Alexander of Aphrodisias: On Aristotle Prior Analytics 1.14-22 9781472551610, 9780715628768

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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 and contingent (possible) 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. Aristotle also invented the theory of the syllogism, and in this volume he extends this theory to include syllogisms containing contingent propositions, the contingent being what may or may not happen. December 1998

R.R.K.S.

Introduction We offer here a translation of Alexander of Aphrodisias’ commentary on chapters 14-22 of Aristotle’s Prior Analytics, in which Aristotle presents what we call his modal logic as applied to contingent as well as necessary and what we call unqualified propositions. In a separate volume we have translated Alexander’s commentary on chapters 8-12, in which Aristotle treats arguments not involving contingent propositions, and also chapter 13, in which Aristotle discusses contingency in general, and part of chapter 17 in which he treats the conversion of 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 of 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).

2

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 time, of knowledge, and of space 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, because any material added by us is put between 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 terminology and with some of our apparatus for representing Alexan-

Introduction

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der’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, where we also deal with chapters 13-22. Finally in a brief fourth section we discuss the treatment of modality by Theophrastus, the associate of Aristotle. 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 1.20-2 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 the major term as predicate and the minor as subject.3 Aristotle

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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. AAA1 2. EAE1 3. AII1 4. 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. Barbara1 2. Celarent1 3. Darii1 4. Ferio1. Aristotle calls these four syllogisms complete (teleios, rendered ‘perfect’ 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 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

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are animals’ is false. Aristotle shows that a first-figure combination of specific premisses XY and YZ yield 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: EE-conversion: XeY  YeX (25a14-17) AI-conversion: XaY  YiX (25a17-19) II-conversion: XiY  YiX (25a20-2)

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Introduction

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 – 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.

Introduction

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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) NEC(AaB) BaC NEC(AaC) Bocardo3(NCU) NEC(AoC) CON(BaC) AoB EA_2(CU_) CON(AeB) 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: EE-conversionn: NEC(XeY)  NEC(YeX) (25a29-31) AI-conversionn: NEC(XaY)  NEC(YiX) (25a32-4) II-conversionn: NEC(XiY)  NEC(YiX) (25a32-4)

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Introduction

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

( C  N )

Introduction

11

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 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:

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Introduction Baroco2(NNN) Bocardo3(NNN)

NEC(AaB) 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 Barbara1(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

Introduction

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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 section IV of the introduction. II.C. N+U combinations (1.9-11) In chapters 9-11 Aristotle takes up the N+U cases, devoting a 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 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.

14

Introduction

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’), and so we postpone considering it until our discussion of the U+C firstfigure cases in section III.E.2.a. 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 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

Introduction

15

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 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

16

Introduction

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 firstfigure 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. (30a23-8)

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.

We prefer the following paraphrase of this argument:

Introduction

17

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 Barbara1(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 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),

18

Introduction

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. Finally, in III.E.1-3 we consider the various combinations involving contingent propositions. 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 nonnecessary’; 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

20

Introduction

next sections we shall emphasize occurrences of this word by including the transliterated Greek. In his comment on 25b14, Alexander says: 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

Introduction

21

Thus we may state ‘Alexander’s diorismos’ as: CON(P) iff (i)  P, and (ii) no impossibility follows from P35 Alexander frequently refers to this strict sense of contingency as 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,

22

Introduction

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: AI-conversionc: CON(AaB)  CON(BiA) II-conversionc: CON(AiB)  CON(BiA) the result is still the equivalence of: (ia) CON(AaB) (ib) CON(AeB) and of all of: (iia) CON(AiB) (iib) CON(AoB) (iic) CON(BiA) (iid) CON(BoA) as well as the implication of any of (iia)-(iid) by either of (ia) or (ib). On the other hand, as we have already mentioned, Aristotle denies EE-conversionc 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(BaC) CON(AaC) CON(AaB) CON(BeC) CON(AaC) CON(AeB) CON(BeC) 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.

Introduction

23

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)

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

24

Introduction 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 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

Introduction

25

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 (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: NEC(AeB)  NEC(BeA) AI-conversionn: NEC(AaB)  NEC(BiA) II-conversionn: NEC(AiB)  NEC(BiA) We recall Aristotle’s justification of EE-conversionn:

26

Introduction 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 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)

Introduction

27

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 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)’.

28

Introduction III.D. Conversion of contingent propositions

III.D.1 Conversion of affirmative contingent propositions (1.3, 25a37-b3); more on Alexander and the temporal interpretation 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  C   N; he later (1.17, 37a9-31) rejects the analogous argument for EE- conversionc: assume  CON(YeX); then ( C   ) 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’

Introduction

29

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) 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   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)

30

Introduction

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 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 propositions 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’:

Introduction

31

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 important 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, 36b35-37a32) 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

32

Introduction

white’ to ( C    ) 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 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

Introduction

33

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 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 Appendix 5 on weak two-sided Theophrastean contingency. We now look at Aristotle’s rejection of *. It begins at 37a14:

34

Introduction 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, (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. (37a20-4)

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

Introduction

35

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). III.E. Syllogistic and non-syllogistic combinations In chapters 14-22 Aristotle deals with CC, U+C, and N+C combinations, treating the first figure in chapters 14-16, the second in 17-19, and the third in 20-2. Alexander, of course, follows this order. However, the order does not seem to serve any clear purpose and it obscures the logical connections established by Aristotle’s practice of reducing second- and third-figure combinations to corresponding first-figure ones. We shall, therefore, discuss the CC combinations together in section III.E.1, then the U+C combinations in III.E.2, and then the N+C combinations in III.E.3. III.E.1. CC premiss combinations (1.14, 17, 20) The CC combinations are the least interesting ones involving contingency. Aristotle has at his disposal terms to show that none of these combinations are syllogistic, but his sense of what is complete and his transformation rules commit him to a large number being syllogistic (e.g., any first- or third-figure CC combination with two universal premisses). We have not found any evidence concerning Theophrastus’ attitude toward the CC combinations, but it is unclear how he could have possibly accepted any if CON is interpreted as  NEC . For any valid syllogism:  NEC( P1)

 NEC( P2)  NEC( P3)

could be used to justify, e.g.: NEC( P3)

 NEC( P2)

NEC( P1)

an apparent violation of the peiorem rule. The same difficulty would arise for any  N + U syllogisms; for if they do not violate the peiorem and have a contingent conclusion they can be transformed into  N + N syllogisms with an unqualified conclusion or N+U syllogisms with a necessary conclusion. III.E.1.a. The first figure (1.14) Aristotle accepts as complete all the CCC analogues of the first-figure

36

Introduction

UUU syllogisms plus four waste cases which they generate. At 33a34ff. he rejects all other first-figure CC combinations. He first uses an abstract argument which Alexander explicates at 170,24-171,13, and then gives terms which take for granted the truth of the following propositions: (i) (ii) (iii)

CON(Animal i White) CON(Animal o White) CON(White a Human) CON(White a Cloak) CON(White e Human) CON(White e Cloak) NEC(Animal a Human) NEC(Animal e Cloak)

Although there are undoubtedly ways of arguing that Aristotle might accept the propositions of (i) without accepting their universal analogues, we are inclined to think that he is rather stuck with (i’)

CON(Animal a White)

CON(Animal e White)

and thus with terms which rule out the combinations he has already accepted. As we mentioned at the end of section III.A, Alexander is inclined to think that the propositions of (i) are made false by the (de re) truth of NEC(Animal i White) and NEC(Animal o White). He offers the following improvement to Aristotle’s terms in the present case: (i)

CON(White i Walking) CON(White o Walking) (ii) CON(Walking a Swan) CON(Walking a Crow) CON(Walking e Swan) CON(Walking e Crow) (iii) NEC(White a Swan) NEC(White e Crow) It seems even clearer in the case of these terms that if one accepts the sentences under (i) as true one will also have to accept: (i’) CON(White a Walking)

CON(White e Walking)

III.E.1.b. The second figure (1.17) We have discussed Aristotle’s rejection of EE-conversionc in section III.D.2. It takes up much of chapter 17, in which Aristotle rejects all second-figure CC-combinations. Aristotle used EE-conversionu to justify three of the four syllogistic UU combinations in the second figure, proving the fourth, Baroco2, by reductio. Aristotle uses the failure of EE-conversionc to argue (37a32-5) against the possibility of reducing Cesare2(CC_) to Celarent1(CCC) in the way Cesare2(UUU) was reduced to Celarent1(UUU). The argument obviously generalizes to Camestres2(CC_) and Festino2(CC_). Aristotle also argues (37a35-7) against

Introduction

37

the possibility of any indirect reduction. Although his text is obscure, it is clear that no indirect reduction of a CC combination is possible; for one of the premisses of the reducing argument will be contingent, so the conclusion will not be necessary, and so will not contradict the other contingent premiss. Aristotle concludes the chapter (37a38-b18) by giving terms to show that no contingent conclusion with B as predicate and C as subject follows from contingent premisses having those terms as subject and a term A as predicate. III.E.1.c The third figure (1.20) Aristotle’s rules commit him to the position that every third-figure CC combination with a universal premiss is syllogistic. He explicitly mentions AA_3 (Darapti3; 39a14-19) and its consequence EE_3 (39a26-8), but not its consequence AE_3; he also mentions EA_3 (Felapton3; 39a19-23). As for the cases with one particular premiss he mentions AI_3 (Datisi1; 39a31-5), but not its consequence AO_3; IA_3 (Disamis3; 39a35-6) and its consequence OE_3 (39a38-b2), but not its consequences OA_3;55 and IE_3, and EI_3 (Ferison3; 39a36-8) and its consequence EO_3 (39a38-b2). Alexander does not mention any of the cases, not mentioned by Aristotle. Aristotle rejects all the CC combinations with particular premisses taking as true, e.g., CON(Animal i White) and CON(Human i White). Alexander raises no doubt about Aristotle’s taking these premisses to be true, even though, as we have just mentioned, he did find CON(Animal i White) problematic in the rejection of certain firstfigure CC combinations, and if the whiteness of swans falsifies CON(Animal i White), it presumably also falsifies CON(Human i White) by verifying NEC(Human o White) and thus falsifying CON(Human o White). III.E.2. U+C premiss combinations (1.15, 18, and 21) Given the apparent impossibility of distinguishing between permitted interpretations of contingent and unqualified propositions, it seems clear that Aristotle’s syllogistic is in as much difficulty over the U+C cases as it is over the CC cases. Aristotle’s discussion of the U+C cases is, however, much more difficult. III.E.2.a. The first figure (1.15) As one expects, Aristotle focuses on the analogues of the complete UUU cases. He takes the CU analogues to be complete, but thinks that the UC analogues need to be proved to be syllogistic. Aristotle gives two rather complicated illegitimate indirect arguments for Barbara1(UC_) and Celarent1(UC_), which, he says, can also be used for Darii1(UC_) and Ferio1(UC_). The arguments are, in fact, essentially the same, but for reasons which are never specified, Aristotle apparently holds (33b25-

38

Introduction

33) that Celarent1(UC_) and Ferio1(UC_) yield a  NEC conclusion, which is not contingent in the way specified, whereas Barbara1(UC_) and Darii1(UC_) yield a standard contingent conclusion. Alexander points out very explicitly at 198,5-199,15 that the conclusion of Barbara1(UC_) is  NEC (AaC) rather than CON(AaC),56 but in the sequel of the commentary he continues to speak as if the conclusion were CON(AaC). His (and our difficulties) are only increased by Aristotle’s parallel, but importantly different treatment of the NC cases and by Aristotle’s later willingness to forget the distinction he drew between the affirmative and negative first-figure UC cases. We represent the four standard first-figure cases as follows: Barbara1(UC‘C’) AaB Celarent1(UCN) AeB Darii1(UC‘C’) AaB Ferio1(UCN) AeB

CON(BaC)  NEC (AaC) CON(BaC)  NEC (AeC) CON(BiC)  NEC (AiC) CON(BiC)  NEC (AoC)

For Barbara1 57 Aristotle assumes  CON(AaC), infers NEC (AaC), i.e., NEC(AoC), and purports to derive an inconsistency. In the case of Celarent1 Aristotle assumes  CON(AeC), infers NEC (AeC), i.e., NEC(AiC), and purports to derive an inconsistency. In connection with Celarent1 Aristotle points out that he has not really established CON(AeC), but only  NEC(AiC) (i.e.,  NEC (AeC)), the negation of the assumption he made for the reductio argument: Thus the conclusion of this syllogism is not a proposition which is contingent in the way specified, but is ‘of none by necessity’;58 for this is the contradictory of the hypothesis which was made, since it was posited that A holds of some C by necessity, and the conclusion of a syllogism by means of the impossible is the opposite of the hypothesis. (34b27-31)

Obviously the same point can be made about the indirect derivation of Barbara1(UC_): the conclusion which is shown to follow is, if anything, NEC(AoC), i.e., NEC (AaC).59 On the basis of what we have said up to now one might suppose that all that the treatment of the CU combinations involves is a certain incongruity in Aristotle’s treatment of contingency, insofar as it sometimes corresponds to the notion we represent by CON and sometimes to the notion we represent by  NEC. But Aristotle’s attempts to justify Barbara1(UC‘C’) and Celarent1(UC N) are not just incongruous; they are fallacious. It will suffice to focus on Barbara1. Before trying to establish its validity, Aristotle’s argues (34a5-24) for something like the following true proposition: If P implies Q and it is possible that P, it is possible that Q.

Introduction

39

Alexander’s discussion of this material is somewhat vitiated by a failure to distinguish clearly between the assertion that P implies Q and the conditional ‘If P then Q’, but it includes important material relating to Hellenistic treatments of the conditional and implication. (See especially 177,25-182,8 and 183,34-185,18.) Alexander takes the upshot of his discussion to be the corollary which Aristotle announces sloppily at 34a25, but which Alexander understands correctly as: If P implies Q and it is possible that P, then Q may be false but it cannot be impossible. Aristotle argues for the validity of Barbara1(UC‘C’) as follows: Assume  CON(AaC), ‘i.e.’, NEC (AaC), i.e., NEC(AoC); and assume BaC; then (Bocardo3(NUU)) AoB, contradicting AaB, the major premiss. The obvious problem with this argument is the assumption of BaC, when the given premiss is CON(BaC). Aristotle apparently tries to justify this move – which we shall call U-for-C substitution – on the grounds that if CON(BaC), no impossibility should result from the assumption that BaC, or more generally that no impossibility follows from the assumption that a contingency holds, although a falsehood may. The problem is that, even if CON(BaC), an impossibility may result from BaC and other assumptions, in this case AaB and NEC(AoC). We may take AaB as unproblematic because it is given as a premiss. The issue, then, is this: given the assumption that CON(BaC), is the fact that NEC(AoC) and BaC are inconsistent with AaB to be ‘blamed’ on NEC(AoC) or on BaC? Alexander does his best to justify putting the blame on the former, but he is not really able to do so.60 At 34b7-12 Aristotle mentions interpretations which show that the premisses of Barbara1(UC_) are not syllogistic, interpretations which take the following assertions to be true: Animal a Moving CON(Moving a Horse) NEC(Animal a Horse)

Human a Moving CON(Moving a Horse) NEC(Human e Horse)

Aristotle dismisses these interpretations by saying the unqualified premiss shouldn’t be taken as true at a time, that is, one shouldn’t interpret the unqualified premiss to make it true at one time false at another. As we have indicated in section III.D.1, Alexander realizes that Aristotle violates this restriction relatively often, and so he chooses to interpret Aristotle’s statement by developing the point that ‘All moving

40

Introduction

things are humans’ cannot be true at a time when ‘All horses are moving’ is true,61 connecting it with the U-for-C substitution in the justification of Barbara1(UC N ). Indeed, he insists that this is all Aristotle has in mind in ruling out the above interpretation: For he does not call a proposition which is always unqualified – since what is always true is ipso facto necessary – but rather he calls unqualified a proposition which can remain unqualified universal affirmative and true even when the contingent affirmative universal premiss which has been taken along with it is transformed into an unqualified premiss. For in this way the proof by impossibility which he uses to show that this combination is syllogistic is preserved. He might say either that it is necessary that the universal unqualified premiss not be taken in such a way that it is possible for its truth to be restricted temporally by the contingent premiss added to it or rather that the contingent premiss added to it must not be such as to restrict temporally the truth of the unqualified premiss taken before it. (189,33-190,6)

It is clear that Alexander’s interpretation is unfortunate both as a piece of hermeneutics – Aristotle does not mean what Alexander claims he does – and as logic – to relativize the truth of CON(P) to other propositions taken as unqualifiedly true is to totally change the notion of contingency. At 191,14 Alexander finally raises the queston whether Aristotle’s interpretations don’t show that Barbara1(UC_) yields no conclusion, but again he backs off: It should be asked whether perhaps the setting down of terms and proof that with true premisses the first term holds of all of the last by necessity and holds of none by necessity does not rather show that the combination is non-syllogistic. I have also discussed this elsewhere. (191,14-18)

In fact, the terms do show that Barbara1(UC_) is not syllogistic, and it is clear enough that they can be applied to any first-figure UC combination since it seems that if one can take ‘Animal a Moving’, ‘Human a Moving’ and ‘CON(Moving a Horse)’ as true, one can also take the corresponding e-, i-, and o-propositions as true. Given the apparent indistinguishability of contingent and unqualified propositions, the same terms would suffice to show that there are no syllogistic firstfigure CU or CC combinations either. Each of the syllogistic UC cases generates its own waste case, in order, AEA1(UC‘C’), EEE1(UC N ), AOI1(UC‘C’), and EOO1(UC N ). Thus Aristotle is committed to six of the eight U+C combinations with universal premisses being syllogistic. At 35a20-4 Aristotle gives terms for rejecting the remaining two combinations AE_1(CU_) and EE_1(CU_). Alexander recognizes that the terms are problematic and is unable to offer fully satisfactory ones, but he insists that the combina-

Introduction

41

tions are not syllogistic (201,3-24). Aristotle uses terms to reject all of the remaining cases with one universal and one particular premiss and all the other cases at 35b8-19. III.E.2.b. The second figure (1.18) Aristotle’s treatment of these cases is a source of considerable difficulty for Alexander, partly because Aristotle begins to lose sight of the distinction between kinds of contingency and partly because Alexander produces justifications of syllogisms which Aristotle rejects. We begin with the first issue. Aristotle accepts: Cesare2(UC‘C’)

AeB

CON(AaC)  NEC (BeC)

(37b23-8)

and describes its simple reduction to Celarent1(UC N ) when the negative premiss is converted. He then says that the situation is the same with: Camestres2(CU‘C’) CON(AaB) AeC

 NEC (BeC)

(37b29)

Alexander points out that the situation in this case is not exactly ‘the same’. For after converting AeC to CeA and invoking Celarent1(UC N ) to get  NEC (CeB), one must do a further conversion to get  NEC (BeC). This is, of course, a perfectly legitimate step, but Alexander is quite right to wonder (231,29-232,9) why Aristotle doesn’t remark that the step is all right because the conclusion is not contingent in the way specified. Another passage in which distinction between  NEC and CON arises is 38a3-4, where Aristotle apparently accepts: IEO2(CU‘C’)

CON(AiB) AeC

 NEC (BoC)

which he would presumably justify by converting AeC to CeA and invoking Ferio1(UC N ) to get  NEC (CoB). If the conclusion here were CON(CoB), this could be transformed to CON(BoC), but no corresponding transformation is possible for  NEC (CoB). Alexander discusses the issues involved here at 233,34-234,12; he concludes by suggesting that Aristotle may not have intended to accept IE_2(CU_). This problem with OO-conversion‘c’ arises again in connection with third-figure U+C combinations, but by the time Alexander gets to discussing those he has quite rightly abandoned any hope of understanding what Aristotle is up to with contingency.62 It seems best to consider some of the relevant material on those combinations at this point. At the beginning of chapter 21 Aristotle says of first-figure U+C syllogisms, ‘in the first figure when one of the premisses signified

42

Introduction

contingency, the conclusion was also contingent’ (39b14-16). In his comment Alexander quotes Aristotle’s introductory remark on U+C first-figure cases in chapter 15: If one premiss is taken to be unqualified and the other contingent, when the premiss relating to the major extreme signifies contingency, all syllogisms will be complete and their conclusion will be contingent in the way specified which has been described; but when the premiss relating to the minor extreme signifies contingency, all of them are incomplete, and the conclusions of the privative syllogisms will not be contingent in the way specified but rather their conclusions will be that something holds of none by necessity or does not hold of all by necessity. For if something holds of none by necessity or does not hold of all by necessity, we say that it is contingent that it holds of none or not of all. (33b25-33)

And he remarks: So perhaps what he says in the present passage concerns affirmative premisses . Or perhaps he was being extremely precise when he said that the conclusion, which he said was of none by necessity, is not contingent, since ‘of none by necessity’ is different from ‘by necessity of none’, as he showed then. Nevertheless, thereafter he also classifies a negative proposition of this kind among contingent negative propositions because it is not directly unqualified. For in the case of a mixture in the second figure of a negative unqualified major and a contingent minor he already said63 that the conclusion is contingent in the way specified. (245,28-35)

Thus Alexander’s conclusion is that although originally Aristotle distinguished between, e.g.,  NEC(AiC) and CON(AeC), and only considered the latter to be a truly contingent proposition, he later started calling  NEC( P) contingent because it doesn’t imply P. But, of course, this does nothing to solve the formal problems. We refer the reader to the commentary on chapter 21 for details of how Alexander tries to cope, and return to his discussion of second-figure U+C cases. We have seen that Aristotle accepts Cesare2(UC‘C’) and Camestres2(CU‘C’). On the other hand, he rejects: Cesare2(CU_) Camestres2(UC_)

