei frammenti e delle testimonianze (Pubblicazioni del Centro di studio per la storia della storiografia filosofica, 2), Bologna: Il mulino. Ross, W.D. (1949), Aristotle’s Prior Analytics: A Revised Text with Introduction and Commentary, Oxford: Clarendon Press. Sharples, R.W. (1982), ‘Alexander of Aphrodisias: problems about possibility I’, Bulletin of the Institute of Classical Studies 29, 91-108. Sharples, R.W. (1987), ‘Alexander of Aphrodisias: scholasticism and innovation’, in Haase, Wolfgang (ed.), Aufstieg und Niedergang der römischen Welt, part II, vol. 36.2, Berlin and New York: Walter de Gruyter, 1176-1243. Simplicius, in Cael. (Heiberg, J.L. (ed.), Simplicii in Aristotelis De Caelo Commentaria (CAG VI, 1894)). Smith, Robin (trans.) (1989), Aristotle Prior Analytics, Indianapolis and Cambridge: Hackett. Stephanus, in Int. (Hayduck, Michael (ed.), Stephani in Librum Aristotelis De Interpretatione Commentarium (CAG XVIII.3, 1895)). SVF (Arnim, Hans von (ed.), Stoicorum Veterum Fragmenta, 4 vols., Leipzig: B.G. Teubner, 1903-1924). [Themistius], in An. Pr. (Wallies, Maximilian (ed.), Themistii Quae Fertur in Aristotelis Analyticorum Priorum Librum I Paraphrasis (CAG XXIII.3, 1894)). Wallies, Maximilian (ed.), Alexandri in Aristotelis Analyticorum Priorum Librum I Commentarium (CAG II.1, 1883). Wehrli, Fritz (1955), Eudemos von Rhodos (Die Schule des Aristoteles, vol. VIII), Basel: Benno Schwabe.

English-Greek Glossary This glossary lists most of the terms in the Greek-English index (which should be consulted in connection with this glossary), but eliminates some which occur infrequently and includes some not there when they are sometimes used to translate an English word in this glossary. absolutely: pantôs; haplôs absurd: atopos accident: sumbebêkos accidentally: kata sumbebêkos additional condition: prosdiorismos affirm (v.): kataphaskein affirmation: kataphasis affirmative: kataphatikos; katêgorikos affirmative proposition: kataphasis agree (v.): homologein; sunkhôrein alter (v.): allassein alteration: parallagê anaphoric reference (or use): anaphora antecedent: hêgoumenon antithesis: antithesis ask (v.): epizêtein; episkeptein assertion: phasis; legomenon; eirêmenon assume in advance (v.): prolambanein assumed, be (v.): keisthai categorical: katêgorikos cause: aition censure (v.): aitiasthai clear: dêlos; enargês combination: suzugia; sumplokê complete: teleios complete (v.): teleioun completion: teleiôsis conclusion: sumperasma; sunagomenon condition: diorismos conditional: sunêmmenon confirm (v.): pistousthai conflict (v.): makhesthai

congruous: katallêlos conjectural: stokhastikos consequent: hepomenon; lêgon consideration, under: ekkeimenos, prokeimenos, keimenos contain (v.): periekhein contingent, be (v.): endekhesthai contingent in the way specified: kata ton diorismon endekhomenon contradiction: antiphasis contradictory (adj.): antiphatikos contradictory (n.): antiphasis contrary: enantios conversely: anapalin convert (v.): antistrephein co-predicate (v.): proskatêgorein correct: alêthês; orthos counter-example: enstasis counterpredicate (v.): antikatêgorein deduce (v.): sullogizein define (v.) horizein definition: horismos; horos demand (v.): apaitein demonstration: apodeixis demonstrative: apodeiktikos denial: arsis deny (v.): anairein; apophanai; apophaskein destroy (v.): anairein; phtheirein destruction: anairesis destructive: anairetikos determinate: diôrismenos determination: diorismos determine (v.): horizein differ (v.): diapherein difference: diaphora different: diaphoros

166

English-Greek Glossary

directly: autothen; euthus disjoin (v.): apozeugnunai dissimilar in form: anomoioskhêmôn divide (v.): diairein division: diairesis do away with (v.): anairein; also do away with (v.): sunanairein ekthesis: ekthesis encompass (v.): perilambanein equally: ep’ isês; homoiôs equivalent, be (v.): antakolouthein; isodunamein; ison dunasthai; ison sêmainein; isos einai. establish (v.): kataskeuazein; elenkhein; deiknunai evident: phaneros extension, of wider: epi pleon extension, have a greater (v.): huperteinein external: ektos; exôthen extreme: akros fall (outside) (v.): piptein (ektos) falling under: hupo false: pseudês figure: skhêma find (v.): heuriskein follow (v.): akolouthein; hepesthai; sunagesthai for the most part: epi to polu genus: genos go through (said of a proof) (v.): proeinai; proerkhesthai; prokhôrein hold (of) (v.): huparkhein hold fixed (v.): phullattein holding: huparxis; huparkhein hypothesis: hupothesis hypothesize (v.): hupokeisthai immediately: eutheôs imply (v.): sunagein implication: akolouthia; akolouthêsis impossible: adunatos in general: holôs; katholou at 164,31 and 180,4 in itself: haplôs incomplete: atelês incongruous: akatallêlos

indefinite: aoristos indemonstrable: anapodeiktos indeterminate: adioristos indicate (v.): dêloun; deiknunai; endeiknunai; episêmainein indication: sêmeion individual: atomos; kath’ hekaston induction: epagôgê inference: sunagôgê infrequent: ep’ elatton inquire (v.): zêtein instrument: organon interchange (n.): hupallagê interchange (v.): metatithenai; allattein investigate (v.): episkeptein investigation: exetasis ipso facto: hêdê justification: pistis keep, keep fixed (v.): têrein known: gnôrimos last (term): eskhatos major: meizon material terms: hulê mean (v.): sêmainein minor: elattôn mixture: mixis modality: tropos name (n.): onoma name (v.): onomazein necessary: anankaios (‘be necessary’ sometimes represents dei) necessity: anankê negation: apophasis negative proposition: apophasis negative: apophatikos non-syllogistic: asullogistos objection: enstasis oppose (v.): antidiastellein opposite (be the opposite of) (v.): antikeisthai part: meros; morion particular: en merei; kata meros; epi merous peculiar feature: idion peculiarly qualified (individual): idiôs poiôs per se: kath’ hauto

English-Greek Glossary posit (v.): tithenai possible, be (v.): dunasthai possible: dunatos potentially: dunamei predicate (v.): katêgorein predication: katêgorêma premiss: protasis preserve (v.): phullattein; têrein; sôzein privative: stêretikos proof: deixis proposed (conclusion): prokeimenos proposition: protasis prove (v.): deiknunai provide (terms) (v.): euporein put together, be (v.) sunkeisthai quality: poion; poiotês reason: aitia, aition. reasonable: eikos; eikotôs; eulogôs reduce (v.): anagein reductio ad impossibile: eis adunaton apagôgê reduction: anagôgê. refer (v.): deiknunai; semainein reference: deixis refutation: elenkhos refutation, dialectical: epikheiresis refute (v.): elenkhein; epikheirein reject (v.): diaballein; paraiteisthai remain, remain fixed (v.): menein restrict (v.): horizein result (v.): sumbainein; gignesthai separate (v.) khôrizein set down (v.): paratithenai setting down: parathesis show (v.): deiknunai signify (v.): sêmainein similar in form: homoioskhêmôn simple: haplos simply: haplôs (kata psilên at 184,7) sound: hugiês

167

species: eidos specification: diorismos specified, contingent in the way: kata ton diorismon endekhomenon specify further (v.): prosdiorizein statement: axiôma straightforwardly: antikrus strict sense, in the: kuriôs subject (logical): hupokeimenon subject (of study): pragmateia substance: ousia syllogism: sullogismos syllogistic: sullogistikos take (v.): lambanein term: horos thereby: hêdê transform (v.): metalambanein transformation: metalêpsis true: alêthês (‘be true’ may represent alêtheusthai) true together, be (v.) sunalêtheuein under consideration: prokeimenos understand (v.): eidenai; akouein; exakouein; prosexakouein; prosupakouein; hupakouein. understandable: gnôrimos understood: gnôrimos unique opposite, be the (v.): idiôs antikeisthai; idiai antikeisthai universal: katholou (katholikos at 125,27) unqualified: huparkhôn unqualifiedly: huparkhontôs usual: epi to pleiston view (outlook): doxa weaker: kheirôn whole: holos without condition: adioristôs without qualification: haplôs yield a conclusion (v.): sunagein