CON(AeB) AaB

AaC CON(AeC)

(37b19-23) (37b19-23)

All he says in rejecting these is that ‘the demonstration will be the same and use the same terms’. Alexander first cites the failure of EE-conversionc and the impossibility of a reductio argument. He takes Aristotle’s ‘same terms’ to be the ones used at 1.17, 37b3-10 in the rejection of Cesare2(CC_). These verify:

Introduction

43

CON(White e Horse) White a Horse White a Human CON(White e Human) NEC(Horse e Human) and so – according to Alexander – rule out any contingent or unqualified conclusion. Since it is not clear that they rule out a necessary negative conclusion, Alexander adds terms verifying NEC(BaC), taking as true CON(White e Human) White a Human White a Literate CON(White e Literate) NEC(Human a Literate) Once again it seems that these two sets of terms can be used against any second-figure U+C combination. Alexander has an at least partial realization of this point, since after his discussion of Aristotle’s justification of Cesare2(UC‘C’)and Camestres2(CU‘C’), he raises the question (232,10ff.) whether the terms used to reject Camestres2(UC_) and Cesare2(CU_) won’t do just as well for rejecting the two whose validity Aristotle has just affirmed. He defends the terms against certain objections and then says: It seems that he posits these combinations as syllogistic by only paying attention to the conversion of the unqualified negative premiss and by not investigating them using terms. (232,30-2)

But, using terms, Alexander is in a position to reject all the second-figure U+C syllogisms accepted by Aristotle. We refer the reader to the Summary for the other material in chapter 18, which is of less interest. III.E.2.c. Third-figure U+C combinations (1.21) We shall pass over most of the material on U+C third-figure combinations and concentrate on one important passage. At 31b31-9 Aristotle offers a justification of: Bocardo3(CU‘C’)

CON(AoC)

BaC

 NEC (AoB)

assuming  CON(AoB), ‘i.e.’, NEC(AaB). But since BaC, by the nonTheophrastean Barbara1(NUN), NEC(AaC), contradicting CON(AoC). Alexander repeats Aristotle’s proof (247,10-20), and then raises the question why Aristotle didn’t try for a direct derivation for the combination OA_3(CU_). He mentions two possibilities. The first appears to be illegitimate. In the second Alexander apparently justifies the conclusion  NEC (AiB), essentially making use of the fact that OAI3(CU‘C’) is a waste case of Disamis3(CU‘C’). ‘And this proof,’ he says, ‘was thought

44

Introduction

more acceptable. For the reductio ad impossibile is objectionable.’ (247,29-30). Presumably Alexander has in mind Aristotle’s use of the non-Theophrastean Barbara1(NUN). For, after correctly denying the possibility of establishing Bocardo3(UC‘C’), he points out that Aristotle’s proof depends on Barbara1(NUN), which is unacceptable to Theophrastus. He then says that people use Bocardo3 (‘C’U‘C’) to justify: Barbara1(NUN)

NEC(AaB)

BaC

NEC(AaC)

For, if  NEC(AaC), i.e.,  NEC (AoC) then (Bocardo3(‘C’U‘C’))  NEC (AoB), i.e.,  NEC(AaB), contradicting NEC(AaB). It is hard to see how Theophrastus could possibly accept Bocardo3( N ,U,  N ) and reject Barbara1(NUN), and both are equally violations of the peiorem rule. However, Alexander proceeds to give us Theophrastus’ U-for-C proof for Bocardo3(‘C’U‘C’). He changes CON(AoC) into AoC, assumes NEC(AaB) and infers (Baroco2(NUU)) BoC, contradicting BaC. We do not know how to make sense out of Theophrastus’ position – if he had one –, but it seems clear that he was willing to use illegitimate U-for-C argumentation. III.E.3. N+C premiss combinations (1.16, 19, 22) Both Aristotle and Alexander stress the parallels between the N+C and U+C cases. They are helpful sometimes as a mnemonic, but overstressing them can cause one to lose sight of important differences in Aristotle’s handling of them. III.E.3.a. The first figure (1.16) The obvious parallel and the source of many of the others is that Aristotle treats the standard CNC first-figure cases as complete, and says that the NC_ cases need to be justified. Aristotle never actually discusses Darii3(CNC). He asserts the completeness of Barbara1(CNC) at 36a2-7, and Alexander simply affirms Aristotle’s words in his own terms. But when Aristotle affirms the completeness of Celarent1(CNC), he also denies that one could establish Celarent1(CNU) by reductio. For difficulties with this claim and Alexander’s treatment of it see 209,32211,17. At 36a39-b1 Aristotle asserts that Ferio1(CNU) is not valid, and Alexander confirms (213,6-11) that it cannot be established by a reductio. The NC cases are much more difficult. As we have seen, Aristotle, believes he has established all of Barbara1(UC‘C’), Celarent1(UC N ), Darii1(UC‘C’), and Ferio1(UC N ), where, in fact, all the conclusions are at best all of the  N variety, and are established by logically disastrous U-for-C substitutions. Aristotle also thinks mistakenly that he can justify:

Introduction Barbara1(NC‘C’)

NEC(AaB)

CON(BaC)

45 ‘CON’(AaC)

and Darii1(NC‘C’)

NEC(AaB)

CON(BiC)

‘CON’(AiC)

In the case of Barbara1(NC‘C’), Aristotle says only that ‘it will be proved in the same way as in the preceding cases’ (36a1-2). Alexander first gives a U-for-C argument for Barbara1(NC N ), making clear at 207,3-18 that the conclusion established is not contingent in the way specified. The argument uses Bocardo3(NUU). At 207,19 he points out that, if Bocardo3(NC‘C’) were available, one could establish Barbara1(NC N ) by reductio without changing the contingent premiss into an unqualified one. Alexander remarks that Aristotle does not yet have Bocardo3(NC_) available to him, but unfortunately he does not discuss the status of the mood itself, and its status is very unclear; see 252,3254,35 with the notes. Here we remark only that there would seem to be no way in which one could hope to establish Bocardo3(N, N , N ) independently of Barbara1(N, N , N ). After pointing out that Bocardo3(NC_) is not available to Aristotle, Alexander makes the devastating remark that ‘it is necessary to understand that insofar as it involves a reductio ad impossibile using the third figure, in the case of a mixture having a universal necessary major premiss (either affirmative or negative) and a contingent minor it can be proved that there is a necessary and an unqualified and a contingent conclusion; and the conclusions are affirmative if the necessary premiss is affirmative and negative if it is negative. But he himself has said , “But there will not be a syllogism of by necessity not holding” ’ (207,29-34). We understand Alexander to be saying that if one accepts certain third-figure syllogisms, presumably acceptable ones, but ones not yet treated by Aristotle, one can show that each of Barbara1(NC_), Celarent1(NC_), Darii1(NC_), and Ferio1(NC_) yields a contingent, an unqualified, or a necessary conclusion. We shall call such arguments circle arguments because they make use of things not yet established. The arguments for necessary and contingent conclusions make heavy use of Theophrastean contingency, and we have judged it most perspicuous to introduce CONt and Ct as abbreviations for  NEC and  N . (The arguments in the middle column go through without this change.)64

Assume i.e., But

Barbara1(N,Ct,_)

NEC(AaB)

CONt(BaC)

 NEC(AaC) CONt(AoC) CONt(BaC)

 (AaC) AoC CONt(BaC)

NEC (AaC) NEC(AoC) CONt(BaC)

46

Introduction

Therefore, by Bocardo3(Ct,Ct,Ct) CONt(AoB) i.e.,  NEC(AaB) contradicting NEC(AaB)

Bocardo3(U,Ct,Ct) CONt(AoB)  NEC(AaB) NEC(AaB)

Celarent1(N,Ct,_) NEC(AeB)  NEC(AeC) CONt(AiC) CONt(BaC) Disamis3(Ct,Ct,Ct) CONt(AiB) i.e.,  NEC(AeB) contradicting NEC(AeB) Assume i.e., But Therefore, by

Assume i.e., But Therefore, by i.e., contradicting

Assume i.e., But Therefore, by i.e., contradicting

Bocardo3(N,Ct,Ct) CONt(AoB)  NEC(AaB) NEC(AaB) CONt(BaC)

 (AeC) NEC (AeC) AiC NEC(AiC) CONt(BaC) CONt(BaC) Disamis3(U,Ct,Ct) Disamis3(N,Ct,Ct) CONt(AiB) CONt(AiB)  NEC(AeB)  NEC(AeB) NEC(AeB) NEC(AeB)

Darii1(N,Ct,_)

NEC(AaB)

 NEC(AiC) CONt(AeC) CONt(BiC) Ferison3(Ct,Ct,Ct) CONt(AoB)  NEC(AaB) NEC(AaB)

 (AiC) NEC (AiC) AeC NEC(AeC) CONt(BiC) CONt(BiC) t t Ferison3(U,C ,C ) Ferison3(N,Ct,Ct) CONt(AoB) CONt(AoB)  NEC(AaB)  NEC(AaB) NEC(AaB) NEC(AaB)

Ferio1(N,Ct,_)

NEC(AeB)

CONt(BiC)

 NEC(AoC) CONt(AaC) CONt(BiC) Datisi3(Ct,Ct,Ct) CONt(AiB)  NEC(AeB) NEC(AeB)

 (AoC) AaC CONt(BiC) Datisi3(U,Ct,Ct) CONt(AiB)  NEC(AeB) NEC(AeB)

NEC (AoC) NEC(AaC) CONt(BiC) Datisi3(N,Ct,Ct) CONt(AiB)  NEC(AeB) NEC(AeB)

CONt(BiC)

Aristotle’s wavering treatment of contingency makes it difficult to evaluate the force of these arguments against him. The status of all three versions of Bocardo3 with a contingent minor premiss in Aristotle’s syllogistic is uncertain, but he is committed to versions of all of the other third-figure syllogisms used in these arguments. Alexander does not give the arguments for Barbara1(NCtN) or Barbara1(NCtU) in his discussion of chapter 16, but gives the arguments for Celarent1(NCtN) at 213,11-19 and again at 249,20-5 (where he proceeds to give an argument for Cesare2(NCtN)), and for Celarent1(N,Ct,U) at 216,34-217,2. He gives a similar argument for Ferio1(N,Ct,N) at 213,19-25, for Darii1(NCCt) at

Introduction

47

213,36-214,12, and for Darii1(NCtU) at 214,12-18. Obviously this is a devastating situation for syllogistic, even if the problematic Barbara1 cases are left out of account. Unfortunately, Alexander leaves us once again with references to another work for fuller discussion (207,35-6, 213,25-7). The reductions for the first-figure NC cases discussed thus far all argue indirectly from the denial of the conclusion and the contingent premiss. We wish now to look at reductions using the denial of the conclusion and the necessary premiss, focusing on Aristotle’s and Alexander’s treatment of Celarent1(NC_), which proceeds in this way, but with certain complications which we leave out of account in this introduction.65 We begin from the opening statement of chapter 16, which relies on the parallelism between U+C and N+C combinations: But when one premiss signifies holding or not holding by necessity and the other contingency, there will be a syllogism if the terms are related in the same way; and it will be complete when necessity is posited in relation to the minor extreme. If the terms are affirmative, whether they are posited universally or not universally, the conclusion will be contingent and not of holding. But if one is affirmative and the other privative, when the affirmative is necessary, the conclusion will be contingent and not of not holding; but when the privative is necessary, the conclusion will be that it is contingent that something does not hold and that it does not hold, whether the terms are universal or not universal. And one should take its being contingent that something does not hold in the conclusion in the same way as in the preceding. But there will not be a syllogism of by necessity not holding, since not holding by necessity is distinct from by necessity not holding. (35b23-34)66

Alexander understands this last line to say that the conclusion of a first-figure N+C case will be contingent in the same way as the corresponding U+C case. But the contrast Aristotle has made in the preceding lines amounts to saying that the NC cases other than Celarent1(NC_) and Ferio1(NC_) have a contingent conclusion and do not have an unqualified one, whereas those two have both an unqualified and a contingent conclusion. And this is borne out by Aristotle’s handling of Celarent1(NC_). He first (36a7-15) offers an argument for Celarent1(NCU)

NEC(AeB)

CON(BaC)

AeC

and then asserts (36a15-17), ‘And it is evident that there is also a syllogism which concludes that it is contingent that A does not hold since there is one which concludes that A does not hold’, that is, he infers ‘It is contingent that AeC’ from AeC. We shall introduce a new symbol CONu to indicate this kind of case,67 and represent Celarent1(NC_) as follows:

48

Introduction Celarent1(NCCu)

NEC(AeB)

CON(BaC)

CONu(AeC)

It seems reasonably clear that Aristotle suspects something is wrong with his apparent ability to justify inferences to unqualified propositions when his assumptions include a contingent one,68 and is attempting to paper this over by weakening the conclusion from an unqualified one to a contingent one. What is more important for us is that because of what Aristotle says at the beginning of chapter 16 Alexander frequently lumps together CONu(P) and  NEC( P) as cases of contingency not in the way specified,69 although he does distinguish them when Aristotle gives the following summary comparison of the first-figure U+C and N+C cases: It is clear from what has been said that when the terms are related in the same way there is or is not a syllogism with an unqualified premiss and with necessary ones except that if the privative premiss is posited as unqualified the conclusion of the syllogism is contingent, but if it is posited as necessary the conclusion is both of contingency and of not holding. (36b19-24)

Here Aristotle implies that the conclusion of Celarent1(UC N ) is contingent, but Alexander knows perfectly well that he has denied that it is contingent in the way specified. Alexander writes: But how can this assertion be sound? For he also showed that in the combinations in which the negative premiss is unqualified the conclusion is not contingent in the way specified but is ‘A holds by necessity of no C’ or ‘A does not hold by necessity of all C’. Or because the conclusion is still contingent in a way even if it isn’t straightforwardly contingent in the way specified? ‘A holds by necessity of no C’ is contingent in this sense. But in the case in which the premiss is necessary negative, it was proved that the conclusion is straightforwardly unqualified ... . (216,7-13)

Aristotle’s derivation of: Celarent1(NCU)

NEC(AeB)

CON(BaC)

AeC

is confusing but essentially sound. He assumes  AeC, i.e., AiC, converts NEC(AeB) to NEC(BeA), and infers (Ferio1(NUN)) NEC(BeC), contradicting CON(BaC). At 209,4-18 Alexander points to the use of a first- figure NUN case and gives a derivation of Celarent1(NC N ) using Ferio1(NNN). At 209,22-32, he gives a U-for-C derivation for Celarent1(NCU). Aristotle’s treatment of Ferio1(NCU) at 36a34-9 is completely analogous, and so is Alexander’s commentary (212,4-28), although he does not mention the possibility of a U-for-C derivation. Alexander also gives U-for-C justifications of Barbara1(NC‘C’) at 206,22-

Introduction

49

207,3 and of Darii1(NC‘C’) at 213,27- 214,12, but he does not point out that there are straightforward ‘uncontroversial’ indirect derivations of Barbara1(NC N ) (using Baroco2(NNN)) and Darii1(NC N ) (using Celarent2(NNN)). Alexander recurs to the difference between Celarent1(UC_) and Celarent1(NC_) in connection with Aristotle’s concluding remark at 36b19-24, which we have just quoted. He first argues that one cannot establish Celarent1(UCU) by a reductio, analogous to the one used for Celarent1(NCU), although one could give a U-for-C justification of it. He points out again that Aristotle’s justification of Celarent1(NCU) depends on his acceptance of first-figure NUN cases. He offers the circle argument for Celarent1(NCU) which we have already given, but then points out that no such argument can be given for Celarent1(UCU) unless one uses U-for-C substitution. This leaves him with the question of why such a substitution argument doesn’t suffice. He concludes by applying his interpretation of 1.15, 34b7-18 to the present case. The major premiss AeB of Celarent1(UC_) is true only at a time, and ceases to be true when AiC holds and the contingent premiss CON(BaC) is transformed into BaC. If this remark has any force, it is equally devastating against Aristotle’s justification of Celarent1(UC N ). The upshot of this discussion of Celarent1(NC_) should be clear. Aristotle’s acceptance of Celarent1(NCU) violates the peiorem rule. But Alexander has at his disposal a legitimate Theophrastean argument for Celarent1(NC N ), so that when Aristotle uses Celarent1(NCU) to draw unqualified conclusions from N+C combinations in the second and third figure, Alexander is in a position to substitute a justification for a  NEC conclusion. Moreover, he has made clear that the only kind of derivation which Aristotle could use for Celarent1(UCCu) requires the questionable U-for-C substitution. Thus, he is in a position to maintain the peiorem rule, but at the cost of disallowing U-for-C justifications, on which Aristotle’s treatment of the first-figure UC combinations turned. III.E.3.b. The second figure (1.19) There is no reason to doubt that Aristotle intends a perfect parallelism between the second-figure U+C and N+C combinations. He is in general more explicit about the character of the contingency of the conclusion in the latter case than he was in the former. The most interesting combinations are: Cesare2(NCCu) Camestres2(CNCu)

NEC(AeB) CON(AaB)

CON(AaC) NEC(AeC)

CONu(BeC) CONu(BeC)

Since each is proved by reduction to Celarent1(NC_), Alexander is in a position to give a justification of a Cu or a  N conclusion in each case.

50

Introduction

But the text of Aristotle forces him into more elaborate procedures. For Cesare2(NC_) Aristotle proceeds by acting as if the reduction to Celarent1(NC_) yielded the conclusion ‘It is contingent that BeC’. He then adds an indirect derivation that the premisses of Cesare2(NC_) imply BeC.70 Alexander points out (235,21-30) that this derivation depends on Aristotle’s acceptance of Ferio1(NUN), effectively leaving the reader with a Theophrastean justification of Cesare2(NC N ). Aristotle says that Camestres2(CN_) will be handled in the same way. Alexander is concerned about the extra conversion of the conclusion required by Camestres2. He points out (235,33-236,6) that this is all right for Aristotle since he takes the conclusion of Celarent1(NC_) to be CONu(AeC) and that does convert to CONu(CeA). But he also points out (236,11-14) that the conversion is acceptable for those who take the conclusion of Celarent1(NC_) to be  NEC (AeC). Aristotle’s rejections raise even more interesting questions. He offers a very elaborate rejection of Cesare2(CN_)71 and says that there will be a similar rejection of Camestres2(NC_). Alexander describes simple derivations for Cesare2(CN N ) (at 238,29-34) and of Camestres2 (NC N ) (at 238,24-9). Since Aristotle never mentions Festino2(CN_), Alexander doesn’t either. But he obviously can justify Festino2(CN N ) as well. Thus Alexander has derivations for three combinations actually or presumably rejected by Aristotle. His conclusion is the usual disappointing reference to another work: If this is the way things are, then either reductio ad impossibile should be rejected as insufficient to show that a combination is syllogistic, or, if this cannot be rejected, it would seem that material terms are not sufficient to reject a combination as non-syllogistic. I have also said what the solution of this difficulty is in my book on mixtures. (238,34-8)

Our delights, however, do not stop here. For Aristotle also rejects AA_2(NC_) and AA2(CN_) at 38b13-23, but Alexander shows (240,4-10) that he is committed to AAE2(NC N ); and he could also verify AAE2(CN N ). Thus Alexander is in a position to show that Aristotle is committed to saying that every second-figure U+C combination with two universal premisses is syllogistic. There are further similar problems with the combinations involving a particular premiss, but we shall not deal with them here. III.E.3.c. The third figure (1.22) At the beginning of chapter 22 Aristotle gives a general description of the situation with third-figure N+C universal combinations, distinguishing between those that yield a ‘contingent’ conclusion only and those that also yield an unqualified one, and remarks that there will be no necessary conclusions ‘just as there wasn’t in the other figures’. We

Introduction

51

have already seen in section III.E.3.a. that Alexander is in possession of derivations for at least Celarent1(NCtN) and Ferio1(NCtN). He repeats his derivation of the former and adds one for Cesare2(NCtN), presumably to suggest that Aristotle was wrong about both of the other figures. Unfortunately he again leaves us dangling as to what he thinks of this matter: It is worth asking why he says that in a mixture of a necessary negative universal premiss and a contingent affirmative one there is no necessary negative conclusion that X holds of no Y by necessity in any of the figures. For either (i) it is necessary that reductio ad impossibile be rejected; or (ii) the combinations in the third figure through which I produced the reductio ad impossibile must be non-syllogistic; or (iii) it follows that X holds of no Y by necessity. [250,1] As I have said already, I have investigated this and spoken about it at greater length in On the disagreement of Aristotle and his associates concerning mixtures. And I have discussed it at greater length in my notes on logic. (249,32-250,2)

Aristotle accepts Darapti3(NC‘C’) at 40a12 and its waste case AEI3(NC‘C’) at 40a33. He accepts Darapti3(CNC) at 40a16, Felapton3(CNC) at 40a18, and Felapton3(NCCu) at 40a25. Neither Aristotle nor Alexander mentions the EE cases, but EE_3(NC_) would be reducible to Felapton3(NC_), and EE_3(CN_) would stand or fall with: AE_3(CN_)

CON(AaC)