Greek-English Index This index refers to the page and line numbers of the CAG text and covers Alexander’s commentary on Aristotle’s Prior Analytics 1.8-22, translated in this series in two volumes, of which this is the first. The index includes a range of logical and philosophical terms and a few commentator’s expressions used by Alexander. Only the first few occurrences (followed by ‘etc.’) of the most common terms are given. We have sometimes left out of account non-technical uses of a word, and we do not cite occurrences in the lemmas or in Alexander’s quotations of Aristotle. The translations indicated are our usual but not invariant ones. adiaphoros, duplicating (in the phrase adiaphorôs perainontes, a kind of argument considered by the Stoics), 164,30 adioristos, indeterminate (said of propositions which do not specify quantity), 159,22; 160,22; 170,29; 215,4; 234,22; 241,21; 244,32; 248,31; 254,8. The adverb adioristôs is translated ‘without condition’ at 179,32 and as ‘in an indeterminate way’ at 215,17 (a difficult passage). See also diôrismenos adunatos, impossible. We give a few of the many occurrences not in the phrase apagôgê eis adunaton, 128,32; 131,13.14(2).17(2), etc. aitêma, postulate, 126,11 aitia, reason, 130,6; 133,18; 148,23; 160,16; 162,14.23; 164,22; 170,31; 171,19; 202,10; 220,1; 253,12. Cf. aition aitiasthai, to censure, 223,30; 232,27 aition, reason, 120,13; 144,13; 149,28; 159,10; 188,25; 194,19; 211,24; 218,3; 223,26; 251,38; 253,30. We have translated aition ‘cause’ at 163,14 and 178,31. Cf. aitia akatallêlos, incongruous, 250,24

akolouthein, to follow, 130,8; 131,17; 133,14; 135,30; 138,28, etc. We note two occurrences at 157,29 which express the relationship of predicate to subject and have been translated ‘apply to’, and an occurrence at 235,21 which we have translated ‘cohere’. See also epakolouthein akolouthia, implication, 158,18; 176,2(2).24; 178,13.22; 182,10.23.29; 183,8.13.14; 184,22.25; 185,1.12.26.33. We note three difficult occurrences: 161,18; 177,6 and 221,24 akolouthêsis, implication, 184,26 akros, extreme (term), 124,31; 125,20; 148,12; 164,22; 171,19.23; 188,21; 189,6.15; 231,19; 237,4; 239,9; 246,7; 253,33. There are more occurrences of this word in Aristotle’s presentation of modal logic than in Alexander’s commentary on it (excluding quotations). Only at 171,23 does Alexander clearly use the neuter substantive to horon; all other occurrences can or must be read as masculine with an explicit or implicit horos; by contrast Aristotle would seem to use the neuter substantive uniformly. (The entry akron in the

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Greek-English index of Barnes et al. should be corrected.) At 36a21-2 Aristotle refers to a major premiss as the premiss apo (relating to) the major extreme. In the last two uses of akron Alexander characterizes a conclusion in which the minor term is the predicate as apo the minor extreme alêthês, true, correct, 119,24.25; 126,10.11(2).12, etc. alêtheuesthai, to be true, 174,14; 183,2.4.12; 198,30; 203,20; 205,37; 206,5; 211,13.15; 231,36; 233,8. We wish to call attention to the italicized passages in which Alexander uses the expression alêtheuesthai kata such and such to mean something like ‘to be true in the case of’. In 233,8 Alexander uses the preposition epi to the same effect anagein, to reduce (one syllogism to another), 136,3; 137,4.26; 145,28; 149,21; 151,3; 242,26; 243,15; 247,11; 252,19 anagôgê, reduction (of one syllogism to another), 123,17; 136,11.16; 138,5; 145,26; 154,9; 219,22; 242,22; 246,4 anairein, to do away with, to destroy, 131,19; 139,8; 171,26; 172,27, etc. We have rendered anairesis (182,16; 188,10; 192,32; 198,1; 223,31) and anairetikos (172,11.23.26; 195,2, etc.) accordingly. Barnes et al. give ‘to destroy’, ‘to reject’, ‘to cancel’ for anairein and ‘rejection’ for anairesis. (At 174,17; 196,28.29.35; 197,2.6; and 198,29 we translate anairein ‘deny’.) analutika, Analytics. Occurs in references to Theophrastus’ Prior Analytics at 123,19 and 156,29. Otherwise there are no occurrences of forms of analuein, analusis, or analutikos in this section of the commentary anankê, necessity (ubiquitous in the phrase ‘by necessity’ (ex

anankês)), 119,11; 121,25.26.27. 29(2).31.32.33, etc. Reasonably rare otherwise and usually in quotation or paraphrase of Aristotle, it occurs only in the nominative and is often translated ‘It is necessary’. 131,19.30; 157,17; 158,29-31(4), etc. anankaios, necessary, 119,9.10.14.19.23, etc. anankaiôs, necessarily, 119,22.23; 130,8.9.18; 143,30; 149,5; 155,7.24; 197,1; 207,25; 222,11 anapalin, conversely. Alexander uses this adverb when features of two propositions are interchanged, modality (146,12; 187,16; 232,24), quality (229,33; 230,18; cf. 37b11), or both (238,11.17; cf. 37b11). At 182,14 he applies the adverb to the interchange of antecedent and consequent anaphora, anaphoric reference or use. Four occurrences in 179,20-9. The word is used non-technically at 165,5, where we translate ‘application’ anapodeiktikos, indemonstrable, used at 124,6 to refer to a complete syllogism. See the note ad loc. anomoioskhêmôn, dissimilar in form (= quality), 170,28; 254,9. Cf. homoioskhêmôn antakolouthein, to be equivalent, 158,24; 159,4.23.25.31; 160,15.17.23.25; 223,29. See also isos einai anti, instead of; to stand for. We mention this term because Alexander uses it in phrases such as ‘Aristotle says a instead of b’. These phrases can sometimes mean something like ‘the expression a means b’, and sometimes they seem to mean something like ‘he says a, but he means b’, and sometimes it is hard to tell what exactly Alexander has in mind. The

Greek-English Index following are the relevant occurrences. 127,28; 129,9; 144,21; 147,27; 149,6; 152,20.23; 161,24; 180,28; 184,31(2); 185,5; 186,24.34; 200,2; 203,4; 210,28.30; 221,6; 224,30.35; 227,28; 228,24; 228,29; 242,23.24; 249,1; 254,13 antidiastellein, to oppose, 152,24 antikatêgorein, to counterpredicate, 252,12 antikeisthai, to be the opposite of, 121,4.5; 126,33; 127,7, etc. Normally it is clear that the opposite of a proposition is what we would call its contradictory. But sometimes Alexander underlines this fact by speaking of the contradictory opposite (antiphatikôs antikeisthai; 187,8; 188,4; 199,22; 208,24; 237,25; 238,4). The unique (or unique and proper) opposite (idiôs (kai oikeiôs) antikeisthai; 197,24; 199,13; 207,4; 211,16), or the opposite in the strict sense (kuriôs antikeisthai; 198,24). At 237,22-37 Alexander is driven by an obscure remark of Aristotle’s to consider including contraries among opposites. Cf. antiphasis antikrus, straightforwardly, 197,8 (equivalent); 216,11 (contingent); 216,12 (unqualified) antiphasis, contradictory (157,16-30(5); 158,5; 159,29.30; 188,1; 196,15.25); contradiction (141,4; 187,28; 195,20). Alexander prefers antikeimenon, which we have translated ‘opposite’, to antiphasis; see antikeisthai antiphatikos, contradictory, 157,20; 237,30. Alexander uses the adverb antiphatikôs with antikeisthai at 187,8; 188,4; 195,21; 208,24; 237,26.34; 238,4; we have translated ‘be the contradictory opposite’ antistrephein, to convert. Alexander can say that a proposition converts (120,20.22; 126,7; 138,13; 139,4, etc) or that

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it is converted (136,2-20(8), etc.) or that a person converts it (122,13; 130,30; 135,4; 137,15; 142,9, etc.). In discussing relations between propositions P and Q he can say that P and Q convert, that they convert with (dative) each other, that P converts with (dative) Q, that P converts pros Q, and that P converts eis (into; 168,25; 201,34; 203,8; 211,21; 245,11; cf. 239,14) Q. For the most part it is impossible to tell whether saying that P converts with Q is also saying that Q converts with P. In five cases (222,5.7; 232,7; 236,11; 243,16) the relation does not appear to be symmetric, and we have translated the dative by ‘from’. Pros has caused us the most difficulty; we have chosen to translate ‘with respect to’. In two cases (128,16 and 160,13) ‘from’ seems correct (cf. the lexicon in Barnes et al.), but in the remaining cases (220,27; 222,17.21.35; 232,6) ‘to’ or ‘with’ (symmetric) seems likely. See also antistrophê and horos antistrophê, conversion, 120,16.23.25; 121,23; 122,1, etc. Alexander twice (200,25 and 224,25) speaks of conversion of one proposition into (eis) another, but otherwise he just speaks of conversion or of conversion of a proposition. See antistrephein and horos antithesis, antithesis, 160,5. Used by Aristotle at 32a32 in a passage quoted once by Alexander (159,17) aoristos, indefinite (a kind of contingency mentioned by Aristotle at 32b10-11). The word occurs 11 times between 163,1 and 165,18 and again at 169,5 and 183,31. Alexander uses the noun aoristia at 164,21. At 203,16 and 21 he uses the word in connection with 35b11 apagôgê, reductio. Except at 216,16

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this word occurs only in the phrase hê eis adunaton apagôgê, which we translate reductio ad impossibile. 120,28; 121,3; 123,22; 126,29; 127,4, etc. There are very occasional variants, hê di’ adunatou deixis, translated ‘proof by impossibility’ (134,17; 189,36; 191,25; 248,1. Cf. hê deixis hê eis adunaton at 175,24, and dia tou adunatou, translated ‘by means of the impossible’ (175,14-17 (3 times; cf. 34a20), 202,11 (cf. 35a40)). Alexander once uses the phrase apagein eis adunaton (216,14) apaitein, to demand, 220,2 apo. Alexander uses this preposition only 14 times in this section of the commentary. The interesting occurrences are at 246,7 and 253,32 where Alexander characterizes a conclusion as apo tou elattonos akrou, and we translate ‘with the minor extreme as predicate’ apodeiktikos, demonstrative. This word is used only at 164,15 (lemma) and 165,14 (explanation of the occurrence in Aristotle). Cf. apodeixis apodeixis, demonstration. In this section of the commentary all but one occurrence of this word is in a quotation of Aristotle or explanation of a quotation in which it occurs. 130,26; 144,4.21(2); 203,17; 204,12; 204,21; 230,34; 241,22; 249,1.2. At 164,19 and 29 Alexander uses it in connection with a passage in which Aristotle speaks of an apodeiktikos syllogism. The term apodeiknunai (demonstrate) does not occur in this section of the commentary. Alexander prefers deixis (proof) and deiknunai (prove) apodidonai, to give (the reason or definition), 160,17; 167,30; 170,31; 174,15; 175,28; 182,29 apophasis, negation (136,24.26;

157,20-30(8), etc.); negative proposition (138,22; 158,25; 159,28; 160,23.27; 161,12.30; 229,29; 237,20.23.28.33; 238,6.8; 243,33; 245,31, 251,23). See the appendix on affirmation and negation apophanai, to deny, 156,23.25.26 apophaskein, to deny, 138,31; 195,22; 218,22 apophatikos, negative, 120,22.26.27; 121,5.15.17, etc. At 159,21 and 218,21 Alexander uses the term to apophatikon to refer to what we might call the negation operator; there we have translated ‘what negates’. See the appendix on affirmation and negation apozeugnunai, to disjoin, 124,19(2).20; 220,14-21(7) arkhê. In this section of the commentary the word is used only in its ordinary sense of ‘beginning’ (151,16; 168,33; 195,14; 200,31; 216,12; 220,9) or idiomatically in the phrase tên arkhên (‘at all’ 152,12; 169,33; 234,10). Similarly for arkhesthai (167,3; 232,3) arsis, denial, 131,20 asullogistos, non-syllogistic (adjective applied to a pair of premisses which do not yield a conclusion). 125,18; 135,2; 141,25.27; 165,22, etc. See also sullogistikos atelês, incomplete, 173,2.14.18; 174,10; 202,9.10; 206,25; 217,25; 242,21(2); 254,15. Cf. teleios atomos, individual, 122,32.36. The word is translated ‘uncut’ at 184,9-17 (5 occurrences) atopos, absurd, 177,6; 178,25; 218,13; 224,14.17 axiôma, statement (all occurrences in this part of the commentary reflect Stoic usage), 177,31; 179,32; 180,2.13; 181,4.32 autothen, directly, 169,22; 174,7; 245,32

Greek-English Index deiknunai (1), to prove, to show. Alexander uses this verb and the noun deixis (‘proof’) with very great frequency mainly in connection with Aristotle’s validation or rejection of combinations of premisses, 120,15.16.18.25.30, etc. Sometimes ‘prove’ or even ‘show’ seemed too strong. We have translated deiknunai ‘indicate’ at 128,5; 147,19.20; 148,25; 149,7.11; 150,11; 158,18; 165,2. Other variants of no real significance are ‘establish’ at 141,3, 156,4, 159,8 and 238,16, ‘to yield conclusions’ at 165,11, ‘deal’ at 200,23, ‘argue’ at 200,18 and 203,25 deiknunai (2), to refer, 177,28; 178,6.17(2); 179,14.24; 180,33; 181,4.7.11.12; 182,6. deixis (2), reference. 177,32.33; 178,18.19; 179,26; 181,20. These passages all occur in the context of a discussion of Chrysippus’ use of propositions such as ‘If Dion has died, he has died.’ In the first and last of the passages Alexander speaks of the referent as receiving (anadekhesthai) a deixis deiktikos, 122,26; 146,9; 147,20; 149,7.11; 191,17; 215,6; 230,14; 238,16; 250,21. We have translated with forms of ‘prove’, ‘show’, and ‘indicate’; see deiknunai (1) dêlos, clear, 119,12.25; 120,30; 139,16; 143,1, etc. dêlotikos, see dêloun, 125,12; 137,18; 154,11; 159,17; 166,15; 167,15; 192,35; 204,22; 208,16; 247,5 dêloun, to indicate (119,18.23; 120,3; 122,7, etc.); make clear (154,8; 163,29; 175,28; 220,7; 232,29). This verb is translated ‘show’ at 151,17 and ‘express’ at 200,24. See also dêlotikos diaballein, to reject (a proof, a combination, a method of proof; Barnes et al. render ‘to disprove’).

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223,15; 232,23; 238,20.34.36; 249,35. Alexander uses the adjectival form adiablêtos at 238,35 diabolê, showing false, 178,9 diapherein, to differ, be different, make a difference, 119,14; 135,8; 155,29; 156,18; 158,27; 189,30; 198,14; 206,14; 214,4; 245,30 diaphora, difference, 119,13.18(2).25; 124,1; 125,30; 129,19; 135,12; 137,6; 185,13; 194,33; 216,2 diaphoros, different, 170,28 diairesis, division, 141,1; 161,32.33 diairein, to divide, 224,13 diôrismenos, determinate (opposite of adioristos), 159,24; 160,23; 222,3. This is the only form of diorizein used by Alexander in this section of the commentary diorismos, specification, determination, condition. Alexander’s main uses of diorismos are given under kata ton diorismon and meta diorismou. At 156,3 he refers to the modal operators as diorismoi (‘specifications’); at 160,9 and 22 he speaks of the quantifiers as diorismoi (‘quantitative determinations’); and at 189,31 he speaks of adding a temporal diorismos (specification) to a proposition. At 159,24 and 160,23 he refers to quantitatively definite propositions as diôrismenos diphoroumenos, duplicated (a kind of argument considered by the Stoics), 164,29 doxa, view, position, 126,9; 127,15; 140,19; 247,39 dunamei, potentially, 154,8; 179,15; 218,23.33. At 184,26 we have rendered dunamei with the word ‘meaning’. This is the only form of dunamis in this section of the commentary dunasthai, to be possible (sometimes translated by ‘can’), 123,7; 125,28; 129,28; 130,4;