NEC(BeC)

which Aristotle rejects at 40a35. The terms he gives assume the truth of the following propositions: CON(Sleeping a Human) NEC(Sleeping-horse e Human) NEC(Horse-that-is-awake e Human) NEC(Sleeping a Sleeping-horse) NEC(Sleep e Horse-that-is-awake) Alexander treats Aristotle’s two ‘conclusions’ as what he calls necessary on a condition:72 All horses which are in fact sleeping must be asleep All horses which are in fact awake must not be asleep Since Alexander treats such assertions as unqualified, he takes Aristotle’s interpretations to rule out only an unqualified or necessary conclusion, but not a contingent one. Alexander considers the possibility of proving AEO3(CN‘C’). His discussion is affected by his uncertainty over the status of  NEC propositions and whether they are contingent in the way specified. Alexander first converts CON(AaC) to CON(CiA)

52

Introduction

producing the premisses of Ferio1(NC_). For Aristotle the conclusion is then CONu(BoA), which does not convert to CONu(AoB). Alexander continues: However, if not ‘B does not hold of some A’ but ‘It is contingent that B does hold of some A’ were to follow in the case of the mixture under consideration, a syllogism would seem to result because a particular contingent negative proposition converts. (251,35-7)

However, what Alexander has shown to follow in this case (see section III.E.3.a) is  NEC (BoA), and that, too, does not convert. Aristotle’s discussion of the universal/particular combinations is very unclear, and so is Alexander’s treatment of it. We refer the reader to the notes on 252,3-254,9 for an account. IV. Theophrastus and modal logic73 It is clear from Alexander’s commentary that Theophrastus made some suggestions for improving Aristotle’s logic. However, it does not seem possible to extract from the commentary a satisfactory Theophrastean modal logic. We are inclined to think that Theophrastus never worked one out, but merely put forward some criticisms and suggestions. We here try to explain the problem by indicating how one might go about extracting a system from Alexander’s remarks. It is generally agreed that Theophrastus’ modal syllogistic involved three major departures from Aristotle’s: (i) restriction of the relevant notion of contingency to CONt, i.e.,  NEC ; (ii) adoption of the peiorem rule; (iii) rejection of the use of ekthesis to justify Baroco2(NNN) and Bocardo3(NNN). Logically, the peiorem rule disqualifies as non-syllogistic all CtCt and U+Ct combinations because of the following equivalences: CONt(P1) and CONt(P2) yield CONt(P3) to NEC(3) and CONt(P2) yield NEC(1) CONt(P1) and P2 yield CONt(P3) to CONt(P1) and NEC( P3) yield P2 P1 and CONt(P2) yield CONt(P3) to NEC( P3) and CONt(P2) yield P1 If Theophrastus agreed with Aristotle on non-modal syllogistic, then, he might have developed a system satisfying the following conditions: (i) P1 and P2 yield P3 if and only if NEC(P1) and P2 yield P3 and also P1 and NEC(P2) yield P3 and (ii) NEC(P1) and NEC(P2) yield NEC(P3) if and only if

Introduction

53

NEC(P1) and CONt(P2) yield CONt(P3) if and only if CONt(P1) and NEC(P2) yield CONt(P3) Aristotle treats all the first-figure N+U syllogisms as complete and reduces the second- and third-figure syllogisms to them, although his handling of the indirect reductions is unsatisfactory. Theophrastus could have done the same thing unproblematically, although he could also give indirect reductions of all the first-figure N+U syllogisms to UU syllogisms, reducing NEC(P1), P2, P3 to3, P2 P1 (so that  NEC(P1)), and P1, NEC(P2), P3 to  P3, P1,  P2 (so that  NEC(P2)). So the real problem lies with (ii). As we mentioned in section II.B, Alexander says that Theophrastus chose to postpone the justifications of: Baroco2(NNN)

NEC(AaB)

NEC(AoC)

NEC(BoC)

NEC(AoC)

NEC(BaC)

NEC(AoB)

and: Bocardo3(NNN)

until he had the materials for an indirect justification of them. It seems clear that the syllogisms needed are: Barbara1(NCtCt)

NEC(AaB)

CONt(BaC)

CONt(AaC)

NEC(BaC)

CONt(AaC)

for Baroco2(NNN), and: Barbara1(CtNCt)

CONt(AaB)

for Bocardo3(NNN). Aristotle treats Barbara1(CNC) as complete. Theophrastus could have done likewise for Barbara1(CtNCt). That would take care of Bocardo3(NNN). We have seen in section III.E.3.a that Aristotle does not deal very explicitly with Barbara1(NC_), but seems to have had in mind an illegitimate U-for-C justification of what could perhaps be written as Barbara1(NCtCt). The only other option we can see for Theophrastus would be to declare Barbara1(NCtCt) complete. If he did so, it is somewhat surprising that Alexander never mentions the possibility of proceeding in this way.74 As we have mentioned in section III.E.2.c, Alexander does tell us that Theophrastus gave a U-for-C justification of what we will now write as: Bocardo3(CtUCt)

CONt(AoC)

BaC

CONt(AoB)

This information is disconcerting since any CtUCt syllogism violates the

54

Introduction

peiorem rule,75 and Bocardo3(CtUCt) obviously could be used to justify Barbara1(NUN). The violation of the peiorem rule is not surprising since U-for-C argumentation involves strengthening a contingent premiss into an unqualified one. If we ignore Alexander’s remark about Theophrastus’ justification of Bocardo3(CtUCt), we can imagine a lovely Theophrastean system in which the only syllogistic patterns are the standard ones of nonmodal syllogistic, the modal syllogisms all obey the peiorem rule, and the syllogistic combinations are all NN, N+U, or N+Ct. This system could be developed by taking the first-figure N+Ct combinations as complete and reducing the second- and third- figure combinations to them, following the pattern of the nonmodal reductions. One could then reduce the NNN combinations to the N+Ct combinations or follow the Aristotelian pattern except for the reductions of Baroco2(NNN) to Barbara1(NCtCt) and of Bocardo3(NNN) to Barbara1(CtNCt). One could follow the same procedure for the N+U combinations or they could be reduced to UUU combinations. We are very doubtful that Theophrastus worked out such a system. We believe that if he had done so, Alexander’s commentary on Aristotle’s modal syllogistic would have been much clearer. 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 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,

Notes to pp. 7-15

55

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, 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.

56

Notes to pp. 16-25

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 section III.E.2a below. 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 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

Notes to pp. 26-38

57

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 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. 55. It is true that at 39a36-8 Aristotle says, ‘Similarly if AC is negative and BC affirmative, since again the first figure will result by conversion’, a description which fits both Ferison3 and Bocardo3. But Aristotle says nothing to show how one might reduce Bocardo3(CCC) to the first figure. Alexander does the reduction for Ferison3(CCC), and says nothing about Bocardo3. 56. For similar misgivings about the conclusion of Barbara1(NC_) see 207,318. 57. For textual difficulties in Aristotle’s argument for Barbara1(UC‘C’) see the note on 185,32. The argument we present here is perhaps more like what Aristotle ought to say than what he literally says, but we think it must be what

58

Notes to pp. 38-54

he has in mind. See also the note on 187,9 for the special difficulties caused by 34b2-6 (now generally thought to be interpolated, but not questioned by Alexander). 58. mêdeni ex anankês, i.e.,  NEC(AiC). See Appendix 1 on the expression ‘by necessity’. 59. Alexander’s treatment of Darii1(UC‘C’) and Ferio1(UC N ) in his commentary on chapter 15 (202,7-203,1) implies by its silence that the conclusions are contingent in the way specified, but at the beginning of the next chapter (205,26-206,11) he makes clear that this is not so for Ferio1(UC N ). 60. For a suggestion of some misgivings see 188,7-17, where, however, Alexander ends by referring us to another work for fuller discussion. 61. Alexander recognizes that the interpretation rendering NEC(AaC) true does not produce this conflict; see 190,26-191,1. 62. See 246,15-35 63. At 1.18, 37b28 Aristotle describes the conclusion of Cesare2(UC‘C’) as ‘It is contingent that B holds of no C’, but he doesn’t say anything about the nature of the contingency. 64. We note that the arguments given in the text will not go through for UCt combinations because the  NEC statements in the next-to-last lines would not contradict the unqualified premiss. 65. See the note on 208,7. 66. For various textual matters see the textual notes on 35b23, 35b32-3, 35b34, and 35b35 (Appendix 6). 67. Formally this new symbol is totally redundant since CONu(P) is equivalent to P. 68. If there is something wrong, it lies in the acceptance of the first-figure NUN cases, which enable one to give indirect proofs for both Celarent1(NCU) and Ferio1(NCU). 69. See, e.g., 231,35-6 and 232,36-233,12. Alexander’s mistake is due to Aristotle’s misleading assertion at 35b32-4 that ‘one should take contingency in the conclusion in the same way as in the preceding’. 70. For further complications in Aristotle’s treatment of Cesare2(NC_) see the note on 235,3. 71. For the details see 236,15-238,10 with the notes. 72. See Appendix 3 on conditional necessity. 73. On Theophrastus’ modal logic see Bochenski (1947), pp. 67-102 and Repici (1977), pp. 103-31. 74. Further possible evidence against Theophrastus having done this is the reduction of Celarent1(NCtCt) to Darii1(CtNCt) which Philoponus (in An. Pr. 205,13-27, Theophrastus 109A FHSG) ascribes to ‘those around Theophrastus’. However, it is interesting to note that apparently Theophrastus did not reduce Celarent1(NCtCt) to Ferio1(NNN), as Alexander does at 209,9-18. Darii1(NCtCt) and Ferio1(NCtCt) can only be reduced to Celarent1(NNN). 75. That Theophrastus did allow CtUCt or UCtCt combinations is confirmed by 173,32-174,2, which tells us that Theophrastus and Eudemus said that the conclusion of a mixture of a contingent and an unqualified premiss will be contingent.

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)

60

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

61

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 (28a26-30) Since BaC (AI-conversionu) CiB. So (Darii1(UUU)) AiB. Felapton3(UUU) AeC BaC AoB (28a17-26) 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)

62

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 (cf. 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

63

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)

64

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

65

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’) AaB CON(BaC)  NEC  (AaC) (34a34-b2) *Celarent1(UC  N  ) AeB CON(BaC)  NEC  (AeC) (34b1935a2) Darii1(UC‘C’) AaB CON(BiC)  NEC  (AiC) (35a35-b8) Ferio1(UC  N  ) AeB 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 CON(BeC)  NEC  (AaC) (35a3-11) AeB CON(BeC)  NEC  (AeC) (35a11-18) AaB CON(BoC)  NEC  (AiC) (35a35-b8) AeB CON(BoC)  NEC  (AoC) (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?)

66

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) Disamis3(CU‘C’) Ferison3(CUC) Ferison3(UC‘C’) IEO3(UCC) IE?3(CU?)

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

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

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

Waste cases justifiable by transformationc rules AEI3(UC‘C’) EEO3(UC‘C’) EOO3(UC‘C’) OE?3(CU?) tioned)

AaC CON(BeC)  NEC  (AiB) (39b22-5) AeC CON(BeC)  NEC  (AoB) (39b22-5) AeC CON(BoC)  NEC  (AoB) (not mentioned) CON(AiC) BeC  NEC  (AoB)? (not men-

Indirect reductions  NEC  (AoB) (39b31-9) *Bocardo3(CU‘C’) CON(AoC) BaC AO?3(UC?) AaC CON(BoC) ? (39b31-9?)

Summary AO?3(CU?) OA?3(UC?)

CON(AaC) AoC

67

BoC CON(BaC)

? ?

(39b31-9?) (39b31-9?)

? ?

(not mentioned) (not mentioned)

Further waste cases EO?3(CU?) OA?3(CU?)

AeC CON(AoC)

CON(BoC) BaC

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) CON(BaC)  NEC (AaC) (35b3736a2) *Celarent1(NCCu) NEC(AeB) CON(BaC) CONu(AeC) (36a7-17) *Darii1(NC‘C’) NEC(AaB) CON(BiC)  NEC  (AiC) (36a39-b2) Ferio1(NCCu) NEC(AeB) 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) Camestres2(CNCu) CON(AaB) NEC(AeC) CONu(BeC) Festino2(NCCu) NEC(AeB) CON(AiC) CONu(BoC)

(38a16-25) (38a25-6) (38b24-7)

68

Summary

IEO2(CN?)

CON(AiB) NEC(AeC)

CONu(CoB)? (38b24-7?)

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

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

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 *Cesare2(CN_) CON(AeB) *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’) 16) Darapti3(CNC) Felapton3(CNC) Felapton3(NCCu) Datisi3(CNC) Datisi3(NC‘C’) Disamis3(NCC) Disamis3(CN‘C’) b2) Ferison3(CNC) Ferison3(NCCu) *Bocardo3(CN‘C’) Bocardo3(NCCu)

NEC(AaC) CON(BaC)

 NEC  (AiB) (40a12-

CON(AaC) NEC(BaC) CON(AeC) NEC(BaC) NEC(AeC) CON(BaC) CON(AaC) NEC(BiC) NEC(AaC) CON(BiC) NEC(AiC) CON(BaC) CON(AiC) NEC(BaC)

CON(AiB) (40a16-18) CON(AoB) (40a18-25) CONu(AoB) (40a25-32) CON(AiB) (40a40-b2)  NEC  (AiB) (40a40-b2) CON(AiB) (40a40-b2)  NEC  (AiB) (40a40-

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

CON(AoB) (40b2-3) CONu(AoB) (40b3-4)  NEC  (AoB) (40b2-3)5 CONu(AoB) (40b3-4?)

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

Waste cases justifiable by transformationc rules

AE_3(NC‘C’) EEO3(NCCu) AOI3(NC‘C’) IEO3(NCC) EOO3(NCCu) OEO3(NCCu)

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

Summary

69

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

 NEC  (AiB)? (40a33-5) 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: NEC(AiC) CON(BeC) CON(AiB) IEI3(NCC) 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 1.14-22 Translation

Textual Emendations

74

Textual Emendations

1.14-16 The first figure 1.14 Combinations with two contingent premisses1

32b382 When it is contingent that A holds of all B and B of all C, [there will be a complete syllogism that it is contingent that A holds of all C. This is evident from the definition. For we explain3 ‘It is contingent that X holds of all Y’ in this way. 33a1 Similarly, too, if it is contingent that A holds of no B and B of all C, there will be a complete syllogism that it is contingent that A holds of no C. For ‘It is contingent that A holds4 of that of which it is contingent that B is said’ was for none of the things of which it is contingent that they are under B to be left out.] It is clear that it is contingent , since he has just said this.5 He says that the conclusion will be universal affirmative contingent when the premisses are taken in this way as universal affirmative contingent in the first figure. (It is necessary to discuss this figure and the combinations in it first because the combinations in the other figures are proved through them.) The words ‘This is evident from the definition’ indicate that the fact that a universal contingent affirmative conclusion follows in the combination under consideration is clear from the definition of ‘said of all’. For ‘X is said of all Y’ was ‘it is not possible to take anything in Y of which X will not be said’.6 So when A is taken to be said of all B contingently, there will be nothing of B of which it will not be contingent that A is said. But C is under B, so that it is also contingent that A is said of all of C. (33a1) The proof is the same if it is contingent that A holds of no B and it is contingent that B holds of all C. For it follows again that A holds of no C contingently. And this is again clear because of the definition of ‘X is said of no Y’.7 For it is not possible to take any of B of which it will not be contingent that A does not hold. For the inferences in the first figure come about because of these things. 8 Having said that what follows in the case of the first-figure combination consisting of two universal affirmative contingent premisses is evident from the definition – ‘For we explain9 “It is contingent that X holds of all Y” in this way’ – he first posited the next combination, the one which has the major premiss universal negative contingent and the minor universal affirmative contingent, and he then has given the

167,8

10

15

20

25

76

Translation

30 definition of ‘X is said of all Y’, because of which he says the conclusion of the previous combination10 is clear. Consequently what is said seems to be rather obscure. Or is it necessary to understand as added that ‘X is said of no Y’ in a similar way? For if it is contingent that A holds of no B, it will not be possible to take any of B of which it is not 168,1 contingent that A does not hold. And it is possible that he has referred to the definitions of both the affirmative and the negative with the words ‘For “It is contingent that A holds of that of which it is contingent that B is said” ’, and that along with it is necessary to understand ‘or it is contingent that A does not hold’. 5

33a511 When it is contingent that A holds of all B and it is contingent that B holds of no C, [there is no syllogism based on the premisses taken. But if BC is converted in the way appropriate to contingency, the same syllogism as before results. For since it is contingent that B holds of no C, it is also contingent that B holds of all C; this was said previously.12 Thus if B holds of all C and A of all B, the same syllogism results again. Similarly, too, if the negative proposition is posited along with ‘it is contingent’ in relation to both premisses. I mean if it is contingent that A holds of no B and B of no C. There is again no syllogism based on the premisses taken, but again, if the premisses are converted, there will be the same syllogism as before. It is evident then that if the negative is posited in relation to the minor extreme or in relation to both premisses, either there is no syllogism, or there is one but it is not complete since necessity is reached from the conversion.]

He now takes combinations in the first figure, one having the major premiss universal affirmative and the minor universal negative, with both contingent, the other having both premisses universal negative 10 contingent. Both of these combinations were non-syllogistic in the case of unqualified and of necessary premisses. He says that also in the case of contingent premisses nothing will follow syllogistically either if they are of this sort and are held fixed. However, if negatives are transformed into affirmatives (this can be done since it has been shown that they convert with one another; for ‘It is contingent that X holds of all Y’ 15 converts with ‘it is contingent that X holds of no Y’, as was shown), when the negative contingent premisses are transformed into affirmative contingent ones, he says the combinations will be syllogistic, since they will then consist of two contingent universal affirmative premisses: in the case of the first combination described if the minor is transformed since it was negative, in the case of the second if both are transformed into affirmatives, since both are assumed negative. 13But in fact, if only the minor premiss is transformed there will be a syllogism having

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its major premiss universal negative contingent, its minor universal affirmative contingent, and having its conclusion universal negative contingent. It is then clear that the combinations previously mentioned imply no conclusion on the basis of the premisses assumed. However, the combinations do become syllogistic when negative propositions are converted into affirmative ones on the grounds that the latter are true together with the premisses assumed. This is a peculiar feature of contingent propositions, since it is also only in their case that things said negatively are true together with the affirmations. (It should be noted that he left out the proof that the combination consisting of two universal negative contingent premisses is syllogistic even if only the minor premiss is transformed into an affirmative.) It is necessary to understand that when the negative premisses are transformed into affirmative ones the resulting syllogisms do not preserve the notion of the contingent as what is usual, at least not if the negative premisses were originally taken to be contingent in the sense of being usual. For an affirmative which is infrequent converts with a negative which is contingent in the sense of usual. Consequently, when the negative premisses assumed in the combinations are transformed into contingent affirmative ones, what is contingent in the sense of infrequent and the indefinite will be posited. But when the indefinite is assumed there will indeed be a syllogism, but not one having any use, as he himself said before.14 Consequently we, too, will say that these combinations are without qualification syllogistic with respect to their formal validity,15 but they are useless and non-syllogistic in relation to contingency in the sense of what is usually true. (Some syllogisms in certain arts and in certain counsels, choices, and activities proceed on the basis of this notion of contingency.) Perhaps he was considering this fact when he said ‘either there is no syllogism’. For there is no syllogism for a person with an eye on usefulness – or if someone only looks at the premisses.16 ‘or there is one but it is not complete’ because the syllogism does not result from the premisses assumed but from their transformation and conversion (or together with their conversions).

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33a2117 If18 one of the premisses is taken to be universal, the other particular, [and universality is assumed with respect to the major extreme, there will be a syllogism. 33a2319 For if it is contingent that A holds of all B and B of some C, it is contingent that A holds of some C. This is evident from the definition of ‘It is contingent’. 33a2520 Again, if it is contingent that A holds of no B and it is contingent that B holds of some C, it is necessary that it be

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Translation contingent that A not hold of some of the C’s. The demonstration is the same. 33a2721 If the particular premiss is taken as privative, the universal as affirmative, and they are related similarly by position – i.e., if it is contingent that A holds of all B and it is contingent that B does not hold of some C – there is no evident syllogism based on the premisses taken; but if the particular premiss is converted and it is posited that it is contingent that B holds of some C, the conclusion will be the same as before (as in the first cases).]