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134,21, etc. dunatos, possible (sometimes translated by ‘can’). Over 90% of the occurrences of dunatos are in the commentary on chapter 15. We list the few which are not: 139,3; 157,8; 160,8(2); 168,13; 220,24; 224,4; 229,9.10 êdê. Alexander uses this word in its ordinary sense of ‘already’, e.g. at 141,2. But he also uses it to convey a notion of implication and non-implication; for example he might say something like ‘Just because P, not êdê Q.’ We have frequently translated this êdê as ‘thereby’, but we have also used ipso facto, ‘in fact’, and other terms. Examples of this use at 119,24; 125,23; 131,19; 140,10; and 156,22 eidos, species, 122,28; 162,1 eikos, reasonable, 129,28; 160,20. See also eikotôs and eulogôs eikotôs, reasonable, reasonably, 122,25; 123,3; 124,31; 160,20; 174,18; 197,23; 203,12. See also eikos and eulogôs ekkeimenos, under consideration (usually applied to a premiss combination), 121,18; 126,31; 128,23; 137,27; 138,27; 175,30; 187,23; 199,19; 201,13; 227,17; 229,4.6.13; 230,5.7.11; 236,11.37; 237,16; 248,1. The word is applied to the setting out of terms at 237,18. This is the only form of ekkeisthai used by Alexander in this section of the commentary. See also keisthai and prokeimenos (2) ekthesis, ekthesis (a method or methods of proof used by Aristotle), 122,17.24.28.31; 123,4.9.20; 143,14; 144,25.33; 151,26). The term is applied to the setting down of terms at 173,8 and to a combination being considered at 176,7 (cf. ekkeimenos) ektithenai, set out (cf. ekthesis),

121,16; 122,19.27.29; 123,12.14; 227,14 (technical uses only) elattôn (elassôn at 231,24), minor (usually said of a premiss, sometimes of a term), 120,26; 124,23.31.32; 125,10, etc. We have translated ‘less’ at 124,12 and 17, where it is used as a synonym of ‘weaker’ (kheiron) in a description of the peiorem rule elenkhein, elenkhos In this section of the commentary Alexander mainly uses these words in connection with the use of terms as counterexamples. Sometimes we have used ‘refute’ or ‘refutation’ to render elenkhein (129,23; 134,27; 204,11) and elenkhos (171,22; 229,36; 238,13). But sometimes we have translated the former as ‘establish’ (126,13; 129,27; 211,29; 215,28; 230,6) or ‘show’ (139,9) and the latter as ‘way of establishing’ (204,23). At 221,12 we have translated elenkhôn kai deiknus as ‘showing’ en merei, particular, 148,2; 149,27; 150,4; 170,3.6; 203,11.19; 214,29; 215,24; 219,1; 230,3(2); 233,32; 234,11; 241,2; 242,9; 251,37; 252,36. See also kata meros and epi merous enallax, in alternation; in this section of the commentary the word occurs only in a quotation of Aristotle at 204,9. At 230,4 Alexander uses the participle enêllagmenos enantios, contrary, 159,29.30; 221,16-34(6); 227,30.34. At 251,25 Alexander uses the term loosely, and we have translated ‘opposite’ enargês, clear, 171,21 endeiknunai, to indicate, 128,30; 164,25; 165,2; 233,20 endein, to be missing (from the text), 239,1 endeixis occurs just once (133,32) in the commentary, with the sense of ‘proof ’ endekhesthai, to be contingent

Greek-English Index (exceptions noted in translation), 119,10.12.19.24; 120,3, etc. endekhomenôs, contingently, 147,19; 149,10; 167,19.22; 192,34; 194,17; 210,2; 220,18.21.23; 221,2; 241,28 enkhôrein, may, might, 128,29; 138,32; 154,2; 160,20; 165,4; 185,17. This word occurs more often in Aristotle’s discussion of modal logic than in Alexander’s enstasis, counter-example (227,8); objection (247,30). The verb enistasthai does not occur in this section of the commentary ep’ elatton, infrequent, 163,5.7.10.13.18.22; 169,2.4; 183,31 ep’ isês. Alexander uses this phrase four times (163,2.9.18.29) to describe a proposition which is as often true as it is false (by contrast with what is epi to polu or ep’ elatton; in those places we translate it ‘equally balanced’. In the same context (164,1) he speaks of two propositions being ep’ isês (equally) true. At 125,20 he speaks of two terms being ep’ isês (co-extensive). At 163,23 and 28 Alexander refers to equal balance as to hopoter etukhen and to hôs hopoter etukhe

epagôgê, induction, 159,31 epakolouthein, to follow, 129,33; 196,17; 208,24 epharmozein, to apply to, 125,28; 157,5; 161,32; see also harmozein epi merous, particular (of i- and o-propositions and of syllogisms with such conclusions), 120,26.27; 121,5.15.20.22, etc. See also kata meros and en merei, which we also translate ‘particular’. In the modal logic chapters Aristotle uses epi merous once, kata meros 12 times, and en merei 45 times. In his commentary on that text Alexander uses kata meros once. His overwhelming preference is for epi merous, which he uses

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close to 300 times. Many of the 30-odd occurrences of en merei are in quotations or paraphrases of Aristotle epi to pleiston, usually (synonymous with epi to polu), 162,9.13.28; 163,1.4.11.24.32.33; 164,2; 168,32; 169,1(2).7; 183,30. At 163,5 Alexander uses epi pleiston with the same sense, and at 163,17 he uses epi pleiston in a more general sense epi pleon, of wider extension (of terms, 178,17 (pleion).21.32; 188,25; 190,21. Cf. 125,20 where ep’ isês is translated ‘co-extensive’); true in more cases (of propositions, 178,10; 179,7; 263,17); at greater length (of discussions [non-technical]), 188,17; 207,35; 249,38; 150,1). See also huperteinein epi to polu, for the most part (synonymous with epi to pleiston, which we translate ‘usually’), 162,2.6.7.12.30; 163,23; 165,13.14 epikheirein, refute, 180,12 epikheirêsis, dialectical refutation, 180,9 epipherein. Alexander usually uses this word to introduce what Aristotle goes on to say; in these cases we have translated it ‘to add’ (125,14; 137,16; 140,7; 152,25; 153,30; 163,30; 166,2; 222,1; 243,6). In three passages (124,32.33; 174,25) Alexander uses it for the relation of predicate to subject in a conclusion; there we have translated it ‘to apply’ epistêmê, knowledge, science. Aristotle uses the word at 32b18 and Alexander repeats it at 164,18. Otherwise epistêmê occurs only as a term in counterexamples at 124,26-7; 195,31-2; 199,24-5 epizêtein, to ask, 131,29; 144,23; 155,3; 213,11; 217,8; 218,7; 222,16; 232,10; 240,4; 244,26; 249,34; 253,17

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eskhatos, last (term), 124,33; 125,1-2(4); 171,23; 189,14; 191,6.11.16; 202,3; 237,3; 251,14 eulogôs, reasonable, 119,9; 144,23; 158,19; 163,18; 217,8; 218,16; 239,21. See also eikos and eikotôs euporein, to provide, 231,13 euthus/eutheôs, directly, immediately, 129,3; 134,29; 223,17 exetasis, investigation, 125,29; 145,9; 232,32 exôthen, external, from outside, 132,2; 174,27; 175,17; 181,27; 184,8; 187,22; 202,5 genesis, coming to be (163,20; 182,27(2).31.32; 183,25.31); production (136,1) genos, genus, 161,33 gnôrimos, known (121,9; 123,23; 136,24; 143,5; 149,32; 170,20; 174,9; 210,27; 227,17; 233,34; 254,9); understandable (157,14.32); understood (228,31) gnôsis, understanding, 120,32 haplos, simple, 121,10; 184,26; 219,23. Cf. haplôs haplôs, simply (frequently = huparchontôs), 122,19; 128,20.27.28; 130,23; 131,1; 139,9; 140,3; 142,5.13; 143,23.23; 150,25; 152,28; 248,19. Without qualification (often contrasted with meta diorismou; see 139,27-141,7). 151,34; 155,22; 169,10; 179,32.33.34(2); 180,1; 185,30; 191,22; 201,21; 246,13; 251,20. In itself, 158,32; 159,2; 218,21. Absolutely, 217,25. Cf. haplos harmozein, to fit (135,18; 185,1); to apply to (150,29); to verify (230,5); see also epharmozein hêgoumenon, antecedent, 176,3.31; 177,21.22.29, etc. At 178,31 Alexander uses the verb hêgesthai to mean ‘to be the antecedent’ hepesthai, to follow (see also hepomenon), 129,11; 131,17.23; 133,20.23, etc. hepomenon, consequent (sometimes