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Having spoken about combinations of two universal and contingent premisses, he has turned to those having only one universal premiss; and he says that there will be a syllogism if the major is universal, the minor particular, no matter how the premisses are taken to be with respect to being affirmative or negative. However, when the major premiss is universal and affirmative or negative, and the minor is particular affirmative, the conclusions will be directly from the premisses assumed and because of them. And he says this is evident ‘from the definition of “It is contingent” ’. Perhaps he means from the definition of ‘It is contingent that X holds of all Y’ since holding of all was ‘it is not possible to take any of Y of which X will not be said’.22 (And if he meant this, the words ‘of all’ would be missing in what is said since he should say ‘This is evident from the definition of “It is contingent that X holds of all Y” ’.) For if it is contingent that A holds of all B, there will be nothing of B of which it is not contingent that A is said. But something of C is under B, so that it is also contingent that A holds of some C. Or23 does he mean ‘the definition of contingency itself’? The definition of contingency was ‘P is not necessary and if, when P is posited to hold, nothing impossible follows.’ For, in the case of this combination, nothing impossible follows if it is posited that it is contingent that A holds of some C. This is because no syllogistic combination at all results when it is assumed that it is contingent that A holds of some C. For if the contingent particular affirmative premiss BC is converted two particular contingent premisses in the first figure result; and if BC, which is also itself particular affirmative contingent, is added to the conclusion, which is particular affirmative contingent, there results two particular premisses in the third figure; and if the premiss ‘it is contingent that A holds of all B’ is added to the conclusion, two affirmative premisses in the second figure result. But even if we hypothesize the opposite24 of ‘It is contingent that A holds of some C’ and use reductio ad impossibile, nothing impossible turns out. For let the opposite of ‘It is contingent that A holds of some C’ be taken, i.e., ‘A holds of no C by necessity’, and let ‘It is contingent

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that A holds of all B’ be added. It follows in the second figure25 that it is contingent that B holds of no C, which is not impossible when it is assumed that it is contingent that B holds of some C. For if it is contingent that X holds of some Y, X can also hold of no Y, and it can be the case that it is contingent that it holds of no Y. The first interpretation26 of what is said is better. For this proof would show rather that the 15 combination under consideration does not yield a conclusion because nothing impossible followed . (33a25) Similarly, if the major premiss is universal negative contingent and the minor particular affirmative, there follows from the premisses assumed a particular negative contingent conclusion. For the definition of ‘It is contingent that X holds of no Y’ is known, and in the 20 case of this combination it produces the conclusion. (33a27) But if the major premiss is universal and either negative or affirmative,27 and the particular minor is taken to be privative contingent, a syllogism will result, but not on the basis of the premisses assumed; rather it results when the particular negative premiss is transformed into a particular affirmative by converting contingent propositions. [33a3428 If the premiss relating to the major extreme is taken as particular, and that related to the minor is taken as universal, if both premisses are posited as affirmative or both as privative or they are not similar in form or they are both indeterminate or both particular, there will be no syllogism at all. For nothing prevents B from having a greater extension than A and not being predicated equally. Let C be taken as that by which B has a greater extension than A. Then it is not contingent that A holds of all of C or none or some or not of some, since contingent propositions convert, and it is contingent that B holds of more things than A does.] But when the universal premiss is not the major but the major is particular contingent, and the minor is universal contingent, he says there will be no syllogism, whether both premisses are taken to be similar in form (i.e., similar in quality) or dissimilar in form (i.e., different in quality); and there will not be a syllogism if both are taken to be indeterminate or both particular. 29He gives the reason why all such combinations in which the major is particular contingent are non-syllogistic: if it is assumed that it is contingent that A holds of some B, nothing prevents B from having a greater extension than A and being said of more things, for example, if it is assumed that it is contingent that literate holds of something which is asleep; for being asleep is predicated of more things than literate. Thus, if one takes some of the things falling under being asleep by which B, being asleep, has a greater

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extension than A, literate, for example, horse,30 it is not the case that it is contingent that A holds of all horse or contingent that it holds of none. For, if it is contingent that X holds of no Y, it is also contingent that it holds of all Y. But how could it be contingent that literate holds of all 5 horse? For it holds of no horse by necessity. And it is not contingent that literate holds of some horse. Nor is it contingent that it does not hold of some. For if it is contingent that X does not hold of some Y, it is also contingent that X holds of some Y by the conversion of what is contingent, as he himself indicates when he adds, ‘since contingent propositions convert’. So, if the conclusion must be either universal affirmative contingent or universal negative or particular 10 and affirmative or negative , and none of these is possible in the first figure in the case of contingent premisses when the major is particular contingent, the minor universal contingent, there will never be a syllogism when the major is of this kind, as is shown with terms:

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33b331 Furthermore, this is also evident from terms; for if the premisses are related in this way it is not contingent that the first term holds of any of the last and necessary that it holds of all of it. [Common terms for holding of all by necessity: animal, white, human; for not being contingent: animal, white, cloak.]

Having said that in the first figure all combinations of contingent premisses in which the major is particular will be non-syllogistic and also added the reason (for it is possible32 that the middle term have a 20 greater extension than the extreme), he nevertheless also proves that the combination is non-syllogistic by setting down terms, thereby making this sort of proof and this sort of refutation clearer and more striking. He shows through the terms which he sets down both that the first extreme holds of all the last by necessity and that the first term holds of none of the last by necessity, and he does away with all contingent 25 conclusion as well as all unqualified and necessary ones, that is, all of ‘A holds of all C’, ‘A holds of some C’, ‘A holds of no C’, and ‘A does not hold of some C’; for each of the universal necessary conclusions – the affirmative and the negative – does away with all contingent conclusions. He takes animal, white, human as terms for holding of all by necessity. For let it be contingent that animal holds of something white and let it be contingent that white holds of every human; animal holds of 30 every human by necessity. Truer terms would be white, walking, swan. For it is contingent both that white holds of some thing that walks and that it does not hold of some thing that walks, and it is contingent both that walking holds of every swan and that it does not hold of every swan (and also of some); and white holds of every swan by necessity. He sets

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down as terms for holding of none by necessity animal, white, cloak. For it is contingent both that animal holds of something white and that it 172,1 does not hold something white, and it is contingent both that white holds of every cloak and that it does not hold of every cloak (and of none and of some and not of some); and animal holds of no cloak by necessity. Again, truer terms would be white, walking, crow; for the conclusion is that white holds of no crow by necessity and the premisses can be taken 5 in all the ways mentioned. 33b8 It is, then, evident that if the terms are this way there is no syllogism.33 [For every syllogism is either of holding or of necessity or of contingency. But it is evident that there is no syllogism of holding or of necessity. For the affirmative is destroyed by the privative, the privative by the affirmative. It remains that there is a syllogism of contingency. But this is impossible. For it has been shown that when the terms are related this way, it is necessary that the first holds of all of the last and not contingent that it holds of any. Thus there will not be a syllogism of contingency. For the necessary was not contingent.] Having shown by setting down terms that in the combinations under consideration the conclusion is both holding of all and of none by necessity, he says that not only does something contingent not follow but also nothing else, that is, nothing unqualified or necessary. For we have shown34 that holding of all by necessity does away with holding of none by necessity and with holding of none, and holding of none by necessity does away with holding of all by necessity and with holding of all. For using these facts we showed the non-syllogistic combinations with necessary or unqualified premisses . Furthermore, as we have said,35 when the conclusion is necessary or unqualified, it is necessary that one or both of the premisses be necessary or unqualified, but both the premisses in the combination under consideration are contingent. And through this the conclusion would be shown to be neither necessary nor unqualified. Having said that none of these things can follow, he shows next that something contingent cannot follow since holding of all by necessity and holding of none by necessity follow in a combination of this kind. He indicates this with the words ‘and not contingent that it holds of any’.36 For holding of all by necessity does away similarly with all contingency, and with holding of none, with not holding of some, and, furthermore, with holding of none by necessity and not holding of some by necessity. Again, holding of none by necessity itself does away with holding of all or some and with holding of all or of some by necessity, just as it does away with all contingency. He recalls the refutation of contingency we defined the contingent as what is not necessary.37 For

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if A holds by necessity of all C and of no C by necessity, it is not contingent that it holds of all C or of some or of none or not of all.

[33b1838 It is evident that if the terms in contingent premisses are universal, whether they are affirmative or privative, there is always a syllogism in the first figure, except that the syllogism is complete if they are affirmative, incomplete if they are privative. 33b22 It is necessary not to include contingency among necessary things, but in the said way specified – this is sometimes forgotten.]

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(33b18) Having said these things, he reminds us of what has been shown , namely, that in the first figure if both premisses are universal there is a syllogism no matter what the quality of the premisses, except that the syllogism is complete if both are affirmative. Similarly, if only the major premiss is negative. And (as he showed) the syllogisms by conversion of the contingent premiss are incomplete, both the one consisting of two negative premisses and the one with a negative minor. (33b22) He also reminds us that it is necessary to take contingency in the premisses as non-necessity. For the contingency which is predicated of what is necessary is homonymous and does not convert.39 He says that such a thing is often forgotten. For we say of what holds of none by necessity that it is contingent that it holds of none in cases in which it is not possible to say that it is contingent that it holds of all.40 But even he himself in setting out the terms previously mentioned took it that it is contingent that animal holds of something white.41 However, it holds of something white by necessity. And for this reason perhaps he added what he also says elsewhere:42 ‘But the terms ought to be taken in a better way.’ He has proved that in the first figure there are four from two contingent premisses if both premisses are universal; that two are complete, the one from two affirmative premisses and the one from a negative major only; the one from a negative minor only and the one from two negatives are incomplete; similarly there are again four syllogisms when one premiss is particular if the minor premiss is particular; but also two of these are complete, those in which either both premisses are affirmative or only the minor is; but those in which only the minor is negative or both premisses are negative are incomplete.

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1.15 Combinations with an unqualified and a contingent premiss43

33b25 If one premiss is taken to be unqualified and the other contingent, [when the premiss relating to the major extreme signifies contingency, all syllogisms will be complete and their conclusion will be contingent in the way specified which has been described; but when the premiss relating to the minor extreme signifies contingency, all of them are incomplete, and the conclusions of the privative syllogisms will not be contingent in the way specified but rather their conclusions will be that something holds of none by necessity or does not hold of all by necessity.44 For if something holds of none by necessity or does not hold of all by necessity, we say that it is contingent that it holds of none or not of all.] it would be cohesive to discuss first the second and third figure, and then mixtures. But since syllogisms in those figures result from conversions with respect to terms and he has not yet spoken about this kind of conversion in the case of contingent propositions,45 he first discusses mixtures in the first figure of a contingent and an unqualified46 or a contingent and a necessary premiss,47 since what is proved in this figure does not need conversion. Furthermore, he will also show that in the second figure nothing follows from two contingent premisses because nothing follows from two affirmative premises,48 but that something does follow in the case of a mixture of a contingent and an unqualified premiss.49 But first he had to speak about mixtures of an unqualified50 and a contingent premiss in the first figure; and he has made the mixture of an unqualified and a contingent premiss primary. 51 His associates, Theophrastus and Eudemus, say that also in a mixture of a contingent and an unqualified premiss the conclusion will be contingent no matter which of the premisses is taken to be contingent, since again the contingent is weaker than the unqualified. But Aristotle does not say this. Rather he says that when the major is contingent and the minor unqualified, the conclusion will be contingent in the way specified52 – i.e., ‘

if P is not necessary and nothing impossible follows when it is posited to be.’ And he says that the syllogisms in such a combination are complete, that is, they directly prove the proposed conclusion, since the proof for the syllogism is by means of ‘said of all’ and of ‘said of none’, and syllogisms which have their conclusion known through these things are complete. But he also says that all syllogisms in which the minor is contingent and the major unqualified will be incomplete; and furthermore that those which imply a negative conclusion will imply one which

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is contingent not in the way specified but in the sense in which what is merely a negation of necessity is called contingent. For these will imply either ‘A holds by necessity of no C’53 or ‘A does not hold by necessity of all C’,54 and contingency is true when these things are, but not the contingency of which we have given the definition, since what is contingent in that sense does not yet hold.55 For these things signify that something does not hold by necessity without denying that it holds.56 He will make what he means evident as he proceeds. However the associates of Theophrastus57 also call such things contingent and so, reasonably, say that the conclusion in such combinations is contingent. 58 Syllogisms in which the minor is contingent are not complete because in these cases it is not possible to prove the conclusion using said of all. For suppose A holds of all B and it is contingent that B holds of all C; then since C is not yet some of B (because the contingent does not yet hold), it is not the case that if A is said of all B and it is not possible to take anything of B of which A will not be said, that thereby and because of this we also have that A applies to C; for C is not some of B, if it is contingent that B holds of it and B does not actually hold of it. Consequently, since these syllogisms need something external for their proof, they are not complete; for they are proved by reductio ad impossibile. Having said these things, he first proves that syllogisms having the major contingent are complete and have a conclusion which is contingent in the way specified:

33b33 For let it be contingent that A holds of all B, and let it be assumed that B holds of all C.59 [Since, then, C is under B and it is contingent that A holds of all B, it is evident that it is also contingent that it holds of all C. Thus there is a complete syllogism. Similarly if the premiss AB is privative and BC is affirmative and the former is taken to be contingent, the latter unqualified, there will be a complete syllogism that it is contingent that A holds of no C. It is evident then that if holding is posited in relation to the minor extreme, the syllogisms are complete.]

Invoking the notion of said of all and its definition, he proves that the conclusion is contingent and the syllogism complete. For since it is 175,1 contingent that A holds of all B, it is not possible to take any of B of which it will not be contingent that A holds; but C is some of B, if, indeed, B holds of all C; therefore, it will also be contingent that A holds of all C. And if the major premiss is posited to be privative contingent and the minor unqualified universal affirmative, the proof that the conclusion

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will be contingent negative universal is similar. For it will be by means 5 of the definition of ‘it is contingent that X holds of no Y’. Q) and P is possible, Q is possible> 34a2 [It is necessary to show by means of the impossible] that when the premisses are related in the contrary way there will be syllogisms.60 [At the same time it will be clear that these syllogisms are also incomplete since the proof is not from the premisses taken.] He has proved in the case in which both premisses are universal that when the major premiss is contingent the conclusion is contingent and the syllogisms are complete (the same proof works for the case in which 10 the syllogism is particular and the conclusion affirmative or negative).61 He turns to the combinations in which the major premiss has been taken to be unqualified, the minor contingent. For the phrase ‘related in the contrary way’ signifies for him the interchange of the premisses. He says that it will be proved by means of the impossible that combinations of this kind are syllogistic. But if by means of the impossible, it is clear that these syllogisms are not complete. Therefore 15 he adds the words ‘At the same time it will be clear that these syllogisms are also incomplete.’ For complete syllogisms are based on the premisses assumed and need nothing external in addition, but proof by means of the impossible is not through the premisses which have been taken and assumed. 34a5 It should be said first that if when A is it is necessary that B is, then if A is possible,62 B will also be possible by necessity.63

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He is going to prove using reductio ad impossibile that the combination of an unqualified major and a contingent minor is syllogistic; but in the proof ad impossibile in the present cases he hypothesizes and takes not just the opposite of what he wishes to prove follows; rather he also 25 transforms the contingent premiss into an unqualified one; and this is not impossible (for if it is contingent that P come to be, it is not impossible to hypothesize that P is (as the definition which has been given makes clear); but nevertheless false. Consequently in order that someone not think that the impossibility which follows from taking the opposite of what one wishes to prove in the combination under 30 consideration and from the transformation of the contingent premiss into an unqualified one (which is false but not impossible) follows from the transformation of the contingent premiss into an unqualified one

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but think that it follows from the hypothesized impossibility (which is the opposite of the conclusion he wishes to prove), he shows first that something impossible cannot follow from something possible, but that it is necessary that that from which an impossibility follows be impossible in the case of every necessary implication.64 (A necessary implication is not one which holds temporarily, but one in which it is always the case that the consequent follows from what has been taken as antecedent being the case.65 For the conditional ‘If Alexander exists, Alexander is conversing’ or ‘If Alexander exists, he is so and so old’ is not true – even if Alexander is so and so old when the proposition is spoken.) When it is shown , then, in the case of the combination set out, if the conclusion which results from the assumed premisses is impossible, it does not result from the contingent premiss having been transformed into an unqualified one (since the unqualified proposition is false but not impossible), but from the opposite of the conclusion (which is impossible) being taken. At the same time by means of what is now proved he might also be thought to establish that proof by means of reductio ad impossibile is sound. For if it is not agreed that the impossible follows from the impossible, then reductio ad impossibile would not appear to have force either, since a hypothesis would not be absolutely destroyed as impossible when something impossible was inferred . 66 Or can someone make use not of the impossibility but of the falsehood of the conclusion which follows from the hypothesis to do away with the hypothesis as false on the grounds that something false cannot follow from true things? Or is it the case that a conclusion of this kind would not result more from the hypothesis than also from the transformation of the contingent into the unqualified? He shows that a possibility always follows from a possibility and that an impossibility cannot follow from a possibility in the following way: 34a767 For let it be the case that this is the situation and that A is possible, B impossible. [Then if what is possible could come to be when it is possible and what is impossible could not come to be when it is impossible, and if A were possible and B impossible at the same time, then it would be contingent that A comes to be without B, and if comes to be then is . For what has come to be is when it has come to be.]

The words ‘this is the situation’ mean what we have said previously: ‘when A is it is necessary that B is’. If this implication holds and B is 25 the consequent of A by necessity, let it be hypothesized that A is possible and B impossible. Since A is possible, it could come to be when it is possible for it to come to be. Similarly also B, if it is impossible, could

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not come to be when it is impossible for it to come to be. And if the one is possible, the other impossible, at the same time, when A is, B – insofar as it is impossible – would not be, but – insofar as it is necessary that B be when A is – B would be. Therefore B would be and not be at the same time, which is impossible. Therefore, if the antecedent is possible, its consequent by necessity, i.e., B, will also be possible. 34a10  and if68 A were possible and B impossible at the same time, then it would be contingent that A comes to be without B, [and if comes to be then is . For what has come to be is when it has come to be.] He takes as hypothesis that A is possible, B impossible, on the assumption that B follows from A,69 and taking as universal that ‘what is possible could come to be when it is possible and what is impossible could not come to be when it is impossible’. He adds that B follows from A and shows the absurdity. For suppose B is impossible at the time when A is possible (for this is what is meant by ‘at the same time it would be contingent that A comes to be without B’). What can come to be might come to be at some time, but the impossible, which B is, could not come to be; thus if A is (for if it has come to be, it also is), B will not be; but it was assumed that if A is, B is. It could also be shown from the definition of possible that it is not possible that B, which is impossible, follow from A, which is possible. For, if (i), when that which is possible is hypothesized to be, nothing impossible results because of it, and (ii) when A is hypothesized to be there results because of the hypothesis an impossibility, namely that B both is and is not (is, since it was assumed that it follows from A, and is not, because it is impossible), then if (iii) something from which an impossibility follows is impossible,70 it is either not possible or not sound for it to be taken that the impossible B follows from A, which is itself possible. 71 Aristotle, then, shows that an impossibility cannot follow from a possibility on the grounds that in a true conditional the consequent72 must follow from the antecedent by necessity. If X follows from Y by necessity, then it always follows from it. And the impossible will always follow from its antecedent, so that, if it is possible for its antecedent to come to be, the impossible will follow from it when it comes to be. But the impossible will be at the time when it follows from the antecedent. It will then be possible for the impossible to come to be; but this is impossible. Although Chrysippus says that nothing prevents an impossibility also following from a possibility,73 he says nothing against the proof stated by Aristotle; rather he tries to show that this is not the case through some examples which are not soundly constructed. For he says

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that in the true conditional ‘If Dion has died, he (referring to Dion) has 30 died’, the antecedent ‘Dion has died’ is possible since it can at some time come to be true that Dion has died, but ‘He has died’ is impossible; for, if Dion dies, the statement74 ‘He has died’ is destroyed since the recipient of the reference no longer exists; for reference is to what is living and with respect to what is living. So if when Dion has died the word ‘he’ is not possible and Dion does not exist again75 so that it is possible 178,1 to say of him ‘He has died’, then ‘He has died’ is impossible. It would not be impossible if later, after the death of Dion, of whom when Dion was alive earlier ‘He has died’ in the conditional were predicated, it were possible for ‘he’ to be predicated again. But since this is not 5 possible, ‘He has died’ is impossible. Chrysippus also sets down another example similar to this one: if it is night, this (referring to the day) is not day. For in this conditional which – he thinks – is true, the antecedent is possible, the consequent impossible. Showing that the conditionals are false76 proves that what Chrysip10 pus says is not sound. For ‘If Dion has died, he has died’ is not a true conditional. For if ‘Dion has died’ is said and said truly in more cases than ‘He has died’ is, and if ‘he’ is not said of that of which ‘Dion’ is said, then ‘He has died’ would not follow from the antecedent ‘if Dion has died’.77 For an implication in which the antecedent can be at a time when the consequent is not is not sound. For, just as if Dion78 were a 15 homonym, the proposition ‘If Dion has died, he has died’ would not be true because ‘Dion has died’ could be said of someone else and not of the referent , so too, if the name making the reference is of wider extension than the Dion referred to, and it is not possible for the reference to be to everyone to whom the name applies, the proposition ‘If Dion has died, he has died’ will not be true. For it would be possible 20 that ‘Dion has died’ be said of something of which ‘He has died’ is not also said. And ‘Dion’ does have a wider extension if it is also said of a dead person but ‘he’ is only said of a living person. As I said,79 an implication in which the antecedent can be at a time without what is taken to follow from it is not sound. For there is nothing absurd about the consequent in a true conditional being when the 25 antecedent is not. For it is not necessary that the antecedent follow from the positing of the consequent. Therefore, it is possible that the consequent be when the antecedent is not. But it is impossible for the antecedent in a true conditional to be when the consequent is not.80 For it is not the case that the conditional is sound if the consequent does not follow from the antecedent because something has been destroyed. For the conditional is false because the consequent does not follow, but not because of the way in which it does not follow. 30 Furthermore, there is no other cause of the destruction of the consequent than the coming to be of the antecedent. But how could what is

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destroyed by its hypothesized antecedent having come to be follow from the antecedent? For Dion81 does not have a wider extension than ‘he’ in this way.82 For a person who takes it that every triangle has its angles equal to two right angles, has also taken this for the scalene triangle, since it is impossible for every triangle to have its angles equal to two right angles unless the scalene also does. The person who says that ‘He has died’ is impossible when ‘Dion has died’ is possible is not making Dion the universal over ‘he’. For if ‘he’ were universal and ‘he has died’ impossible, ‘Dion has died’, which also encompasses the impossible ‘he has died’, would not still be possible. Furthermore, how could the consequent be because the antecedent is – and this is the way we judge the true conditional – if, when the antecedent is, what is taken to follow from it is destroyed? If, then, ‘Dion has died’ is true in more cases than ‘He has died’ and does not always follow from it, the conditional is not true, as has been shown. But if ‘he’ applies to the same things as ‘Dion’, the conditional will be true, but it will no longer be the case that an impossibility follows from a possibility; rather the antecedent will be possible or impossible in the same way as the consequent is. For – if it is necessary to be precise in discussing names – if ‘Dion’ is the name of the ‘peculiarly qualified’83 and what is peculiarly qualified is a living thing, the person who mentions Dion would be mentioning a living thing. For the name refers to what is named in the same way as ‘he’ does. But, if that is so, then also ‘He has died’ in ‘If Dion has died, he has died’ would be potentially contained in ‘If Dion has died’ – at least if Dion is the name and sign of a living thing. In this way the conditional will be true, but the antecedent will no longer be possible. For it is similarly impossible for a living thing to have died and for ‘him’ to have died. But if they84 were to say that ‘Dion has died’ can be true because it is said by anaphoric reference to the living thing (on the grounds that the person who says that ‘Dion has died’ is possible is not saying that the still living Dion has died but that what was Dion has), it will also be the case that ‘He has died’ is possible. For the latter does not signify that the one who is he has died, but that the one who was he has. For the custom of using ‘he’ anaphorically is also ubiquitous. For we say, referring to the corpse, ‘He has died’ and someone looking at a corpse says ‘He is the father or brother of him’. And we do not just use anaphoric reference to the past but also to the future. We say of the house which is still being built or the cloak which is still being woven ‘This belongs to that person’ with anaphoric reference to the future house or cloak. But also we say of someone who is fatally ill ‘He is dying’, but if the ‘he’ who was dying dies and ‘he’ was dying, then ‘he’ would also have died. In general, if they were to say that ‘If Dion has died, he has died’ is a

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true statement without qualification or condition, what they say would 180,1 not be true. For if the conditional were true without qualification, the consequent would follow from the antecedent without qualification, by necessity, and always; for that is the way it is with what is necessary without qualification. But if it is not without qualification but on the condition ‘when the living thing is alive’, the statement will be true and the antecedent and the consequent will be equally impossible. For, when he is alive, ‘Dion has died’ is impossible. In general,85 if Q follows because P is hypothesized to be, Q follows 5 because P is. (For from its being light if it is day and it being hypothesized that it is day, it follows that it is light.) And if Q does not follow from P being, it is clear that Q will not follow from P being hypothesized to be either. But ‘He has died’ does not follow from Dion’s having died; therefore neither will it follow when it is hypothesized that Dion has died. The same argument applies to ‘If it is night, this is not day’. And 10 Aristotle has also used this kind of dialectical refutation.86 For having shown that B does not follow from A holding, he shows that B will not follow if A is hypothesized to hold.87 A more dialectical88 refutation is to show that ‘He has died’ is not even impossible. For if an impossible statement is always false – just as a necessary one is always true – what is not always false is not 15 impossible. But ‘He has died’ is not always false, but only when Dion is alive; for when he has died, no longer is, and if it no longer is, it will not be false, and ‘He has died’ will not be impossible. Furthermore, if they take ‘He has died’ to mean ‘He is not’ (which is equivalent to ‘He who is is not’), the proposition ‘He has died’ will be impossible, but it will not follow from ‘If Dion has died’. For 20 that he who is is not does not follow from Dion’s having died, just as also ‘This is not day’ does not follow from ‘If it is night’. For ‘This is not day’ is equivalent to ‘The day which is is not day’, and this does not follow from ‘It is night’. But ‘It is not day’ follows from ‘If it is night’, and ‘This is not day’ would follow from ‘If it is night, this being day’, but that 25 is no less impossible than the consequent. Similarly too ‘He has died’ would follow from ‘If Dion, who is alive, has died’, but that is as impossible as ‘He has died’; for it is impossible that Dion, who is alive, has died. 89 If they take ‘He has died’ to stand for ‘His soul and body have been separated’,90 then, according to them, ‘He has died’ would not be impos30 sible. For that which can at some time become a true predication is not impossible; and, according to them, ‘His soul and body have separated’ (referring to Dion) can become true after Dion’s death. For they maintain that after the conflagration all things in the world come to be again 35 and are the same in number and that even what is peculiarly qualified comes to be again in that world and is the same thing as in the preceding one; Chrysippus says this in his On the world.91 But if

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this is so, Dion too will be again at some time, so that at that time ‘He has died’ would become true of him; for his soul and body were separated and were conjoined again. But if this is so, then, according to them, ‘He has died’ is not impossible. For they say that the statement ‘This has been destroyed’ (said of fingers which are closed together and referred to), although false at the time, is not impossible because it is possible that ‘This has been destroyed’ be true when the fingers are separated (which is the destruction of their being closed together) and again closed together and referred to; for was destroyed earlier when the fingers were separated. So too ‘He has died’ will be true of the Dion who has come to be again because his soul and body were previously separated (as is also the case with the closing together of the fingers). For, just as in the case of the fingers, the change has only been with respect to number and what is referred to later differs from what is referred to earlier only in number, so too in the case of Dion, at least if the later Dion is the same as the earlier one. But if they were to say that in the case of the fingers the separated and again closed together fingers are the same in number, but that in the case of Dion the conjoined soul and body are not the same in number, this is irrelevant to the argument so long as it is assumed that the later peculiarly qualified individual is the same as the earlier one. (The question how the combination of a soul and a body which are not the same in number could become the same thing is presumably difficult for those who say that the same peculiarly qualified individual does come to be.) For the same person receives this same reference. For it is not the case that both the later Dion is the same as the earlier one and that the word ‘he’ will not be predicated of the same thing.92 But if , both ‘He has died’ and ‘His soul and body have been separated’ will be true of him. But if ‘He has died’ can become true at some time, it is not impossible. For they say that ‘Dion has died’ is possible for this reason as well: that it is true at some time. 93(And they also say that there are alterations in later peculiarly qualified individuals as compared with their predecessors only with respect to certain external accidents – the kind of alterations which also happen to a Dion who lives and remains the same and do not make him another person. For he does not become another person if, having earlier had spots on his face he later has them no more. And they say that these are the kind of alterations that happen in the case of peculiarly qualified individuals in one world and in another.) But if ‘He has died’ is neither an impossible nor a perishable statement, they should also agree that ‘If Dion has died, he has died’ is not a true conditional.94 For it is not the case that when ‘Dion has died’ is true, ‘He has died’ is also true and furthermore not destroyed. The proposition ‘If it is night, this is not day’ is similar to this. For in the case of this , just as it is possible that

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it is night, so too it is possible that this is not day. For when there is change they will not be.95 For if the person who says ‘This is not day’ means that the day which is, when it is, is not, the conditional is not true. For ‘The day which is, when it is, is not’ does not follow from ‘It is 5 night’; what follows is ‘It is not day’. But if the person means that the which is now and is being referred to is not , then the conditional is true and the consequent and antecedent are equally possible. And the other things which have been said previously could also be said in the case of this proof. It is necessary to understand the following implications. If the ante10 cedent in a necessary implication is possible, it is necessary that its consequent also be possible, as has been shown; and if, again, the consequent is impossible, it is necessary that the antecedent also be impossible. But it is not the case that if the consequent is possible it is necessary that the antecedent also be possible since one does not establish the antecedent by positing the consequent, but conversely; nor is it the case that if the antecedent is impossible, it is necessary that the 15 consequent is also impossible since the destruction of the antecedent does not also do away with the consequent, but vice versa. This is why he also says, ‘if A is possible, B will also be possible’; for ‘If you are a bird, you are an animal’ is a true conditional although the antecedent is impossible and the consequent possible. 20

34a12 And it is necessary to take possibility and impossibility96 not just in the case of coming to be but also in the case of truth and holding [and in as many other ways as one speaks about possibility; for the situation will be the same in all cases.]

Having asserted and shown that in implications if the antecedent is possible, the consequent is also possible, he adds that if the antecedent 25 is unqualified the consequent will also be unqualified, and similarly if the antecedent is necessary, so is the consequent. He indicates this by saying that it is necessary not to take what has been said as just being said about possibility in the case of coming to be. Possibility and contingency97 in the case of coming to be is what is not yet but can come to be – he has given the definition of this,98 and his proof concerning implication took place with respect to it. For taking it that A is possible, 30 B impossible, he said ‘Then if what is possible could come to be when it is possible.’99 Here he indicated possibility in the case of coming to be with the words ‘could come to be’. It is not only the case that ‘It is possible that B comes to be’ follows from possibility in the case of coming to be, i.e., from ‘It is possible that A comes to be’ (for ‘It is impossible that B comes to be’ does not follow from ‘It is possible that A comes to be’). But also, if possibility in the 35 antecedent is taken as holding (since possibility also applies to holding),

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its consequent will not be ‘It is impossible that B holds, but ‘It is possible that B holds’, i.e., possibility as holding. But if the antecedent is taken to be possible in the sense of being true, it is necessary that the consequent also be true, since truth follows from truth. What is said to be necessary would also be possible in the sense of being true since the person who says that the necessary is always speaks truly. He says that the consequent of what is possible in this way will be necessary and always true, and possible in the same way as the antecedent. It is also possible that the term ‘impossibility’ has been used to refer to the converse implication from the consequent. For, if the consequent is impossible, the antecedent is also impossible, and in whatever way impossibility is taken , the antecedent is also impossible in the same way. For, if it is impossible that the consequent come to be, it is also impossible that the antecedent do so; and if it is impossible that the consequent holds, it is also similarly impossible that the antecedent do so, and if it is impossible that the consequent ever be true, it is also similarly impossible for the antecedent. For as we have said before,100 the implication is in the reverse direction. For the possible in all its meanings has its implication from the antecedent, the impossible from the consequent. For if the antecedent is impossible, the consequent is not prevented from being possible, as in the case of ‘If you are a centaur, you are an animal’. But if the consequent is impossible, it is necessary that the antecedent also be impossible, whatever meaning of impossibility is taken. For one also speaks about impossibility in the case of falsehood, so that also if the consequent is false, the antecedent is also false, but it is not the case that the consequent is false if the antecedent also is. For just as what is true is possible, what is false is impossible.101 He may have added impossibility to possibility in order to show that in whatever meaning of possibility the antecedent is taken to be possible, impossibility in the sense of the opposite of that possibility cannot be the consequent of that possibility, as we have said before.102 An example of possibility in the case of coming to be is ‘If Dion has died, a human has died’103 if it is said of a Dion who is still alive. An example of possibility in the case of holding is ‘If it is day it is light’, if it is said when it is day; for antecedent and consequent are equally unqualified. An example of necessity is ‘If there are gods, there is a world’. The words ‘and in as many other ways as one speaks about possibility; for the situation will be the same in all cases’ may be said of what is for the most part, of the indefinite, and of the infrequent (these things fell under possibility in coming to be104) and perhaps also of necessity (if he previously took possibility as truth into consideration105); but if he included necessity then, he would now be speaking about what is true, since what is true is also possible.

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He may also be speaking about possibles and about what is called 184,1 Diodorean , namely what either is or will be. For Diodorus posited that only what either is or ever will be is possible. For, according to him, it is possible for me to be in Corinth if I were in Corinth or ever were going to be there, but if I were not going to be there, it would not be possible for me to be there either. And it is possible for a child to 5 become literate, if it ever will be literate. Diodorus propounded his Master Argument in support of this doctrine. Similarly he may be speaking about Philonian possibility. This is what is called simply the suitable for the subject, even if it has been prevented from coming to be by some external necessity. Accordingly Philo said that it is possible that chaff lying in unmown wheat107 or in the depth of the sea be burned where it is, even though by necessity it is prevented by its surroundings. 10 What Aristotle says is intermediate between these two.108 For that which can come to be if it is not prevented is possible, even if it should not come to be. For it is possible that chaff which is not in the unmown wheat or, in general, prevented by something from being burnt be burnt even if it never is burnt; and the reason is that it is not prevented . But it is not possible that the chaff in the unmown wheat 15 be burnt because its burning is prevented by something. Consequently nothing impossible follows if it is hypothesized that the chaff not in the unmown wheat is burnt. But if someone were to hypothesize that the chaff in the unmown wheat is burnt, an impossibility will follow for him, namely that what cannot be affected is affected. ( at least if it is hypothesized that the unmown wheat cannot be affected.) 106

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34a16 Furthermore, one should not interpret the proposition ‘if A is B is’ as meaning that B will be if some one thing A is. For nothing is by necessity if some one thing is. There must be at least two things, [for example when the premisses are related in the way described in the case of the syllogism. For if C is said of D and D of F, C is said of F by necessity; and if each is possible, the conclusion is also possible. Consequently, if someone were to posit A as the premisses and B as the conclusion, it would result not only that if A is necessary, B is at the same time also necessary, but also that if A is possible, so is B.]

He has shown that every necessary implication is such that from the manner of holding with respect to possibility in the case of the antecedent the similar holding of its consequent follows. He now transfers 25 what has been shown universally to syllogistic implication – he also constructed the proof for the sake of this. The meaning of what he says is this: it is necessary not to take this as if it were said of simple implications only but also as said of syllogistic implications; for in these

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too the conclusion follows from the premisses; for if the premisses are, so is the conclusion, so that the conclusion will be possible, if the premisses are, whatever meaning possibility is taken to have. Indeed, 30 if A is taken in place of the premisses, B in place of the conclusion, the proof previously stated would also fit implications of this kind. 185,1 It is necessary to understand the words ‘for nothing is by necessity if some one thing is’ as saying that nothing is by necessity syllogistically; for nothing is by necessity syllogistically if some one thing is. The words ‘consequently, if someone were to posit A as the premisses and B as the conclusion’ are equivalent to ‘and if someone were to take A in place of 5 the premisses and B as conclusion’.109 34a25 This having been proved, it is evident that if one hypothesizes something false and not impossible what results because of the hypothesis will be false and not impossible. [For example, if A is false but not impossible and if when A is, B is, B will also be false but not impossible. For, it has been shown that if when A is B is, then when A is possible B will be possible; but it is assumed that A is possible; so B will also be possible; for if it were impossible, the same thing would be possible and impossible at the same time.] Furthermore he makes evident that for the sake of which he showed that what is possible follows from what is possible. For, since it was shown that in necessary implications of whatever sort the antecedent is with respect to holding, the consequent is also of the same sort, it is evident that the false and the impossible are not the same, but there is a difference between them. For it is not the case that if something is false, it is thereby also impossible. For there is such a thing as a possible falsehood. For the person who hypothesizes what is contingent but is not yet posits a falsehood but not an impossibility. But if the antecedent is false but possible, the consequent will in general be false but not impossible. (For the conclusion may be true when the premisses are false, as he will show in book 2.110) But even if the conclusion is also false, it will be false in such a way as not to be impossible. For if it were impossible in the case of a false but not impossible premiss or premisses, again an impossibility would follow from a possibility. But if in some syllogism one premiss which is false but not impossible is hypothesized and the consequence is impossible, what follows will be impossible not because of the false but not impossible premiss, but it is impossible because of the other one of the premisses from which it follows. For, as he said,111 in syllogistic implications something does not follow by necessity from one assumed premiss. He says ‘for if it were impossible, the same thing would be possible and impossible at the same time’112 because if the antecedent is possible

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and the consequent impossible, the consequent is possible insofar as it follows from an antecedent which is possible and when the antecedent 30 is it must be possible, and impossible insofar as it is hypothesized to be impossible; for what is impossible without qualification is also impossible when its antecedent is possible.

34a34113 These things having been specified, let A hold of all B, [and let it be contingent that B holds of all C; then it is necessary that it is contingent that A holds of all C. For let it not be contingent, and let it be assumed that B holds of all C. (This is false, but not impossible.) Then if (i) it is not contingent that A holds of C, but B holds of all C, (ii) it is contingent that A does not hold of all B. For there is a syllogism through the third figure. But (iii) it was hypothesized that it is contingent that A holds of all B. Therefore, it is necessary that it is contingent that A holds of all C. For when something false but not impossible is posited the result is impossible.] Having discussed and reached an understanding of the implication 186,1 relation of the conclusion to the premisses, a relation of which he is going to make use, he returns to the business at hand, and discusses mixed combinations of an unqualified and a contingent premiss in the first figure, the major being unqualified, the minor contingent. His proof that 5 combinations of this kind in which the major is unqualified are syllogistic is by reductio ad impossibile. This is why he also said114 that the syllogisms of this kind are not complete. He takes it that A holds of all B and that it is contingent that B holds of all C, and says that it is contingent that A holds of all C. For if this is not the case, the opposite is the case, and the opposite of ‘It is contingent that A holds of all C’ is 10 ‘It is not contingent that A holds of all C’, which is equivalent to ‘A does not hold of some C by necessity’. Positing this, he transforms the contingent universal affirmative premiss BC into an unqualified universal affirmative, which is false but not impossible (since it is assumed that it is contingent that B holds of all C, and if it is contingent that B holds of something it is not impossible to take it that B holds of it). From 15 ‘A does not hold of some C by necessity’ and ‘B holds of all C’ there follows in the third figure115 that it is not contingent that A holds of all B, which is impossible, since A was assumed to hold of all B. He says ‘Then if it is not contingent that A holds of C’ rather deficiently, since he leaves out the word ‘all’. For ‘it is not contingent that A holds of all C’ is the opposite of ‘It is contingent that A holds of all C’. He says ‘It is contingent that A does not hold of all B’ as 20 equivalent to ‘A does not hold of all B’. For this was shown to be the

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conclusion in the third figure in mixtures of a necessary and an unqualified premiss in which the minor is universal affirmative unqualified and major particular negative necessary; for the conclusion is particular negative unqualified. However, he also says ‘But it was hypothesized that it is contingent that A holds of all B’ instead of ‘It was hypothesized that A holds of all B’; for the universal affirmative premiss AB is unqualified. From this it is also clear that in referring to the conclusion he says ‘It is contingent that A does not hold of all B’ as equivalent to ‘A does not hold of all B’. In this way too what is inferred turns out to be impossible, if, when it is assumed that A holds of all B, it is inferred that it does not hold of some. For it is impossible that what holds of all does not hold of some. But what he seems to take because of the words is not impossible. For it is not impossible, when it is assumed that it is contingent that A holds of all B – and he says this –, that it is also contingent that A does not hold of all – and he also seems to have said that –, since it can be the case that it is contingent that the same thing holds of all and contingent that it holds of none. But, as I said, he uses the expression ‘it is contingent that X’ instead of ‘X holds’ in both cases, since contingency is also predicated of what holds. Having found and proved what is inferred to be impossible in the case of the premisses under consideration (namely, the hypothesis of the opposite of the conclusion and the transformation into something which is false but not impossible), he makes use of what has been shown and says that the impossibility has not followed from the falsehood. For was false but possible, and what follows from a possibility is possible, not impossible, as was shown. Therefore the impossibility follows from the other assumption, the hypothesis , so that it is impossible, since it is necessary that one of the premisses is impossible, and the other one is not impossible. Therefore the opposite of the hypothesis is true, i.e., ‘It is contingent that A holds of all C’. For ‘It is contingent that A holds of all C’ is the contradictory opposite of ‘It is not contingent that A holds of all C’. 34b2116 It is also possible to produce the impossibility through the first figure, by positing117 that B holds of C. [For if B holds of all C, and it is contingent that A holds of all B, then it would be contingent that A holds all C; but it was assumed that it is not possible for it to hold of all.] What he asserts and proves is the following. He says that the hypothesis ‘It is not contingent that A holds of all C’ (for this is how the conclusion is made a hypothesis ) is impossible in the case of the premisses assumed, namely ‘A holds of all B’ and ‘It is contingent that B holds of all C’, and he says that he proves it by means of the first figure.