‘consequence’), 156,20; 157,7; 165,16; 176,24.31, etc. Cf. lêgon heuriskein, to find, 125,19; 131,28; 145,11.15.16; 155,17; 186,35; 190,30; 197,22; 198,15; 205,30; 230,12; 236,23; 248,26 holos, whole, 121,32; 126,8; 158,32; 162,1; 196,21; 248,36 holôs, in general (oud’ holôs, never, not at all), 119,21; 125,8; 155,15; 163,15; 165,10; 171,12; 184,13; 185,17; 218,24; 229,5; 236,34; 238,10; 239,21 homoioskhêmôn, similar in form, 166,9; 167,3.5; 170,27. Aristotle uses homoioskhêmôn six times in the Prior Anaytics to apply to the premisses in a combination. In five of them (27b11 and 34; 33a37; 36a7; 38b6) it means ‘having the same quality’, but at 32b37 it means ‘having the same modality’. In commenting on this passage at 166,5ff. Alexander uses the term in the same way, once (166,9) applying it to combinations rather than their premisses; there we have translated it ‘with premisses similar in form’. Cf. anomoioskhêmôn. See also 86,12 with the note in Barnes et al. homologein, to agree, accept, 176,12; 181,32; 212,20; 218,25; 247,29 horismos, definition. Aristotle uses this word at 32b40 and 33a25, and it occurs in the commentary on the relevant passages (167,17-30(4) and 169,27-30(3)). Otherwise it occurs at 158,19; 170,19; 174,15.33; 175,6.28; 177,12 horizein, to define (156,12-13(3); 157,13; 161,3.12; 172,28), to restrict (temporally) (188,21-193,15(22); 217,19; 232,20; 233,5); to determine (163,16.17); to specify (161,31) horos, term, 120,1.4; 124,32; 125,21; 127,30, etc. We note five places in which Alexander distinguishes

Greek-English Index ordinary conversion, which involves interchanging subject and predicate terms, from such things as AE-transformationc, 159,14 (antistrophê kata tous horous); 164,14 (hê tôn horôn hupallagê); 173,23 (antistrophê kata tous horous); 221,3 (antistrephein tois horois); 222,7 (antistrephein kata tous horous). We have translated horos ‘definition’ at 182,28 hugiês, sound. This is usually a fairly general word of commendation for a statement or piece of reasoning (122,16(2); 125,33; 127,3.15(2); 139,9 (wrong, ouk hugiôs); 144,17; 147,23; 155,3.4; 157,30 (true); 159,9; 176,11; 177,27; 178,8; 196,20 (wrong, ouk hugiôs); 209,6; 216,7; 223,15.21). It is applied to conditionals or ‘implications’ at 178,13-29(3) and 196,12 hulê. We have signalled occurrences of this word (standardly translated ‘matter’) with the phrase ‘material terms’. It occurs most commonly with the preposition epi, which we have usually rendered with some form of the verb ‘to use’. 124,21; 125,4.16.19.23.25; 126,13; 145,9.15.16; 198,16; 198,29; 203,34; 204,22; 208,18; 215,24; 222,8; 236,26; 237,3.28.32. The occurrences without the preposition epi are 125,28; 190,8; 215,15.23; 238,36. At 164,30 Alexander refers to an obscure Stoic argument called apeiros hulê (infinite matter) hupallagê, interchange (of terms in conversion, 164,13; 220,7; of the modalities of two premisses, 175,13) huparkhein, to hold (of), 119,11(2).12.22.23(2).24, etc. huparkhontôs, unqualifiedly, 124,27; 129,25; 130,16.18.20; 132,8; 133,24; 134,29.31; 143,30; 144,9.18; 145,2; 146,6; 147,24.28;

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149,3.19; 155,7; 166,21; 232,14 huparkhôn, unqualified, 119,18.22; 120,3.5.7.14, etc. huparxis, holding, 184,23.24; 185,11; 197,2(2) huperteinein, to have a greater extension (of terms), 170,32.35; 171,19; 190,30.32; 191,3 hupokeisthai, to hypothesize, 127,2; 131,1; 133,26; 176,25; 184,18, etc. Cf. hupotithenai. We have translated hupokeimenos as subject at 122,25; 126,5; 129,34; 138,31; 184,7; 222,3.4; 252,11 hupothesis, hypothesis, 126,10; 130,24; 131,16; 134,11.14, etc. See also hupotithenai and hupokeisthai hupotithenai (We use ‘hypothesize’ or some closely related expression for this verb, which Alexander consistently uses in connection with arguments by reductio and arguments resembling them), 121,6; 131,10.12.14; 132,3, etc. One problematic exception is at 228,4 where we have used ‘suppose’. See also hupokeisthai and hupothesis idion, peculiar feature (Barnes et al. use ‘proper characteristic’), 152,13; 157,2.5.8; 158,24; 159,33; 161,7.14; 168,27; 222,4. There are informal uses of idiôs at 152,27 (‘just’) and 214,11 (‘properly’) and idiai at 167,4 (‘on its own’) idiôs poios, peculiarly qualified (individual), 179,11.12; 180,34; 181,17.18.26.30. For a discussion of this Stoic notion see Long and Sedley (1987), vol. I, pp. 166-79 idiôs antikeisthai, to oppose uniquely (i.e., be the contradictory of), 197,23.26; 198,11; 199,12; 207,4; 211,16. Cf. 223,26 (idiai antikeisthai) and 214,11 (idiôs sunagomenon) isodunamein, to be equivalent to, 160,7; 205,36; 234,18. See also isos einai ison dunasthai, to be equivalent to

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(with dative), 136,26; 140,9; 196,15.27; 197,8.32; 200,3; 205,25.32; 214,9; 218,7.8; 223,7; 226,8; 228,27; 229,18; 239,25; 243,33. See also isos einai ison sêmainein, to be equivalent, 151,12. Cf. 166,31 ison kai tauton sêmainein, ‘to mean the same thing as’. See also isos einai isos einai, to be equivalent to (with dative), 122,26; 136,28; 147,22; 160,3; 164,10 etc. See also isodunamein, ison dunasthai, ison sêmainein, isos kai autos. isos is translated ‘equal’ only at 163,10.23 and 178,33-5. The phrase ep’ isês is translated ‘co-extensive’ at 125,20 isos kai autos, equivalent to and the same as (with dative), 194,22; 195,13; 197,15. See also isos einai kata meros, particular, 230,1 (said of a conclusion). See also epi merous and en merei kata sumbebêkos, accidentally, 163,14; 223,33 kata ton diorismon (endekhomenon), (contingent) in the way specified. Alexander’s way of referring to contingency as characterized by Aristotle at 32a18-20. We list all the occurrences of this stock phrase, excluding those in quotations of Aristotle: 161,5.11; 174,5.12.30; 190,29; 191,9; 194,12; 196,7.13.20.25; 197,3; 198,6.15.20; 199,5.8.12; 200,35; 205,8.20; 208,5; 209,10; 210,3.7; 211,9; 212,34; 216,6.8.10; 219,28; 220,18; 221,5; 222,8; 226,13.17; 231,35; 232,2.37; 233,2.18; 234,3.5.33.35; 235,8.10; 236,7; 239,7.11; 240,23; 242,10; 243,1.4.10; 245,5.35; 246,23.34; 249,12; 250,22.23.38; 253,27; 254,21. We note also endekhomenon ek tou diorismou at 223,11 katallêlos, congruous, 129,9; 250,25 kataphasis, affirmation (157,19160,30(14); 164,7; 168,28;

196,30.31; 206,2; 207,1; 218,24; 222,24.26; 229,18). Affirmative proposition (158,25.28; 160,23.27; 161,29; 218,26; 220,8; 229,28; 237,25.30.31.33; 238,6.7; 239,34.37; 251,24). See the appendix on affirmation and negation kataphaskein, to affirm, 218,22 kataphatikos, affirmative, 120,21(2); 121,7-8(3), etc. See the appendix on affirmation and negation kataskeuazein, to establish, 131,18; 134,1.8; 176,11; 182,13; 188,11; 195,11; 197,27; 198,12. The noun kataskeuê occurs at 184,5 and 221,8 katêgorein, to predicate, 126,4; 130,2-21(10); 146,17.24; 156,15.27; 160,26.30; 170,34; 173,5; 178,3-5(3); 180,30; 181,21; 186,34; 188,27.29; 190,31; 205,20; 209,2; 231,34; 234,6; 236,4; 238,3; 242,13; 249,13; 251,33; 252,6.7.10.11. We note an unusual use at 228,19 katêgorêma, predication, 180,30 katêgorikos, affirmative (136,14; 166,19; 173,1; 213,30); categorical (119,16), This word is much more common in Aristotle than in Alexander, who prefers kataphatikos. For Aristotelian uses of this word to mean affirmative see, e.g. 1.9, 30a36. Barnes et al. translate katêgorikos ‘predicative’ kath’ hauto (hautên), per se, 163,15; 203,29; 223,33 kath’ hekaston, individual, 141,4; 160,4.24; 162,16. The expression is used non-technically at, 120,8.10; 121,16; 123,29.31; 165,21 katholikos, universal, 125,27 katholou, universal, 121,7.8(2).21.23.24.30, etc. keisthai, to be assumed, 122,10; 123,5.6; 126,33; 127,11, etc. Alexander frequently uses the participle keimenos to refer to a