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15 For at this point it has been proved by means of the third figure. Now he proves it by taking the two premisses conversely, the unqualified premiss as contingent and the contingent one as unqualified. He takes it that it is contingent that A holds of all B (but this premiss was unqualified universal affirmative), and that B holds of all C (and this premiss was that it is contingent that B holds of all). When the pre20 misses are taken in this way, nothing impossible will be assumed. For it is not impossible that what holds be contingent nor that what is contingent holds. The impossible conclusion which is hypothesized in the case of these assumed premisses and which he allows from outside is ‘It is not contingent that A holds of all C’. For in the combination under consideration, which has a universal affirmative contingent major and 25 a universal affirmative unqualified minor, a universal affirmative contingent conclusion follows, as was proved shortly before this;118 for it was proved that when the major is contingent and the minor is unqualified, the conclusion is contingent. But it is impossible that it is contingent that A holds of all B and not contingent that it holds of all; for this is a contradiction. Therefore the opposite of what is hypothesized, that is, ‘It is contingent that A holds of all C’ will be the conclusion. 119 30 He transformed the major unqualified premiss into a contingent one, but not because the conclusion is not impossible even if both premisses are unqualified; for when both premisses are unqualified it follows that A holds of all C, and, if this is true, the conclusion as hypothesized ) ‘It is not contingent that A holds of all C’ 188,1 is proved impossible. Rather he wanted to give a proof making what is assumed the contradictory of what is inferred, which is a prima facie clear and indisputable impossibility. And furthermore, this is also what is required to be proved in the case of a hypothesis which is hypothesized as impossible. For ‘It is contingent that A holds of all C’ is the contradictory opposite of ‘It is not contingent that A holds 5 of all C’. But if the major premiss AB is posited to be contingent affirmative, it is inferred that it is contingent that A holds of all C. 120 But it is possible that he has again used ‘It is contingent that X’ instead of ‘X holds’, in connection with the premiss AB and the conclusion, as he did shortly before.121 122 But this proof did not proceed by reductio ad impossibile. For one of the assumed premisses was not added to a hypothesis which was the opposite of what was proposed to be proved, an impossibility inferred, 10 the hypothesis destroyed by the denial of this impossibility and its opposite established.123 Rather two premisses were taken, one of which was true and assumed,124 the other125 a transformation into something false but not impossible in the case of these premisses. 126 But if this is impossible, the opposite of the hypothesis is not introduced by means 15 of it. 127 Thus something different is inferred from the assumptions because in the case of things

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assumed in this way the impossibility does not result entirely from the hypothesis, as we have investigated elsewhere at greater length.128 34b7129 It is necessary not to take ‘A holds of all B’ with a temporal restriction, e.g., ‘A holds of all B now’ or ‘A holds of all B at such and such a time’; one must take it without qualification. [For we make syllogisms through such premisses, since there will not be a syllogism if the premiss is taken with respect to a time. For presumably nothing prevents human holding of all that moves at a certain time, if, for example, nothing else were to be moving. And it is contingent that moving holds of all horses. But it is not contingent that human holds of any horse. Furthermore, let the first term be animal, the middle term moving, and the last term human. Then the premisses will hold in the same way, but the conclusion will be necessary rather than contingent. For a human is an animal by necessity. Thus it is evident that one should take a universal proposition without qualification, not with a temporal specification.] He says that in the mixture under consideration it is necessary not to take the first extreme holding of the middle as a matter of holding at a restricted time. What he means is the following. Those unqualified universal propositions are temporally restricted if they can be taken to hold universally at a given time but not always, for example, the proposition which posits that animal holds of all that moves or that human holds of all that moves. 130The reason that the unqualified premiss is temporally restricted is that in the case of unqualified premisses of this kind the middle term is hypothesized to have a wider extension in that it is possible that it holds of other things of which what is predicated of the middle universally cannot hold; for at the time when the middle is taken to hold of that of which it is contingent that it holds (and that is different and does not fall under the major term which is predicated of the middle), the universal unqualified proposition which is the major cannot be true. For if we hypothesize that animal holds of all that moves, because it is possible that moving also holds of non-animals, when it does hold of some of those things, then the proposition which says that animal holds of all that moves will no longer be a true unqualified universal. Thus this proposition holds with respect to a restricted time. For so long as moving holds of nothing else, it will be possible to hypothesize that animal holds of all that moves. But it is impossible that a combination in which the major is such a proposition be syllogistic. For it is necessary that the affirmative minor be contingent or unqualified or necessary;131 and if it is taken that it is contingent that the middle holds of something of which the major extreme cannot also possibly hold, the combination will not be syllogistic because when

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the middle holds of that of which it has been taken that it is contingent that it holds the major premiss does not remain universal unqualified.132 Thus it is possible133 for the unqualified premiss and the contingent premiss to both be true at the same time, but their combination is not syllogistic.134 For the combination of such premisses is not proved syllogistic in any way but only by the reductio ad impossibile in which the contingent premiss is transformed into an unqualified one. But then cannot be true at the same time if the middle is taken to hold of all of a last term which falls outside the major extreme. For then, the major premiss cannot still be universal affirmative unqualified, but it becomes particular. Thus a universal unqualified proposition of this kind, the truth of which is restricted by the contingent premiss taken with it, is useless for a syllogism. For the major premiss is true as long as the minor remains contingent, but when the minor becomes unqualified, the major premiss is no longer true. 135For when it is contingent that moving holds of stone and it does hold of stone, then it is no longer possible for ‘Animal holds of all that moves’ to be taken, but rather its negation, ‘Animal does not hold of all that moves’, which is particular negative, becomes true then. Thus it is not possible for ‘Animal holds of stone’ or ‘It is contingent that animal holds of stone’ to be taken, since animal holds of no stone by necessity. For in the first figure when the major premiss is particular, there cannot be a syllogism involving the major premiss at that time. Thus he says it is necessary to take an unqualified premiss of such a kind that its truth is not restricted by a contingent premiss which is added to it, but of such a kind that it remains universal affirmative unqualified even when the contingent universal premiss is transformed into an unqualified one. For it doesn’t make any difference whether one adds some specification to the unqualified premiss to restrict it temporally or co-ordinates with it a contingent premiss of such a kind that, when it is changed to an unqualified one, it does away with the unqualified premiss. For he does not call a proposition which is always unqualified – since what is always true is ipso facto necessary – but rather he calls unqualified a proposition which can remain unqualified universal affirmative and true even when the contingent affirmative universal premiss which has been taken along with it is transformed into an unqualified premiss. For in this way the proof by impossibility which he uses to show that this combination is syllogistic is preserved. He might say either that it is necessary that the universal unqualified premiss not be taken in such a way that it is possible for its truth to be restricted temporally by the contingent premiss added to it or rather that the contingent premiss added to it must not be such as to restrict temporally the truth of the unqualified premiss taken before it. The person who says that it is necessary that the truth of the

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unqualified premiss not be restricted temporally must not be thought to make the question of whether the combination is syllogistic or non-syllogistic turn on material terms, as it might seem to some, but rather presumably on form.136 For if is restricted temporally and it changes, it makes the major premiss in the first figure particular negative,137 and if this happens, the combination is non-syllogistic. He puts forward as an indication that the combination in which the universal affirmative unqualified major is restricted temporally is non-syllogistic, the fact that when there is a first-figure combination in which the universal affirmative unqualified major is restricted temporally by the minor contingent premiss, to take terms similarly for both ‘A holds of all C by necessity’ and ‘A holds of no C by necessity’, while both premisses are true. For he shows that premisses related in this way are non-syllogistic by setting down terms and showing that A holds of no C by necessity and that A holds of all C by necessity . For ‘A holds of no C’ : ‘Human holds of all that moves’, ‘It is contingent that moving holds of every horse’, and ‘Human holds of no horse by necessity’. For since moving has a wider extension than human, if one takes it that it is contingent that it holds of every horse, one restricts the unqualified premiss temporally. For at the time when moving holds of every horse, human cannot still hold of all that moves. For ‘A holds of all C’ : ‘Animal holds of all that moves’, ‘It is contingent that moving holds of every human’, and ‘Animal holds of every human by necessity’. What proves and makes the combination non-syllogistic is not primarily showing in it holding of all by necessity since ‘It is contingent that X holds of all Y’ is true when ‘X holds of all Y by necessity’ is – at least in the case of what is not contingent in the way specified;138 rather it is finding in it terms for holding of none by necessity, and this occurs by taking – when the middle term has a greater extension than what is predicated – the added contingent premiss in such a way that it temporally restricts the truth of the major premiss. For it is not possible to take terms for ‘A holds of no C’ unless the middle has a greater extension and the contingent premiss is taken in this way. For example, let walking hold of everything literate; and it is contingent that literate holds of every human; but the premiss which posits that walking holds of everything literate does not get restricted temporally; for the major premiss is not destroyed by the universal affirmative contingent premiss co-ordinated with it, if the premiss is taken to be true . And in the case of the terms from which it follows that A holds of all C by necessity, necessity does not follow because the middle has a greater extension than the major, since the contingent premiss is not taken in such a way as to restrict the truth of the unqualified premiss temporally (for some non-animal was not taken so as to make the major

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5 premiss particular and restrict it temporally).139 Rather the necessity follows because the last term is taken to be contained by the first and to be in it. This is how human is related to animal. And again when the contingent premiss is taken to be of this kind, the conclusion is neces10 sary, but not contingent in the way specified. But if the unqualified premiss is not restricted temporally by the contingent premiss and the last term is not taken in such a way as to be under the first, the conclusion would seem to be contingent. For let being asleep hold of every animal, and let it be contingent that animal holds of all that moves; then it will also be contingent that being asleep holds of all that moves. It should be asked whether perhaps the setting down of terms and 15 proof that with true premisses the first term holds of all of the last by necessity and holds of none by necessity does not rather show that the combination is non-syllogistic. I have also discussed this elsewhere.140

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34b17 Thus it is evident that one should take a universal proposition without qualification, not with a temporal specification.

He means the universal affirmative unqualified premiss, since that’s what he has discussed; for he showed that if this premiss is not taken as true without qualification but with a temporal restriction, the combination is non-syllogistic because when it is taken with a temporal qualification it is possible for both ‘A holds of all C by necessity’ and ‘A holds of no C by necessity’ to be inferred in its case. As a result some 25 people have said certain things against the proof by impossibility141 and the transformation of the contingent premiss into an unqualified one, namely that the conclusion is also not impossible – rather it also is false but not impossible –, so that, if the conclusion is also of this sort, it becomes so from the transformation of the contingent premiss into an unqualified one but not because of the hypothesis which says that A does not hold of some C by necessity. This is the sort of thing 30 that was said: The assumed premiss was that A holds of all B. It was inferred in the third figure142 – the one used for the reductio ad impossibile – that A does not hold of some B. And this was shown to be false, but not impossible as follows:

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then ‘A does not hold of some B’ will be false but not impossible; rather, since what holds of all is taken not to hold of some, ‘A does not hold of some B’ is false but not impossible because it is contingent that it does not hold. Against this we say that it is not impossible to transform any contingent particular negative proposition into a particular negative unqualified one and that it is true that if A holds of all B it is contingent that it does not hold of some B. However, at the time when A holds of all B, the transformation is impossible, since, if it is assumed that A holds of all B, the transformation is impossible. It is necessary for ‘A holds of all B’ to remain unchanged if there is to be a syllogism through it. For it is assumed to be such that it is neither temporally restricted nor destroyed in the transformation of a contingent proposition into an unqualified one. But it is impossible for ‘A holds of all B’ to remain fixed if the particular contingent negative143 is transformed into an unqualified proposition. 144 But it was said back against this that, when B holds of no C, the transformation of ‘it is contingent that B holds of all C’ to ‘B holds of all C’ is impossible. For when B holds of no C, it is contingent that it holds . For if it in fact already held , the proposition would no longer be contingent. But ‘it is contingent that B holds of all C’ has been transformed into ‘B holds of all C’ at a time when it is contingent that B holds of all C. But when it holds of none, it is taken to hold of all; but if this is impossible, the transformation of the contingent proposition into an unqualified one is also impossible. 145 This problem could be dissolved by that the person who transforms a contingent proposition into an unqualified one without preserving it – this was contingent but is unqualified negative – makes the transformation of it into an unqualified proposition; for such a transformation is impossible. For if it is contingent that B holds of all C because it holds of none of it, the person who transforms ‘it is contingent that B holds of all C’ into ‘B holds of all C’ does not preserve it as B holds of no C and at the same time also transform it into ‘B holds of all C’. For it is impossible to take it that what holds of none – at the time when it holds of none – holds of all. But as it is possible for a contingent proposition to become a true unqualified one, so the person makes it a hypothesis, assuming in advance what will be the case about it. But it is not possible for ‘it is contingent that B holds of all C’ ever to change into ‘B holds of all C’ if the universal unqualified negative proposition remains fixed; rather the transformation of it into an unqualified proposition is the destruction of the opposite unqualified negation. For the hypothesis146 is not said to be false because B is hypothesized to hold of all C when it holds of none, but because B’s holding of no C is such that it contingently does not hold .

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35 (What indicates this is that it is contingent that what holds of none 193,1 holds of all.) Because ‘B holds of no C’ could change into ‘B holds of all C’, it is hypothesized to have changed already; as it were, what will be tomorrow is today, that is today B has ceased holding of no C. And this becomes false by being assumed in advance, but it is not impossible; for 5 it is taken that what will hold holds already. For one does not at the same time both preserve it as contingent and take it as unqualified, but this would be done if it is taken that B holds of all C at the time when it holds of none. Roughly what happens is a transformation of what does not hold into what does hold by means of contingency. It is clear how what is transformed could be preserved as the same 10 and that the transformation is not impossible. The transformation147 being of this kind and the hypothesis being made, what follows, ‘A does not hold of some B’, is impossible when A holds of all B. For A does not also come to hold of all B in the change of the contingent premiss BC into an unqualified one; for this would have been true even without the 15 change. But it is assumed now that A holds of all B not as holding in a temporally restricted way nor as changing; consequently, the impossibility is thought148 to have been caused to appear in the change of the contingent proposition into an unqualified one, because of this change, and through it. Consequently, when the contingent premiss truly changes and becomes unqualified, A will hold of all B, but it follows that it does not hold of some. And if it is impossible at that time, 20 it is clear that it is also impossible now. For then and now are similar and the same. But these things should be investigated in a better way.

34b19149 Again, let AB be a privative universal premiss, [and let it be taken that A holds of no B, but let it be contingent that B holds of all C. If these things are posited it is necessary that it is contingent that A holds of no C. For let it not be contingent, and let it be assumed that B holds of all C, just as in the preceding. Then, it is necessary that A holds of some B; for a syllogism results through the third figure. But this is impossible, so that it will be contingent that A holds of no C; for when this is posited as false, the result is impossible.] He also proves – again by means of reductio ad impossibile – that the combination of an unqualified negative universal major and a contin25 gent universal affirmative minor in the first figure is syllogistic. For if A holds of no B and it is contingent that B holds of all C, it is contingent that A holds of no C. For if not, the opposite ‘It is not contingent150 that

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A holds of no C’, which transforms into ‘A holds of some C by necessity’, . (For him this move is signified by ‘For let it not be contingent’.) But also let B hold of all C; for again let the contingent be 30 transformed into the unqualified. And again in the third figure it results that A holds of some B; for it was proved that in the third figure consisting of a necessary particular affirmative major and an unqualified universal affirmative minor a particular affirmative unqualified conclusion follows. But it is impossible that A holds of some B, since it was assumed to hold of none. Again, this impossibility did not follow because we transformed the 194,1 universal contingent affirmative premiss into an universal affirmative unqualified one, since even if the latter is false, it is not impossible. However, it was shown151 that the impossible is a consequence of the impossible. Consequently the impossibility comes from the hypothesis . So the opposite of the hypothesis ‘It is not contingent that A holds of no C’ will follow, that is ‘It is contingent that A holds of 5 no C’. 34b27 Thus the conclusion of this syllogism is not a proposition which is contingent in the way specified, but is ‘of none by necessity’;152 [for this is the contradictory of the hypothesis which was made, since it was posited that A holds of some C by necessity, and the conclusion of a syllogism by means of the impossible is the opposite of the hypothesis. 34b31 Furthermore, it is evident from terms that the conclusion will not be contingent.153 For let A be raven, B reflective, and C human. Then A holds of no B, since nothing reflective is a raven. But it is contingent that B holds of all C, since it is contingent that reflective holds of every human. However, A holds of no C by necessity. Therefore, the conclusion is not contingent. But it is not always necessary either. For let A be moving, B knowledge, and C human. Then A will hold of no B, and it is contingent that B holds of all C †, and the conclusion will not be necessary†.154 For it is not necessary that no human is moving, but not necessary that some are. It is clear then that the conclusion is that A holds by necessity of no C.155 35a2 But the terms ought to be taken in a better way.] What he means is this. He is denying that the conclusion in the case of the combination of a negative unqualified universal major and an 10 affirmative contingent universal – which was proved a moment ago by reductio ad impossibile to be that it is contingent that A holds of no C – is contingent in the way specified; the conclusion is rather that A holds by necessity of no C. But holding of none by necessity is not equivalent to holding by necessity of none;156 the latter is what he says

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15 follows. For holding by necessity of none is different from ; for example, walking holds by necessity of no animal, and this is not the same thing as ‘No animal walks by necessity’. For some things walk and walk contingently, and some also do not walk by necessity, and ‘Walking holds by necessity of none’ is true of all of these. 157 He says that the reason why this follows is that it was hypothe20 sized that A holds of some C by necessity since ‘It is not contingent that A holds of no C’ was transformed into this, and ‘A holds by necessity of no C’ – which is equivalent to and the same as ‘It is not the case that A holds of some C by necessity’ – is the opposite of ‘A holds of some C by necessity’. Consequently, because the opposite of impossibility which has become evident is inferred in reductio ad impossibile arguments, what would be inferred in this case is ‘A holds by necessity of no C’; and 25 this is not equivalent to ‘It is contingent that A holds of no C’ because ‘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. It is clear that this is 30 true from the fact that ‘It is contingent that no animal walks’ is not true. But it is not true that A holds of some C by necessity when A does not hold of some C by necessity. And there is a difference between ‘A holds by necessity of no C’ and ‘It is contingent that A holds of no C’. For a person who says that it is contingent that A holds of no C does away with all necessary propositions, both affirmative and negative, at least 195,1 if ‘It is contingent that A holds of no C’ converts with ‘It is contingent that A holds of all C’.158 But although ‘A holds by necessity of no C’ does away with affirmative necessary propositions, it does not do away with negative ones. For ‘A holds by necessity of no C’ can be true even though ‘A does not hold of some C by necessity’ is true. An example is ‘No 5 animal is by necessity walking’ and ‘Some animal is not walking by necessity’. Similarly in the case of ‘no animal by necessity laughs (or talks)’ or anything of this kind. 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 10 former.159 Thus the opposite of that into which is transformed is established, and he proves that the combination is syllogistic, that is, he establishes ‘A holds by necessity of no C’. For he did not transform ‘It is not contingent that A holds of no C’ itself into something equivalent to and the same as it, but into something of such

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a kind as if he originally did not use ‘It is contingent that A holds of no C’ but took it that what is implied by the combination under consideration is ‘A holds by necessity of no C’. For if it is not the case that A holds by necessity of no C, the opposite of this , ‘A holds of some C by necessity’; but an impossibility follows when this is hypothesized; therefore, this ; therefore, its opposite . (34b31) His setting down of terms makes what he says unclear;160 for he does not prove through the terms set down that A holds by necessity of no C but that A holds of no C by necessity, which in no way makes a contradiction with ‘A holds of some C by necessity’. For both are necessary affirmations, but it is necessary that the contradictory opposite of ‘A holds of some C by necessity’ deny necessity. The terms which he sets down are raven for A, reflective for B, and human for C. In the case of these terms, raven holds of nothing reflective, and it is contingent that reflective holds of every human; and raven holds of no human by necessity, but it does not hold by necessity of none161 which is what he wanted to show follows. He encountered this difficulty by taking the major premiss AB not as unqualified negative but as necessary;162 for raven holds of nothing reflective by necessity. Perceiving this difficulty, he again sets down other terms with which, he says, the conclusion does not become necessary negative. What he sets down are moving for A, knowledge for B, and human for C. For moving holds of no knowledge, and it is contingent that knowledge holds of every human; and it is contingent that moving holds of no human, that is, moving holds by necessity of no human, even if it holds. (35a2) But perceiving that in the case of neither the first terms nor the second which he set down did he get an unqualified negative premiss but a necessary one, he says further, ‘But the terms ought to be taken in a better way.’ Then let the terms being angry, laughing, human be taken. Let being angry hold of nothing laughing, and it is contingent that laughing holds of every human; it is contingent that being angry holds of no human, not because it does not hold but it is contingent that it does hold (for it holds of many humans) – since then the conclusion would be contingent in the way specified –, but because it holds by necessity of none.163 The situation would be still clearer if walking were taken to hold of nothing at rest, it being contingent that being at rest holds of every animal. For walking holds by necessity of no animal, although it does not hold of some by necessity. Consequently it is not true that it is contingent that walking holds of no animal because it is not also true that it is contingent that it holds of every animal. 164 One might inquire how his assertion that the conclusion is not ‘contingent in the way specified’ can be sound. For he hypothesized the opposite of ‘It is contingent that A holds of no C’, which is ‘It is not contingent that A holds of no C’. He transformed this into ‘It is neces-

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15 sary that A holds of some C’ as its equivalent. But if the contradictory of ‘It is contingent that A holds of no C’ were ‘It is not contingent that A holds of no C’ and this were equivalent to ‘A holds of some C by necessity’, but an impossibility followed from the hypothesizing of ‘A holds of some C by necessity’, it is clear that the opposite of this would be true. But ‘A holds of some C by necessity’ is equivalent to ‘It is not contingent that A holds of no C’ and ‘It is contingent that A holds of no C’ is the opposite of this. How then does it not turn out that the 20 conclusion is contingent in the way specified? Or is he wrong to transform ‘It is not contingent that A holds of no C’ into ‘It is necessary that A holds of some C’, and is the whole proof fallacious165 because the opposite of contingency which had to be hypothesized in the reductio ad impossibile, is not taken? Or if the opposite is what is hypothesized and this is impossible, in what way is the conclusion not contingent in the way specified? 25 It is worth seeing how ‘A holds by necessity of no C’ is the contradictory of ‘A holds of some C by necessity’, which was obtained by transformation of the negation ‘It is not contingent that A holds of no C’ as being equivalent to it. The proposition that A holds by necessity of no C is taken either (1) as denying with necessity the proposition that A holds of some C or (2) as denying the proposition that it is necessary that A holds of some C.166 Taken in the first way, ‘A holds by necessity 30 of no C’ is not a negation but a necessary affirmation which is not the opposite of ‘A holds of some C by necessity’, which itself is also an affirmation. Taken in the second way, ‘A holds by necessity of no C’ is a negation and is the opposite of the affirmation which says ‘A holds of some C by necessity’. But still, how is the proposition which says this not contingent since a negation of necessity is contingent? Or is the proposition ‘A holds by 35 necessity of no C’, taken as denying the affirmative necessary proposi197,1 tion,167 not contingent because, even if A holds of all C (but not necessarily), it is true even if it does not hold of some by necessity; for it announces the denial of a modality of holding, not the denial of holding. But the proposition ‘It is contingent that A holds of no C’, when it is contingent in the way specified, was not posited among things which hold in fact, but neither can it be true in the case of things of which the particular necessary negative proposition is true because it 5 converts with ‘It is contingent that A holds of all C’;168 for, as I said,169 a universal contingent proposition denies all necessary propositions. But also the transformation of ‘It is not contingent that A holds of no C’ into ‘A holds of some C by necessity’ is a transformation into something which is true together with it and not into something which is straightforwardly equivalent.170 For just as ‘It is not contingent that A holds of no C’ is true when the proposition which says that A holds of 10 some C by necessity is, so too it is true when the proposition which says