Greek-English Index premiss; in these cases we have translated ‘assumed premiss’. 124,12; 131,27; 132,27; 135,2; 145,20.27; 168,24.26; 169,3.13.23(2); 170,18.23; 175,16; 185,25; 187,22; 200,19.29; 210,9; 216,1; 231,32; 243,22; 246,14.26; 251,4. In some cases the word seems to mean something closer to ‘to be established’, but, even then, we have stuck with ‘to be assumed’; see, for example, 211,24 and 217,28. There is an interesting use of keimenon to refer to the proposition P in the proposition NEC(P) at 196,28-30. Other less significant exceptions to our practice are 125,7 (to apply), 132,13, 139,21, 170,16 and 202,35 (to be under consideration; cf. 215,24 and ekkeimenos), 158,19 (to lay down), 164,12 (to be placed), 205,4(2) (to play a role; cf. 216,1). See also hupokeisthai, lambanein, tithenai kheirôn, weaker (universal proposition than particular, negative than affirmative, necessary than unqualified than contingent), 124,12; 174,2. See also elattôn khôrizein, to separate, 130,4; 132,26(2).27.29(2); 220,23. These are the cases with a logical sense. The word is applied to the separation of soul and body five times between 180,28 and 181,23 koinos, general, common, 130,10; 152,27; 215,5 kuriôs, in the strict sense (applied to contingency at 156,19-21(2), and 222,18, and to the opposition of propositions at 158,31-159,2(2), and 198,24) lambanein. We have translated this frequently occurring word as ‘to take’ as often as seemed at least minimally feasible, 120,14; 121,20-30(3); 122,9-36(9), etc. It frequently means something like ‘to assume’ and we have so

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translated it at 165,11; 188,15.16; 210,36; 224,14.16; 236,36. Other variations occur at 141,22 (to add), 152,24 (to use), 158,18; 232,35.36 (to consider), 159,27 and 161,12 (to assert), 196,2 (to get), 229,10 (to obtain). See also hupokeisthai, keisthai, tithenai lêgon, consequent (Stoic term; see Philoponus, in APr. 243,1-10), 177,21; 178,28; 179,34. Cf. hepomenon lexis is used to refer to the text or verbal formulation. We have translated it as, e.g. ‘text’, ‘what is said’, ‘what he says’, ‘words’, 129,9; 167,31; 169,26; 170,15; 186,30; 195,7; 200,8; 204,27; 210,21; 221,7.20.24; 225,1; 228,24; 239,2.27; 249,1; 250,25; 254,22. Barnes et al. translate ‘expression’. See also phônêi logikos. This word occurs three times in each of two brief passages (198,32-199,1 and 224,13-15) as an example, where it is translated ‘rational’. At 250,2 there is a controversial reference to a work called ‘logical notes’ (scholia logika). Finally at 180,12 Alexander characterizes an argument he is about to give as more logikos; there we have translated ‘dialectical’ logos is of no particular interest. We have translated it most often as ‘argument’ or ‘discussion’, 121,28; 122,18; 123,21; 124,17; 125,27; 133,15; 134,10.20; 135,29; 149,27; 157,19; 164,30; 165,3.6; 166,10; 180,8; 181,16; 184,5; 191,21; 197,27; 214,6; 219,35; 222,23; 232,3. It is translated ‘account’ at 119,9 and 157,5, and ‘definition’ at 160,19 makhesthai, conflict with (with dative), 152,1 meizon, major, used of a premiss (120,27; 124,5.22; 126,19.35, etc.) or a term (124,31.33(2); 125,1.2, etc.) menein, to remain, remain fixed,

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136,10.13; 150,1; 181,28; 189,19.29.34; 192,9.12.29; 200,15; 203,13; 211,27; 217,21; 222,28; 234,7; 251,9.39 meros, part, 122,10.12.28; 123,5.8; 130,3; 162,1. At 157,30-2 we have paraphrased a sentence in which Alexander refers to a ‘part’ of a contradiction, meaning one of a pair of contradictory propositions. See also en merei, epi merous, kata meros, para meros, morion meta diorismou (anankaios), (necessary) on a condition, 140,18.21.23.34; 155,25; 180,1; 202,22; 251,22. We note also anankaios meta prosdiorismou at 155,22 and the one occurrence of ex hupotheseôs anankaios at 141,6. See Appendix 3 on conditional necessity metabolê, change, 193,17 metalambanein, to transform. The word is applied particularly to what are sometimes called complementary conversions of contingent propositions, e.g., at 168,13-31(7). Examples of other uses of the word are at 121,22 (to transform), 131,26 and 141,7 (to take instead), 143,11 (to turn to (a subject)), and 144,20 and 147,26 (to change (terms)) metalêpsis, transformation, 169,3.14; 175,32; 176,19; 191,26, etc. See metalambanein metapiptein, to change, 190,10; 192,31; 193,1.2.16(3).18 metaptôsis, change, 193,16(2) metatithenai, to interchange (applied to the assignment of terms to letters at 144,9.10; 149,16; and to the modalities of two premisses at 202,6) mixis, mixture (a combination of two premisses of different modalities), 121,9; 123,23(2).30.32, etc. morion, part, 121,26.31; 122,15; 123,5. At 157,16-18 we have paraphrased a sentence in which Alexander refers to a ‘part’ of a

contradiction, meaning one of a pair of contradictory propositions. See also meros oikeios, appropriate, proper, 119,28; 199,13 onoma, name. This word occurs six times in a discussion of Stoic ideas (178,17-18(2) and 179,11-16(4)), where the name in question is ‘Dion’ and once at 238,3, where Alexander says that Aristotle applies the name phasis to protaseis onomazein, to name, 179,14 organon, instrument, 164,31; 165,1 ousia, substance, 130,4.11 para meros, at alternating times, 161,21 paradeigma, example, 177,27; 183,25 paraiteisthai, to reject, decline, 121,12; 125,28.29; 164,23.27; 165,3.8 parallagê, alteration, 181,26.27 parathesis, setting down, 129,30; 139,30; 143,18; 144,22; 149,1, etc. See also paratithenai paratithenai, to set down, 130,24; 134,29; 135,18; 138,30; 147,12, etc. Normally it is terms which are set down as an interpretation of propositions, but Alexander speaks four times (230,22.23; 237,21.28) of setting down a conclusion or the setting down (parathesis) of a conclusion (meaning giving terms which make it true) perainein, to conclude. The word occurs only at 164,30 in the phrase adiaphorôs perainontes, a kind of argument considered by the Stoics periekhein, to contain, 129,4; 179,15; 191,7. At 184,10 periekhontôn is translated ‘surroundings’ perilambanein, to encompass, 179,3 phaneros, evident, 122,21; 132,26; 141,7; 153,5; 157,15, etc.

Greek-English Index phasis, assertion, 238,1-3(3). The words apophansis, apophainein, and apophantikos do not occur in this part of the commentary phônêi, verbally, 198,14 (only occurrence of the noun) phullattein, to preserve, hold fixed, 161,5; 168,12.32; 193,9. See also sôzein and têrein piptein (ektos), to fall (outside), 189,14 pistis, credibility (125,32); justification (135,7) pistousthai, to confirm, 125,5; 127,3 poion, quality (the quality of a proposition is its being affirmative or negative), 123,32; 170,27-8(2); 172,33; 215,5; 234,23. See also poiotês and idiôs poios poiotês, quality, 233,23; 241,21. See also poion pragmateia, subject (of study), 164,25 prodêlos, prima facie clear, 123,22; 142,14; 188,2; 237,13 proeinai, to go through (said of a proof), 210,21; 234,15; 247,39. See also proerkhesthai and prokhôrein proêgoumenôs, primarily, 190,27 proerkhesthai, to go through (said of a proof), 134,10; 227,3. See also proeinai and prokhôrein prokeimenos. (1) Alexander uses to prokeimenon to refer to what we call the proposed conclusion. Some of his uses of this term occur in discussions which show the importance he attaches to the order of terms in a conclusion; see, e.g., 146,23; 148,13; 234,6; 244,8; 251,3; other occurrences at 122,4; 131,18; 137,7; 141,27; 174,7; 231,31; 235,35; 236,9.13; 244,26; 246,9.19; 252,34. Cf. 188,9 and four occurrences of protithenai at 166,13; 167,6; 199,24 (2). We also translate this word ‘under consideration’ when it is applied to combinations of premisses. 123,20; 144,8; 167,15; 172,8.17; 186,35; 188,20; 195,15; 208,34; 211,11; 212,6.23; 213,34;