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that A does not hold of some C by necessity is, as he himself will also show shortly.171 For if A does not hold of some C by necessity, it is not contingent that it holds either of all or of none. Since, then, ‘It is not contingent that A holds of no C’ is also true when A does not hold of some C by necessity, he transformed it – as I have already said172 – into ‘A holds of some C by necessity’, not as into something the same and equivalent but because the impossibility was proved with this proposition but not if the negative was taken. For when ‘A does not hold of some C by necessity’ was taken and ‘B holds of all C’ added, there resulted in the third figure the conclusion that A does not hold of some B, which was not impossible since it was assumed that A holds of no B. Since, then, when this was taken, nothing was proved, but when the necessary affirmative particular was taken something impossible followed, he transformed ‘It is not contingent that A holds of no C’ into this, which is no less true when ‘It is not contingent that A holds of no C’ is; and, finding that something impossible follows from the transformation of ‘It is not contingent that A holds of no C’ into this, he reasonably says that there will follow what is the unique opposite of that into which ‘It is not contingent that A holds of no C’ was transformed, but not the proposition from which the latter was transformed.173 The unique opposite of ‘A holds of some C by necessity’ is ‘A holds by necessity of no C’, which is equivalent to ‘It is not necessary that A holds of some C’. So this is what has been established by the argument, and not ‘It is contingent that A holds of no C’; for the opposite of this, ‘It is not contingent that A holds of no C’, was just as true when it was necessary that A holds of some C as it is when it is necessary that A does not hold of some C; but nothing impossible followed if the latter of these was taken. And so ‘It is contingent that A holds of no C’ does away with both ‘A holds of some C by necessity’ and ‘A does not hold of some C by necessity’; and the negation ‘It is not contingent that A holds of no C’ is true when either of these is. An indication that ‘It is not contingent that A holds of no C’ was not transformed into ‘A holds of some C by necessity’ as being an equivalent is that when ‘A holds of some C by necessity’ is done away with, he does not say that the opposite of that from which this was transformed is posited but that the opposite of this is, since this opposite is different from the opposite of ‘It is not contingent that A holds of no C’, namely ‘It is contingent that A holds of no C’.174 175 Another question is whether by making use of these things it is possible to say that in the case of the previously discussed combination of affirmations the conclusion is not contingent in the way specified either. For in the case of that combination the transformation was from ‘It is not contingent that A holds of all C’ into ‘A does not hold of some C by necessity’, since the impossibility is proved from the transforma-

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tion into this. For if ‘It is not contingent that A holds of all C’ is transformed into ‘A holds of some C by necessity’, nothing impossible results, just as it doesn’t in this case176 if the conclusion is transformed into ‘It is necessary that A does not hold of some C’. But if what would be established is the unique opposite of ‘A does not hold of some C by necessity’, namely ‘A does not hold by necessity of no C’,177 which is equivalent to ‘It is not the case that A does not hold of some C by necessity’. 178If this differs only verbally from ‘It is contingent that A holds of all C’, the conclusion would be contingent in the way specified. But if in the case of certain material terms it is possible to find that A holds of some C by necessity and it is not the case that A does not hold of some C by necessity, then ‘A does not hold by necessity of no C’ would be true in that case, but ‘It is contingent that A holds of all C’ would not be true – at least if when it is contingent that A holds of all C it is contingent that A holds of none, but it is false to say of what holds of some C by necessity that it is contingent that it holds of none. And thus what is contingent in the way specified would not follow in the case of that combination either. 179The proposition which says ‘No animal is by necessity not a breather’ would not be of this sort; for it is true because some animal is a breather by necessity, and some animal is not a breather by necessity, and it would be opposite in the strict sense to ‘It is contingent that being a breather holds of no animal’. ‘It is not contingent that being a breather holds of no animal’ is of this sort, since such a proposition (‘It is not contingent that being a breather holds of no animal’) is always similarly true both when being a breather does not hold of some animal by necessity and when it does hold of some by necessity. And also in the case of the assumed material terms the negation of ‘It is contingent that being a breather holds of every animal’, i.e., ‘It is not contingent that being a breather holds of every animal’ would be perfectly true since ‘It is not contingent that being a breather holds of no animal’ would be true in the same way. For each of them180 is true in either case, that is when being a breather holds of some animal by necessity and when it does not hold of some by necessity. But we should consider whether it is not possible that the proposition which says ‘Nothing rational by necessity does not think’181 or ‘Nothing which has reason by necessity does not think’ is of this sort. For if someone admits that the divine is also rational, then thinking will hold of it by necessity and not thinking will hold by necessity of nothing rational. It would seem that ‘no natural body by necessity does not move’ is of this kind. For if rotation is motion, motion will hold of the rotating body by necessity, but there will be no body of which by necessity motion does not hold. But if these things are true, then also in the case of that combination

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what follows would not be contingent in the way specified but would be 5 the aforementioned opposite of ‘A does not hold of some C by necessity’ into which the negation of contingency was transformed.182 For if someone were to say, as the associates of Theophrastus183 do, that these things are contingent in the way specified, it would no longer be true that contingent affirmative and negative propositions convert with one another. And so the combination in which the major is affirmative 10 unqualified, like the combination in which the major is negative , will not have a conclusion which is contingent in the way specified. For whatever is uniquely and properly opposite to that into which has been transformed is shown to follow because the reductio ad impossibile posits nothing other than the opposite 15 of the hypothesis from which the impossibility followed. 34b31 [Furthermore, it is evident from terms that the conclusion will not be contingent. For let A be raven, B reflective, and C human. Then A holds of no B, since nothing reflective is a raven. But it is contingent that B holds of all C, since it is contingent that reflective holds of every human. However, A holds of no C by necessity. Therefore, the conclusion is not contingent. But it is not always necessary either. For let A be moving, B knowledge, and C human.] Then A will hold of no B, and it is contingent that B holds of all C †, and the conclusion will not be necessary†. For it is not necessary that no human is moving, but not necessary that some are.184 [It is clear then that the conclusion is that A holds by necessity of no C. But the terms ought to be taken in a better way.] Since he wants to show with terms that in the case of the combination under consideration the conclusion is not ‘A holds of no C by necessity’ 20 but that it is ‘A holds by necessity of no C’ and since in the case of the first terms which he set down what followed was ‘A holds of no C by necessity’ (and also the major premiss, instead of being unqualified, was necessary negative but not unqualified186), he tries to show what he proposed to show by setting down other terms. The terms are moving, knowledge, human. Moving holds of no knowledge, and it is contingent that knowledge holds of every human; the conclusion is that moving 25 holds by necessity of no human. He indicates that this conclusion is different from ‘Moving holds of no human by necessity’ and explains how it is different when he says, ‘For it is not necessary that no human is moving, but not necessary that some are’. Here through the words ‘For it is not necessary that no human is moving’ he denies that the conclu- 30 sion is necessary negative, that is, he shows that it is not ‘A holds of no C by necessity’; and he shows that the conclusion is ‘A holds by necessity of no C’; for this is ‘but not necessary that some are’; 185

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200,1 it is as if he said, ‘The conclusion is not the former, but the latter, that is, it is not ‘it is necessary that none’ but ‘It is not necessary that none’.187 But instead of saying ‘But not necessary that none are’ he says, ‘But not necessary that some are’, as being equivalent to ‘Holds by necessity of none’, which itself is expressed by ‘Moving holds by necessity of no human’ and does away with the proposition that motion holds of human 5 by necessity. After this he says, ‘And it is contingent that B holds of all C; for it is not necessary that no human is moving’ and makes his proof less clear; for, as far as what he says is concerned, it seems that he is not saying these things about the conclusion but about the premiss BC which he has just mentioned.

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35a3188 If the privative is posited in relation to the minor extreme and signifies contingency, [there will be no syllogism from the premisses taken themselves. But there will be if the contingent premiss is converted, as in the preceding. For let A hold of all B, and let it be contingent that B holds of no C; if the terms are this way there will be no necessity. But if BC is converted and it is taken that B holds of all C,189 a syllogism results as before. For the terms are related similarly in position. 35a11 It is the same way if both intervals are privative, that is, if AB signifies that A does not hold of B, BC that it is contingent that B holds of no C. In no way does necessity result through the premisses taken themselves, but there will be a syllogism if the contingent premiss is converted. For let it be taken that A holds of no B, and that it is contingent that B holds of no C; then no necessity results through these premisses. But if it is taken that it is contingent that B holds of all C – which is true – and the premiss AB remains the same, again there will be the same syllogism.]

He has shown what follows when the major is universal negative unqualified, the minor contingent affirmative; he now discusses the combination in which the major is universal affirmative unqualified, 15 the minor contingent negative. He says that nothing will follow if the contingent negative minor remains fixed, but there will be a syllogism if it is transformed into the affirmative – since if it is contingent that B holds of no C it is contingent that it holds of all (for these convert with one another). It was argued in this way in the preceding190 where nothing followed from the assumed premisses but resulted when the contingent negative premiss was transformed into a 20 contingent affirmative one. For there will be a universal affirmative

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contingent conclusion when there is a combination of a universal affirmative unqualified major premiss and a universal affirmative contingent minor. This is the case which seems to have been proved.191 (When he said ‘But if BC is converted and it is taken that B holds of all C’ he could have meant not only the conversion of the negative contingent proposi- 25 tion into an affirmative one but also the transformation of the contingent proposition into an unqualified one, by means of which the reductio ad impossibile occurred. This is the meaning of his words ‘And it is taken that B holds of all C’.) (35a11) And if the major premiss is universal negative unqualified, the minor universal negative contingent, nothing will follow from the assumed premisses; but if the negative contingent premiss is trans- 30 formed into an affirmative, ‘there will be the same syllogism’ as there was when originally the minor was assumed contingent universal affirmative and the major was universal negative unqualified. Perhaps he adds the words ‘there will be the same syllogism’ to indicate that the conclusion in the case of this combination will not be contingent in the 35 way specified either, just as it was shown not to be in the other case. 35a20192 But if it is posited that B does not hold of all C (and not that it is contingent that B does not hold of all C), [there will not be a syllogism in any way, whether the premiss AB is privative or affirmative. Common terms for holding by necessity: white, animal, snow; for not being contingent: white, animal, pitch.]

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He transforms the minor premiss into an unqualified negative and says there will be no syllogism whether the major is contingent affirmative or contingent negative. And again, as is his custom, he shows that this 5 is so by setting down terms and showing both holding of all and of none by necessity. For the case of A holding of C by necessity he sets down the terms white for A, animal for B, snow for C. For it is contingent that white holds of every animal and it is contingent that it holds of none, animal does not hold of snow, and white holds of snow by necessity. For 10 holding of none he sets down white, animal, pitch. For again it is contingent that white holds of every animal and of none, animal does not hold of pitch, and white holds of no pitch by necessity. The premisses relating to the combination under consideration would be truer if in the case of holding of all we were to take moving, white, walking. For it is contingent that moving holds of everything white and of nothing white, and let white hold of nothing that walks; 15 and moving holds of everything that walks by necessity. In the case of holding of none moving, white, standing still. For again it is contingent that moving holds of everything white and of nothing white, and let white hold of nothing standing still; and moving holds of nothing standing still by necessity. And nothing prevents

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20 transforming the terms for the sake of making the exposition clearer since he himself has said previously193 that ‘the terms ought to be taken in a better way’. But perhaps with these terms the conclusions are not necessary without qualification, but necessary on a condition; for moving holds of everything that walks by necessity – so long as it is walking – and again of nothing standing – so long as it is standing. But in any case the combination is non-syllogistic. 25

35a25 Thus it is evident that when the terms are universal [and one of the premisses is taken as unqualified, the other as contingent, when the premiss relating to the minor extreme is taken to be contingent a syllogism always results, except that sometimes it is from the premisses themselves and sometimes when the premiss is converted. We have said when and why each of these is the case.]

[35a30194 But if one of the intervals is taken as universal and the other as particular, when the interval relating to the major extreme is posited as universal and contingent – whether negative or affirmative – and the particular as affirmative and unqualified, there will be a complete syllogism just as when the terms are universal. The demonstration is the same as before. 35a35195 But when the interval relating to the major extreme is universal and unqualified (not contingent) and the other is particular and contingent, whether both are posited as negative or affirmative or if one is posited as negative and the other as affirmative, there will in every case be an incomplete syllogism. However, some of these will be proved by means of the impossible, others through conversion of the contingent premiss,196 as in the preceding. There will be a syllogism by conversion when the universal is posited in relation to the major extreme and signifies holding and the particular premiss is privative and assumes contingency, for example if A holds or does not hold of all B and it is contingent that B does not hold of some C. For a syllogism results if BC is converted with respect to contingency.]

Having reminded us that he said there is a syllogism in the combinations in the first figure in which an unqualified and a contingent premiss are mixed and both premisses are universal, he turns next to discussing combinations having one premiss universal, the other particular. (For 30 he calls premisses intervals.197) 198When he says ‘from the premisses themselves’ he does not mean that when the minor premiss is contingent

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the proof comes from the premisses alone – for all such combinations are proved by reductio ad impossibile – but he says this because an affirmative is proved without conversion; for when this is posited as negative, it is first necessary for it to be converted to affirmative. (35a30) He says that when the major premiss is universal contingent and either affirmative or negative and the minor is particular affirmative unqualified there will be complete syllogisms of which the conclusion is either that it is contingent that the first term holds of some of the last or that it is contingent that it does not hold of some of the last. These syllogisms are complete because again what they yield as conclusion is evident from the said of all or of said of none and they need nothing external to be proved. (35a35) But if the modalities of the premisses are interchanged and the major is unqualified universal, the minor particular contingent, and both are affirmative or both negative or one or the other is affirmative, the other negative, he says there will be a syllogism, but an incomplete one. And he adds the reason why all of them are incomplete when he says ‘However, some of these will be proved by means of the impossible, others through conversion’; this is equivalent to saying that all will be proved by means of the impossible, but those which have a contingent particular negative minor by means of conversion also, since the negative contingent premiss is transformed into an affirmative contingent one. Moreover, they also require reductio ad impossibile because the combinations in which the major premiss is unqualified are proved syllogistic by reductio ad impossibile. 199 The method of proof and of the reductio ad impossibile is the same and just what they were when both premisses were universal and the major unqualified. For if A holds of all B and it is contingent that B holds of some C, it will also be contingent that A holds of some C. For if not, the opposite , and the opposite of ‘It is contingent that A holds of some C’ is ‘It is not contingent that A holds of some C’, which is equivalent to ‘A holds of no C by necessity’. So A holds by necessity of no C.200 Let it also be taken that B holds of some C; for in the previous cases the contingent premiss was transformed into an unqualified one, which is false but not impossible. There results in the third figure a major which is necessary universal negative, a minor which is particular affirmative unqualified; the conclusion is particular negative necessary. Therefore A does not hold of some B by necessity, which is impossible, since it was hypothesized to hold of all. And even if the conclusion is not necessary but unqualified particular negative, what follows is impossible. For it is impossible that what holds of all not hold of some. 201 The proof is similar if A holds of no B and it is contingent that B holds of some C; for it is contingent that A does not hold of some

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C. For if this is not the case, the opposite , namely ‘It is not contingent that A does not hold of some C’, which is equivalent to ‘A holds of all C by necessity’. But also B holds of some C. Therefore A will 35 hold of some B by necessity; for this conclusion was proved to result in the third figure in the case of the combination under consideration. But this is impossible since it was hypothesized that A holds of no B. Similarly, in the case of this combination 203,1 even if the conclusion is not necessary but unqualified, the result is also impossible. It is also clear what the situation with conversions will be like if the minor premiss is taken to be particular negative contingent. He said ‘There will be a syllogism by conversion’, instead of ‘For there will be a 5 syllogism by conversion’.202 For he is speaking about nothing different from what he spoke about before. For he says about combinations in which the major is universal unqualified and the minor particular negative contingent that if the contingent negative is converted into the particular affirmative contingent. 10

35b8203 But when [the particular premiss posited] assumes not holding of something,204 [there will not be a syllogism. Terms for holding: white, animal, snow; for not holding: white, animal, pitch. One must take the demonstration by means of the indefinite.]

He says quite reasonably there will not be a syllogism when the minor premiss, which is particular negative, is taken as unqualified. For no syllogism resulted in the first figure when the minor was negative. For the unqualified negative premiss remains fixed, since, unlike a contingent premiss, it cannot be transformed into an affirmative. 15 He shows through terms that the conclusion is both ‘A holds of all C’ and ‘A holds of no C’ and adds ‘One must take the demonstration by means of the indefinite’.205 He says this because, using terms, he takes the particular negative unqualified minor in such a way that it is also true universally, not just particularly. For animal holds of no snow and 20 of no pitch, but it was assumed not to hold of some. 206But since ‘B does not hold of some C’ is true both when B holds of no C and when B does not hold of some C and holds of some C, he says that it is necessary to prove by the indefinite that a combination of this kind is non-syllogistic, taking the universal negative instead of the particular negative (since the latter is true if the former is); things 25 were proved in this way in several cases earlier.207 He has used the same terms a moment ago208 when he showed that the combination of a universal affirmative contingent major and a universal negative unqualified minor is non-syllogistic. He says that it is necessary for this to be proved in this way because when the particular negative minor is true per se and not because the universal is true, it turns out that B

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holds of some C; when the particular affirmative unqualified 30 minor was of this sort and the major was contingent universal and either affirmative or negative, complete syllogisms resulted, one with the conclusion that it is contingent that A holds of some C, the other with the conclusion that it is contingent that it does not hold of some C. Since these things were proved syllogistically, it is impossible to take material terms of this kind for holding of all and of none by necessity. 35 35b11209 But if the universal is posited in relation to the minor extreme [and the particular in relation to the major, whether either is privative or affirmative, contingent or unqualified, there will be no syllogism at all. Nor will there be a syllogism when the premisses are posited as particular or indeterminate whether they assume the contingent or the unqualified or each in alternation. The demonstration is the same as in the preceding. Common terms for holding by necessity: animal, white, human; for not being contingent: animal, white, cloak.] He has discussed combinations which are mixtures of an unqualified and a contingent premiss in the first figure of which some were syllogistic. He is now speaking about those which are non-syllogistic, and he says that if the minor premiss is universal and the major particular there will be no syllogism, however the premisses are taken with respect to being unqualified or contingent, affirmative or negative. Similarly, if both premisses are particular, not only if one is contingent, the other unqualified – this is what is meant by ‘in alternation’ – but also if both are contingent or both unqualified.210 He adds this because the same terms can be used when the premisses are taken in the former way and when they are taken in this way.211 For again he refutes these combinations arranged in this way by setting down terms, the demonstration he uses for non-syllogistic combinations. He shows that sometimes it can follow that A holds of all C by necessity and sometimes that it holds of no C by necessity. He shows it for holding of all by necessity with the terms animal, white, human. For let animal hold or not hold of something white or let it be contingent that it holds or be contingent that it doesn’t hold of something white; and let white hold or let it be contingent that it holds of every human or let it hold of some or let it be contingent that it does not hold of some212 – the terms can be taken to be related to one another in whatever way one wishes. And animal holds of every human by necessity. Terms for holding of none by necessity: animal, white, cloak. For again the premisses will be related in the same way as the previous ones, but animal holds of no cloak by necessity. The words ‘The demonstration is the same as in the preceding’213 indicate only that the way of establishing that the combinations discussed are non-syllogistic uses

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material terms. And having said this, he next gives the argument; for he says ‘common terms’. 35b20 It is then evident that there is always a syllogism when the premiss relating to the major extreme is posited as universal, [but there is never one when the premiss relating to the minor extreme is taken in that way.]