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214,18; 228,22; 251,35.36. See also the more general uses of this word at 165,9, 175,24 and 186,2 and the use of prokeisthai at 144,8. And see ekkeimenos prokhôrein, to go through (said of a proof), 135,16; 195,10; 235,25. See also proeinai and proerkhesthai prolambanein, to assume in advance, 157,15; 192,29; 193,4 prosdiorismos, additional condition, 155,22 prosdiorizein, to specify further, 155,17 proskatêgorein, to co-predicate, 119,28 proskeisthai, to be added or attached, 144,5; 151,15; 154,15; 155,12; 159,21; 205,14; 218,22 proslambanein, to add (as a premiss), 121,6; 128,24; 132,3.10.14; 134,1.5.12; 139,24; 170,4.6.11; 188,9; 197,17; 206,32; 208,26; 209,30; 210,18.29.35; 214,14; 216,35 proslêpsis. This word occurs in the phrase kata proslêpsin at 166,18; see the note ad loc prosrhesis, adjunct, 156,17 prosthêkê, addition. Alexander refers to the addition of a modal operator at 119,27 and at 155,11-12(2); for its non-technical sense see prostithenai prostithenai, to add. Alexander mainly uses this word to remark that Aristotle adds something to what he has already said. He applies it to adding a premiss or a diorismos at 189,28-190,31(5), and at 155,11 he speaks of adding a modal operator. Cf. prosthêkê. The word prosthesis does not occur in this part of the commentary protasis, proposition (119,10.12.19.26.26, etc.) or premiss (120,10.29; 121,3.6.15 etc.). Note that we have often supplied the word ‘proposition’ or ‘premiss’ where Alexander simply has a nominalization of feminine

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or neuter adjectives; for example where he speaks of a combination of a contingent and a necessary, we will translate ‘a contingent and a necessary premiss’ pseudês, false (sometimes falsification), 119,25; 126,13; 128,10.20.27, etc. The word is translated ‘fallacious’ at 196,22 rhêthêsetai, will be said. At 24,27-30 Alexander paraphrases Aristotle’s formulation of the dictum de omni et nullo by substitution of rhêthêsetai for lekhthêsetai. We only wish to signal this fact, since Alexander consistently uses the verb he substitutes when he cites the dictum, and he uses it in the same way twice in a discussion of ekthesis. We suggest that insofar as rhêthêsetai is a technical word for Alexander it is used to express the relation of predication between a universal and a particular under it as opposed to the relationship of two universals. We give the relevant passages, 24,30.33; 25,2.19; 32,18; 54,7.15.18; 55,5.6.10; 60,24.25.29; 61,26; 122,33; 123,1; 126,5; 130,1; 167,18; 169,26; 174,24 (technical uses only) sêmainein, to mean (150,18; 151,12; 152,29; 157,1, etc.); to signify (126,27; 129,33; 140,6; 155,11; 157,1, etc.); to refer (150,12; 205,14 (only occurrence of sêmantikos)) sêmeion, indication (134,28; 145,8; 190,13; 198,1; 236,23); sign (179,16) sêmeiôteon, it should be noted, 122,17; 128,32; 149,5; 168,28; 240,32. Barnes et al. translate ‘Note!’ Their entry under sêmeioun is incorrect skhêma, figure (of a syllogism), 120,8-26(6), etc. The word is translated ‘form’ at 190,9

sôzein, to preserve, 189,36, see also phullattein and têrein stêretikos, privative, 135,23-136,4(4), etc. See the appendix on affirmation and negation stokhastikos, conjectural, 165,9 sullogismos, syllogism, 119,9.13.14.16.17.26.27, etc. sullogistikos, syllogistic (adjective applied to a pair of premisses which yield a conclusion), 120,12; 121,2.24.33; 123,20, etc. See also asullogistos sullogistikôs, syllogistically. Normally used with sunagein (141,29; 168,13; 230,32.34; 231,11; 232,11; 237,37; 239,33; 240,5.33; 241,6; 242,33), once with akolouthein (229,8), once with deiknunai (243,22). There are two other occurrences at 124,10 and 185,3 sullogizein occurs only at 165,9 where it is translated ‘deduce’ sumbainein, to result. Alexander rarely uses this word except when quoting or paraphrasing Aristotle. The following are passages where he uses it more or less on his own: 123,22; 157,10; 177,13.14; 198,10; 216,28; 217,11; 236,39; 248,17 sumbebêkos, accident, 181,27 sumperasma, conclusion (an extremely common word in the commentary), 121,5; 122,5; 123,32; 124,3.4, etc. sumplokê, combination (of two premisses; Barnes et al. translate ‘conjunction’). In general there seems to be no difference between sumplokê and suzugia. But we note two occurrences of sumplokê where it seems to mean something like formal validity, 164,27 and 169,10. Standard occurrences, 120,10; 121,7.24, etc. Alexander occasionally uses participles of the verb sumplekein to the same effect (155,29; 156,5.6) sunagein. This is Alexander’s usual

Greek-English Index term for expressing the relation of implication (Barnes et al. translate it ‘to deduce’). We have usually translated the active forms as ‘yield a conclusion’ (e.g., 120,18; 125,19; 135,8; 142,19; 208,10) or ‘imply’ (e.g., 125,21; 127,1.10; 128,26; 138,7) and the passive ones as ‘follow’ (e.g., 121,9; 123,24; 124,14; 125,11.23(2)). But sometimes we have used ‘be inferred’ (e.g., 131,10; 134,6.10; 138,25.26) for the passive forms, and sometimes we have simply spoken of a conclusion (e.g., 132,1; 167,31; 176,10; 186,20) sunagôgê, inference (Barnes et al. translate ‘deduction’). 122,21; 137,7; 167,25; 202,5; 231,31; 232,14.19; 242,4 sunaktikos, yielding a conclusion (Barnes et al. translate ‘deductive’), 217,27; 227,23 sunalêtheuein, to be true together with, 157,31(2); 159,29; 160,2.21; 164,7; 168,26.28; 197,8; 221,20.29. On sunalêtheuein see Lee (1984), pp. 88-92 sunanairein, to also do away with (Barnes et al. translate ‘to reject together with’), 182,15; 223,34. See anairein sunaptein, to attach, 125,2 sunekhês, continuous, 162,25 (picking up on 32b8) sunêtheia, custom, 179,23 sunêthôs, customarily, 155,14 sunistasthai, to compose, 121,18 sunkeisthai, sunthesis, suntithenai. Six of the occurrences of these three words are at 181,4-16, and refer to either the conjunction of soul and body or the closing together of fingers to make a fist. The remaining occurrences are, for sunkeisthai (to be put together), 124,10; 177,27; 212,13; and, for suntithenai (to compose, to conjoin), 121,10 and 217,12 sunkhôrein, to agree, 129,15; 132,7;

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139,5.6; 248,13 sunêmmenon, conditional (a proposition of the form ‘if P then Q’). The word occurs 18 times in this section of the commentary, all between 176,4-182,18 (on 34a5-12) suntattein, to co-ordinate with, 159,19; 189,32; 190,36 sustasis, construction, 119,15.16 suzugia, combination (of two premisses). Apparently interchangeable with sumplokê, but more common. 120,12; 121,18-21(3), etc. taxis, order, ordering, 149,16; 241,27 tekhnê, art, 165,8; 169,8 teleios, complete (Barnes et al. translate ‘perfect’), 169,13; 173,1.13.17; 174,6.9.20.27.29.34; 175,9.15.16; 186,7; 202,3.4.32; 205,4; 206,26.29; 208,4; 210,2.6; 245,23; 250,12.19 teleiôsis, completion (Barnes et al. translate ‘perfection’), 242,24 teleioun, to complete (Barnes et al. translate ‘to perfect’), 217,27; 242,22.26; 253,13; 254,10.12.15 têrein, to keep, keep fixed, preserve, 146,1; 148,8.31; 154,16; 189,9; 192,22.25; 193,6; 207,20; 217,12; 246,19; 250,28; 254,32. See also phullattein and sôzein thesis, positing, 178,25. See tithenai tithenai, to posit, 130,13; 131,20; 132,6.19; 133,31, etc. This word is used with more frequency by Aristotle than by Alexander. We have translated it ‘to assign’ at 150,15, ‘to place’ at 208,15, and ‘to classify’ at 245,32. Like keisthai it sometimes seems to mean something more like ‘to establish’ than ‘to assume’; see, e.g., 199,14 and 249,31. See also thesis, hupokeisthai, and lambanein tropos, modality (rather than the standard ‘mode’, adopted by Barnes et al.), 119,17.26(2).28; 120,21.24; 154,13; 155,11.17;

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159,21; 160,30; 172,5; 197,2; 202,6; 218,21 (technical uses only) zêtein, to inquire, investigate, seek,

161,3; 165,6.11; 188,17; 196,12; 206,12; 207,35; 213,26; 218,14; 247,22; 249,37 zêtêsis, inquiry, 165,7.19

Subject Index a(-proposition), 4 AE-transformationc, 21-2, 40, 96, 98-9, 107-10; App. 5, see also waste cases affirmation, affirmative, 64-5, 108-9, App. 2, App. 4 (p.153) AI-conversionc, 22, 27-31,40 AI-conversionn, 10, 27, 40 AI-conversionu, 8, 39 Andronicus, 98 AO-transformationc, 109 Barbara1(NU_), see NU first-figure combinations Barbara1(UN_), 13-18, 62-5, 68 Baroco2(NC_), 61 Baroco2(NNN), 12-13, 54-8 Baroco2(NU_), 78-81 Baroco2(UC_), 124 n.81 Baroco2(UN_), 13-14, 79-81, 91-2 Bocardo3(CN_), 66-7 Bocardo3(CU_), 61-2 (see n. 46) Bocardo3(NNN), 12-13, 54-8 Bocardo3(NU_), 13-14, 87-8 Camestres2(NU_), 72-7 Celarent1(NU_), see NU first-figure combinations Celarent1(UN_), 17-18, 65-7 Cesare2(UN_), 75-7 circle argument, 61, 66-7, 68-9, 74-5 combination, 4 complete, 5-6; see dictum de omni et nullo completion, 7-8 contingency in the way specified, 19-21 contingency, 9, 12-13, 19-34, 53-4, 93-107, 108-10 contingently (holding), 20, 29-31 contradictories, 7, 94-7 contraries, 7, 97, 108-10

conversion, 7-8, 102; see also AI-conversion, EE-conversion, and II-conversion Darii1(NC_), 74-5 Darii1(NU_), see NU first-figure combinations Darii1(UN_), 16-18 de re/de dicto distinction, 14-15, 22-3, 36 n.22, 52, 124 n.87, 142 n.319 dictum de omni et nullo, 6, 11, 15-16, 57-8, 60-1, 71, 139 n.287 diorismos of contingency, 20-1, 93-5; see also contingency in the way specified e(-proposition), 4 EA-transformationc, 21-2, 40, 95-7, App. 5; see also waste cases EE-conversionc, 25, 27-34, 106-15, App. 5 EE-conversionn, 9-10, 25-7, 40 EE-conversionu, 8, 39 ekthesis, 13, 56-8, 79-81, 88 Eudemus, 15-16, 29, 58-62, 76 extreme, 4 Ferio1(NU_) see NU first-figure combinations Ferio1(UN_), 24-6 n.93, 22-5, App. 1 figure, 4 Herminus, 119 n.35 i(-proposition), 4 II-conversionc, 10-11, 22, 28-31, 40 II-conversionn, 10, 27, 40 II-conversionu, 8, 39 incompatibility acceptance argument, 17-18; see also incompatibility rejection argument

186

Subject Index

incompatibility rejection argument, 17-18, 32, 62-4, 67-70, 73-4, 76-7, 106-8 interpretations, 6-7, 32, 59-60, 64-5, 69, 74-81, 83-4, 85-8, 107, 109, 112, 121 n.51 IO-transformationc, 21-22, 40, 97, 108-10, App. 5; see also waste cases major, 4 major term must be predicate of conclusion, 81-2 middle, 4 minor, 4 necessarily (holding), 29-30 necessity, 59-60, 71, 75-7, 89-90, 91-2, 93-5, 100-1, 105, App. 1, App. 3 negation, negative, 96, App. 2, App. 4 (p. 153) non-syllogistic combinations, 6-7 NU first-figure combinations (i.e., Barbara1(NU_), Celarent1(NU_), Darii1(NU_), Ferio1(NU_)), 13-18, 58-67, 70-1, 75-6, 105, 125 n.93 o(-proposition), 4 OI-transformationc, 21-2, 40, 97, 108-10, App. 5; see also waste cases OO-conversionc, 31-2 opposites, 55, 65-6, 68-9, 96-7, 100-2, 111-15

peiorem rule, 15-16, 59 premiss, 4 propositions, 4 prosleptic propositions, 105 quality, 4 quantity, 4 reductio ad impossibile, 8, 55, 58, 61-2, 65-7, 68-70, 74-5, 110-15 reduction, 77-8 Sosigenes, App. 3 Stoics, 103 temporal interpretation of modality, 23-5, 27-31 terms, 4; see interpretations Theophrastus, 3, 15-6, 29, 58-62, 67, 76, 96, 98, 107, 111, 122 nn.60 & 61, 124 nn.81-3, 125 n.91, 93, 127 n.120, 131 n.166, 139 n.287, App. 3 transformation, 56 n.38 universal (propositions), 14, 30-1, 61, 64-5 unqualified, 9 unqualifiedly (holding), 29-31 waste cases, 21-2 weak two-sided Theophrastean contingency, 32-4, 111-15, App. 5

Index Locorum Bold type is used for references to the pages, notes and appendixes of this book ALEXANDER OF APHRODISIAS

in An. Pr. 24,27-30, 120 n.39 25,26-26,14, 23-5 26,29-27,1, 36 n.33 31,5-6, App. 2 32,32-34,2, 117 n.18 36,7-25, 25-6 36,25-8, 127 n.120 36,25-31, App. 3 37,3-13, 27 37,14-17, 27 37,17-21, 26 37,28-38,10, 19, App. 4 (p. 154) 39,4-11, 31 39,19-23, 20 41,21-4, 107 44,26-7, 38 n.258 54,12-18, 6 60,27-61,1, 6 99,16-100,26, 117 n.18 378,12-379,11, 140 n.290 379,9-11, 120 n.43 in Top. 10,7-12, 139 n.281 On the difference between Aristotle and his associates concerning mixtures 62, 120 n.37

AMMONIUS

in Int. 153,13-154,2, App. 3 221,11-229,11, 141 n.305 245,1-32, 136 n.238, App. 4

[AMMONIUS]

In An.Pr. 39,10-15, 121 n.46 39,10-25, App. 3 n.7 39,31-40,2, 119 n.35 40,9-13, 131 n.166

ARISTOTLE

An. Pr. (outside 29b29-40b15) 24b22-4, 6 24b26-30, 6 24b29-30, 120 n.39 25a1-2, 23 25a2-3, 36 n.33 25a27-36, 10-11, 25-7 25a31-2, 85, 132 n.182 25a29-32, 10, 25 25a32-4, 10 25a37-9, 19 25a37-b3, 21-2, 28 25b3-14, 31 25b3-25, 106 25b14-15, 19 25b14-19, 32 25b15-22, 138 n.263 27a36-b1, 129 n.139 28a24-6, 117 n.18 28b14-15, 117 n.18 28b20-1, 117 n.18, 118 n.22 49b14-32, 120 n.43 52a39-b34, 137 n.244 53b12-16, 137 n.244 Int. 19a23-4, 76 19a23-6, App. 3 23a6-11, App. 4 (pp.153-4) 23a7-20, 19 chs. 12 and 13, App. 4 EUDEMUS (fragments, ed. Wehrli) 12, 128 n.126 16, 141 n.298 PHILOPONUS

in An. Pr. 53,15-24, 137 n.251 53,15-56,5, 141 n.305 123,12-126,29, 118 n.30 123,26-124,1, 120 n.45

188

Index Locorum 124,1-4, 121 n.47 124,9-24, 119 n.32 124,24-8, 119 n.33 124,30-125,18, 121 n.46 126,8-29, App. 3 145,5-6, 133 n.201

STEPHANUS

in Int. 38,14-31, App. 3 n.2 STOICS (dialectic fragments, ed. Hülser) 1169, 139 n.280 SVF (von Arnim) II.259, 139 n.281

(fragments, FHSG) 100B, 136 n.240 100C, App. 3 n.8 100D, 128 n.126 102A, 141 n.298 102B, 142 n.314 102C, 141 n.298 103A, 137 n.254 104, 118 n.24 106A, 118 n.30 106B, 124 n.83 106C, 120 n.45 110A, 120 n.43, 140 n.290 110A-D, 140 n.290

THEOPHRASTUS