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It is necessary to understand as added to what is said the words ‘if both premisses are affirmative and the minor premiss is particular contingent’; for if it, i.e., the minor,215 is unqualified and is taken to be particular negative, the combination is non-syllogistic. 214

1.16 Combinations with a necessary and a contingent premiss216

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35b23 But when one premiss signifies holding or not holding217 by necessity and the other contingency, there will be a syllogism [if the terms are related in the same way; and it will be complete when necessity is posited in relation to the minor extreme. If the terms are affirmative, whether they are posited universally or not universally, the conclusion will be contingent and not of holding. But if one is affirmative and the other privative, when the affirmative is necessary, the conclusion will be contingent and not of not holding; but when the privative is necessary, the conclusion will be that it is contingent that something does not hold and that it does not hold, whether the terms are universal or not universal. And one should take its being contingent that something does not hold218 in the conclusion in the same way as in the preceding. But there will not be a syllogism of by necessity not holding, since not holding by necessity is distinct from by necessity not holding.219]

He turns to the mixture of a contingent and a necessary premiss in the 205,1 first figure and shows which combinations of such premisses in the first figure are syllogistic and which non-syllogistic. He says that in a mixture of this kind those in which the necessary premiss is assumed in the same way as the unqualified premiss was assumed in the combinations just discussed220 will also be syllogistic similarly; and they 5 will be complete in the case of those combinations in which again the major is contingent and the minor necessary. Furthermore, just as in those cases, when both premisses are affirmative and universal or just one is universal, the conclusion will be contingent in the way specified and not unqualified.221 He says that if one of the premisses is negative and the necessary premiss is affirmative, the conclusion will be that it 10 is contingent that something does not hold and not that it does not hold;

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but he says that when the negative premiss is necessary, if either both are universal or one is, the conclusion will be ‘it is contingent that something does not hold and that it does not hold’. The added words ‘when the affirmative is necessary’ refer to the combination which has a contingent negative major since he has already spoken about combinations in which both premisses are affirmative. When he says that in combinations in which the major premiss is universal negative necessary the conclusion will be ‘it is contingent that something does not hold and that it does not hold’, he explains the sense of ‘it is contingent that X does not hold’ ; for ‘it is contingent that X does not hold’ is not taken as signifying one thing and ‘X does not hold’ as signifying another, but ‘it is contingent that X does not hold’ is taken as what is predicated of what does not hold and not in the way specified.222 He indicates this when he says ‘one should take its being contingent that something does not hold in the conclusion in the same way as in the preceding’. By ‘in the preceding’ he means in the cases in which the major was unqualified negative; for he also showed223 in those cases that ‘It is contingent that A holds of no C’ (or that it does not hold of some C) is taken to be equivalent to ‘A holds by necessity of no C’ (or ‘A does not hold by necessity of all C’).224 For if the minor is contingent particular, then ‘A holds of all C by necessity’ is hypothesized in the reductio ad impossibile, since ‘It is not contingent that A does not hold of some C’ is the negation of ‘It is contingent that A does not hold of some C’, and ‘It is not contingent that A does not hold of some C’ is transformed into ‘A holds of all C by necessity’. (For if it is transformed into ‘A holds of no C by necessity’ no impossibility turns out.225) But if ‘A holds of all C by necessity’ is found impossible its negation would be proved, and this is ‘A does not hold by necessity of all C’; if this is equivalent to ‘It is contingent that A does not hold of some C’, then ‘A holds of all C by necessity’ would be the same as ‘It is not contingent that A does not hold of some C’. But if ‘It is not contingent that A does not hold of some C’ is true not only when A holds of all C by necessity but also when A holds of no C by necessity, then ‘A does not hold by necessity of all C’ would not be equivalent to ‘It is contingent that A does not hold of some C’; for then its negation has not been transformed into its equivalent but into something which implies it.226 For ‘It is not contingent that A does not hold of some C’ is true not only when A holds of all C by necessity but also when A holds of no C by necessity because the affirmation which says ‘It is contingent that A does not hold of some C’ is not true when A holds of no C by necessity. 227 †Consequently also its affirmation ‘It is contingent that A does not hold of some C’ is not only true when A does not hold by necessity of all C but also when it is not necessary that A holds of some C. Of these ‘A does not hold by necessity of all C’ does away with the universal

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affirmative necessary proposition, but it is true when A holds by necessity of no C because ‘A holds of all C by necessity’ (which falsifies ‘It is not contingent that A does not hold of some C’) is false then, since ‘It is contingent that A holds of some C’ is true then. And ‘It is not necessary that A holds of some C’ does away with the particular necessary proposition. And both together do away with all necessity, and so does ‘It is contingent that A does not hold of some C’, so that it is equivalent 10 to both together. And ‘A holds by necessity of no C’ and ‘A does not hold by necessity of all C’ are also true if the unqualified non-necessary propositions are true.† At the same time he shows us the nature of such propositions since they are not contingent.228 For this was what was being inquired into. For they had to be either necessary or unqualified or contingent. He says that they are unqualified when he says ‘and that it does not hold’. 15 He indicates the difference between ‘A holds by necessity of no C’ and ‘A holds of no C by necessity’ when he says ‘But there will not be a syllogism of by necessity not holding, since not holding by necessity is distinct’ (and he proved that not holding by necessity also follows in the case of a mixture of an unqualified negative major and a contingent affirmative minor229) ‘from by necessity not holding’. For the former is 20 unqualified, the latter necessary.

35b37230 [It is evident that the conclusion is not necessary] when [the terms] are affirmative. [For let A hold of all B by necessity and let it be contingent that B holds of all C. Then there will be an incomplete syllogism that it is contingent that A holds of all C – it is clear from the demonstration that it is incomplete, since it will be proved in the same way as in the preceding cases.] He proves that in mixtures of a necessary and a contingent premiss the conclusion is not necessary first in the case in which both premisses are affirmative and the major necessary. For let A hold of all B by necessity, 25 and let it be contingent that B holds of all C. Then the syllogism is incomplete; for, as we said,231 the complete syllogisms are those in which the major is contingent. Those in which the major premiss is necessary are also proved by reductio ad impossibile, just as those in which the major was unqualified were. So these aren’t complete either. For if A holds of all B by necessity and it is contingent that B holds of all C, it 30 will be contingent that A holds of all C. For if not, the opposite ‘It is not contingent that A holds of all C’ ; in those combinations this was also transformed into ‘A does not hold of some C by necessity’. If this is assumed and – a contingent premiss again being transformed into an unqualified one – it is added that B holds of all C, the conclusion

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in the third figure will be negative unqualified particular, and A will not hold of some B, which is impossible, since it was hypothesized to hold of all by necessity. Therefore, the hypothesis from which this follows, ‘It is not contingent that A holds of all C’ is impossible; therefore, the affirmation ‘It is contingent that A holds of all C’ is true; and, therefore, the conclusion is contingent. For A will not hold by necessity of all C232 if the minor premiss is contingent. But perhaps again in this mixture what is proved is not ‘It is contingent that A holds of all C’ but the unique opposite of ‘A does not hold of some C by necessity’, namely ‘A will not hold by necessity of no C’.233 For ‘It is not contingent that A holds of all C’ was transformed into ‘A does not hold of some C by necessity’  because ‘It is not contingent that A holds of all C’ is true both when A holds of some C by necessity and when A does not hold of some C by necessity, but if ‘It is not contingent that A holds of all C’ is transformed into ‘A holds of some C by necessity’, nothing impossible follows in this combination. We have spoken about this in the preceding.234 235 Again, ‘A does not hold by necessity of no C’ would do away with ‘A does not hold of some C by necessity’ just as ‘A holds by necessity of no C’ did away with ‘A holds of some C by necessity’; and ‘A does not hold by necessity of no C’ can also be true when A holds of some C by necessity. The proposition which says ‘Moving does not hold by necessity of no body’ is like this since there is a body, namely the rotating body, of which it holds by necessity. We have spoken about this in the preceding.236 237 However, it is also possible to prove the impossibility without transforming the contingent proposition into an unqualified one, but keeping ‘It is contingent that B holds of all C’ fixed. For if A does not hold of some C by necessity and it is contingent that B holds of all C, it follows in the third figure that it is contingent that A does not hold of some B, which is impossible, since A holds of all B by necessity. But since it has not yet been shown what follows in a mixture in the third figure of contingent propositions and propositions which hold necessarily and which combinations in this figure are syllogistic, he uses the transformation of a contingent proposition into an unqualified one. For he has already discussed mixtures of a necessary and an unqualified premiss in all figures. Nevertheless, it is necessary to understand that insofar as it involves a reductio ad impossibile using the third figure, in the case of a mixture having a universal necessary major premiss (either affirmative or negative) and a contingent minor it can be proved that there is a necessary and an unqualified and a contingent conclusion; and the conclusions are affirmative if the necessary premiss is affirmative and negative if it is negative. But he himself has said, ‘But there will not be a syllogism of by necessity not holding’.238 We have investigated these matters at greater length in our book on mixtures.239

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36a2240 Again let it be contingent that A holds of all B [and let B hold of all C by necessity. There will be a syllogism that it is contingent that A holds of all C but not that A holds of all C. And the syllogism will be complete, not incomplete; for it is completed directly through the original premisses.]

He proves that something follows necessarily when the major premiss is universal affirmative contingent and the minor necessary universal affirmative, and at the same time he signals that the syllogism from such a combination is complete since what follows, which is contingent 5 in the way specified, is proved by means of said of all.241 36a7242 If the premisses are not the same in form, [first let the privative premiss be necessary, and let it not be contingent that A holds of any B, and let it be contingent that B holds of all C. Then it is necessary that A holds of no C. For let it be assumed that A holds of all or of some C; but it was assumed not contingent that it holds of any B. Since, then, the privative premiss converts, neither is it contingent that B holds of any A. But A is assumed to hold of all or some C, so that it is not contingent that B holds of †any or of† all C. But it was hypothesized to hold of all originally. And it is evident that there is also a syllogism that it is contingent that A does not hold since there is one that A does not hold.] this: ‘if only one premiss is affirmative’. He shows which combinations with such premisses are syllogistic and that 10 none yields a necessary conclusion. He first takes the combination having a universal negative necessary major premiss – for the words ‘let it not be contingent that A holds of any B’ signify what holds of no B by necessity243 – and a universal affirmative contingent minor premiss BC. He says that in the case of this combination it is necessary that A holds of no C, but he does not mean that A holds of no C by necessity.244 15 For he does not place the necessity in the conclusion; rather he invokes necessity to make clear that ‘A holds of no C’ will be the conclusion. And these combinations which imply something by necessity are syllogistic, and these are the ones in which the same thing results in the case of all material terms. That he takes the conclusion to be unqualified negative is clear from the fact that he hypothesizes as the opposite of this for 20 proving by reductio ad impossibile that this is the conclusion of an unqualified affirmative proposition but not a contingent one (which is the opposite of a necessary one). For he says ‘Let it be assumed that A holds’. (He goes beyond what is required and hypothesizes ‘of all or some C’ to show that something impossible follows from either hypothesis; for the proposition which posits that A holds of some C is the contradictory

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opposite of the one which says ‘A holds of no C’.) Taking and hypothesizing the opposite of the conclusion ‘A holds of some C’, he adds the necessary proposition which converts with the proposition AB, which is that A holds of no B by necessity. And B holds of no A by necessity; but also A is assumed to hold of some or all C. It follows in the first figure when the major premiss is necessary negative universal and the minor is unqualified affirmative and either particular or universal that the conclusion is necessary negative. The result then is that B does not hold of some C or holds of no C by necessity, which is impossible, since it was assumed that it is contingent that B holds of all C. Therefore, in the case of the combination under consideration the opposite of the hypothesis will follow, namely ‘A holds of no C’. And if ‘A holds of no C’ then also ‘It is contingent that A holds of no C’ since contingency is also predicated of the unqualified; for it is true to say that the unqualified is also contingent, but the contingent is not always also unqualified. 245 It is necessary to understand that the preceding proof is sound if it is true that in mixtures of a necessary major and an unqualified minor in the first figure the conclusion is necessary. But if the conclusion is unqualified, nothing impossible follows; for in that case it would follow that B does not hold of some C or holds of none, when it is assumed that it is contingent that B holds of all C; but this is not impossible. 246 Is it then perhaps more correct to say that contingency in the way specified also follows in the case of this mixture? For if it is assumed that A holds of no B by necessity and that it is contingent that B holds of all C, it will follow that it is contingent that A holds of no C. For if not, the opposite , namely ‘It is not contingent that A holds of no C’, i.e., ‘A holds of some C by necessity’; since this is particular necessary affirmative and BA, which comes by conversion of AB, is also necessary negative universal, there results from the two necessary premisses the conclusion that B does not hold of some C by necessity, which is impossible, since it is contingent that B holds of all C. Therefore, ‘A holds of some C by necessity’ is also impossible; therefore the opposite , namely ‘It is contingent that A holds of no C’. Or perhaps the conclusion won’t be contingent in the way specified in this case either since what will follow is the opposite of ‘A holds of some C by necessity’, which is ‘A holds by necessity of no C’, which has been previously shown to be different from a contingent proposition.247 248 The words ‘so that it is not contingent that B holds of all C. But it was hypothesized to hold of all originally’,249 are equivalent to ‘so that it will result that it is contingent that B does not hold of all C’, which he takes to be equivalent to ‘B does not hold of all C’. For this opposite follows from what is proved; for he uses this wanting to show that A holds of no C. 250 What he himself wanted to prove, namely that the conclusion of the combination which has just been discussed is unqualified universal

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negative, is also proved through the third figure by reductio ad impossibile. For if it is not true that A holds of no C, it will be true that it holds 30 of some. If one adds to this the transformation of the contingent premiss into the unqualified ‘B holds of all C’, it will follow that A holds of some B, which is impossible since it was assumed that it holds of none by necessity. 36a17251 Again, let the affirmative252 premiss be necessary, let it be contingent that A holds of no B, [and let B hold of all C by necessity. Then the syllogism will be complete, but it will conclude not that something does not hold but that it is contingent that it does not hold. For the premiss from the major extreme is taken in this way, and it is not possible to do a reductio ad impossibile. For if A is hypothesized to hold of no253 C and it is also assumed that it is contingent that it holds of no B, nothing impossible results from these things.] 35 He in turn takes a combination in which the major is contingent universal negative, the minor universal affirmative necessary; and he 210,1 proves both that the conclusion is contingent, not necessary, and that the syllogism is complete since the conclusion is proved by means of A holding contingently of no B. To show that the conclusion is contingent in the way specified, he says ‘For the premiss from 5 the major extreme is taken in this way’, that is, the major premiss is taken as contingent. And we said254 that when the major premiss is contingent the syllogisms are complete and the conclusion is contingent in the way specified. For it will be contingent that A holds of C since C is among the B of all of which it is contingent that A holds.255 The words ‘and it is not possible to do a reductio ad impossibile’ mean the following. Having said that in the case of this combination it is 10 evident from the assumed premisses that the conclusion is contingent universal negative (for he said ‘But it will conclude not that something does not hold but that it is contingent that it does not hold’), he adds that it is not possible to prove by reductio ad impossibile that the conclusion is unqualified, as we did in the case of the previous combination. For suppose we were to proceed in this case as we did in that 15 and were to take as hypothesis ‘A holds of some C’, the opposite of ‘A holds of no C’, or even ‘A holds of all C’, wishing to prove that the conclusion is ‘A holds of no C’ and not ‘It is contingent that A holds of no C’; to this assumption we would add that it is contingent that A holds of no B; but then the combination would be non-syllogistic since it is in the second figure with an affirmative unqualified particular 20 premiss and a universal negative contingent one, because the contingent premiss is equivalent to an affirmative one,256 and the proof does not go through.

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He has made what he says unclear by being concise. he takes the conclusion to be unqualified negative universal instead of contingent negative universal; and, wanting to prove that this unqualified proposition cannot be shown to follow by a reductio ad impossibile, he does not take the opposite of this and hypothesize it (the opposite is the particular unqualified affirmative ‘A holds of some C’, which, taken together with ‘It is contingent that A holds of no B’ yields no conclusion); instead he omits it and only adds the other premiss of the pair with which it implies nothing that it is known what has to be transformed and hypothesized in place of the universal negative unqualified , namely a particular affirmative unqualified proposition. However in some texts one reads ‘For if A is hypothesized [not]258 to hold of some C’ instead of ‘For if A is hypothesized to hold of no C’. But even if this were the text the proposition that A holds of no C would again have been left out; the opposite of this, ‘A holds of some C’, posited together with ‘It is contingent that A holds of no B’ implies nothing. 259 Moreover nothing impossible will follow even if we add to the hypothesis that A holds of some C the premiss that B holds of all C. For it follows in the third figure260 that A holds of some B, but it is not impossible that it be contingent that A holds of no B and that it holds of some. 261 It is also possible to prove that the conclusion is contingent by means of reductio ad impossibile. For if it is not true that it is contingent that A holds of no C, it will be true that it is not contingent that it holds of none, i.e., that A holds of some C by necessity. But it is also assumed that B holds of all C by necessity. From these premisses it follows in the third figure that A holds of some B by necessity, but this is impossible, since it was assumed that it is contingent that it holds of none (and so of all).262 Therefore it is also impossible that A holds of some C by necessity, so that ‘It is contingent that A holds of no C’ is true. For the conclusion will be contingent in the way specified and will not be that A holds by necessity of no C, as it was in the other cases263 in which ‘It is not contingent that X holds of no Y’ was transformed into ‘X holds of some Y by necessity’. For in the case of the mixture under consideration even if ‘It is not contingent that A holds of no C’ is transformed in the opposite way into ‘A does not hold of some C by necessity’ (‘It is not contingent that A holds of no C’ is true when this is), an impossibility follows in the same way, namely that A does not hold of some B by necessity, when it is assumed to be contingent that it holds of all B.264 So if, when a hypothesis is made with respect to each of these things which alone are such that ‘It is not contingent that A holds of no C’ is true when they are, an impossibility follows, it is clear that the unique opposite of that is proved, namely ‘It is contingent that A holds of no C’. 257

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36a25265 But if the privative is posited in relation to the minor extreme, [when it signifies contingency there will be a syllogism by conversion, just as before; but when it indicates not being contingent there will not be. Nor will there be when both premisses are posited as privative if the minor is not contingent. The terms are the same. For holding: white, animal, snow; For not holding: white, animal, pitch.] He says that if in the mixture of a necessary and a contingent premiss 20 the major premiss is affirmative and the minor negative and the privative minor is contingent, there will be a syllogistic combination when the contingent negative premiss is converted into an affirmative. But he says there will not be a syllogism when the minor 25 is negative necessary and the major is affirmative contingent. The reason is that we assume that it is impossible for there to be a syllogism in the first figure when the minor is negative, but when the negative premiss is taken as contingent, it can be transformed into an affirmative, but if it is taken as necessary it remains negative and does not make a syllogism. Nor will there be a syllogism if both premisses are taken to be negative and the minor is necessary. He establishes that the combinations having the minor necessary 30 negative are non-syllogistic and shows by setting down terms that when things are this way both ‘A holds of all C by necessity’ and ‘A holds of no C by necessity’ can be inferred. The terms for all are white, animal, snow, since it is contingent that white holds of all animal and contingent that it holds of none, but animal holds of no snow by necessity, and white holds of all snow by necessity. Terms for not holding are white, 35 animal, pitch, since the premisses will be related the same way and 212,1 white holds of no pitch by necessity. (It is perhaps more correct to take horse instead of animal since ‘It is contingent that white holds of no animal’ is not true.266)

[36a32267 The situation will be the same in the case of particular syllogisms. For when the privative premiss is necessary, the conclusion will also be of not holding. For example, if it is not contingent that A holds of any B but it is contingent that B holds of some C, it is necessary that A not hold of some C. For if A holds of all C and it is not contingent that it holds of any B, then it is not contingent that B holds of any A either. Thus if A holds of all C, it is not contingent that B holds of any C, but it was assumed contingent that it holds of some.] He says ‘The situation will be the same in the case of particular

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syllogisms’,268 that is, if one premiss is universal and the other particular the situation will be the same as it was when both were universal. For in syllogistic combinations in the mixture under consideration when both premisses are not universal, if the major is universal negative necessary, the minor particular contingent affirmative, he says the conclusion will be particular negative and not contingent but unqualified. He proves this as follows. A holds of no B by necessity; let it be contingent that B holds of some C; then A does not hold of some C. For if that is not so, let the opposite hold and let A hold of all C. But it was assumed that A holds of no B by necessity. The result is a combination in the second-figure put together from a necessary negative major and a universal affirmative unqualified minor, and it was proved that in this case the conclusion is universal negative necessary. For if the necessary premiss is converted, it results that B holds of no A by necessity; but it was also hypothesized that A holds of all C; the result is that B holds of no C by necessity, which is impossible, since it was assumed that it is contingent that it holds of some. Therefore, the hypothesis269 from which this followed is impossible, namely ‘A holds of all C’. Therefore the opposite of this, ‘A does not hold of all C’, is true, and the conclusion is unqualified. 270 But this proof depends on it being agreed that the conclusion of a necessary major and an unqualified minor is necessary. However, it is also possible without this to give an indisputable proof by reductio ad impossibile that the conclusion of the combination under consideration is contingent particular negative. For if ‘It is contingent that A does not hold of some C’ is not true, it will be true that it holds of all by necessity. But it is assumed that A holds of no B by necessity. From these assumptions it follows in the second figure that B holds of no C by necessity, which is impossible since it was contingent that it holds of some. Therefore, it is true that it is contingent that A does not hold of some C.

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[36a39271 But when the particular affirmative premiss, i.e., BC, in a privative syllogism is necessary or a universal premiss AB in an affirmative syllogism is necessary there will not be a syllogism of holding. The demonstration is the same as before.] He now says that, if in a negative syllogism the major negative premiss is not necessary but the minor is particular affirmative necessary (so 30 that the major is universal negative contingent) or again if in an affirmative syllogism the major is universal affirmative necessary and the minor particular contingent, the conclusion (he says) will not still be unqualified; but it is clear that it will be contingent in the way specified. (It is necessary to understand this addition.) 272For if the major 35 is contingent negative and the minor necessary particular affirmative,

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213,1 the conclusion is proved to be contingent particular negative through said of none, because some C is under B and it is contingent that A holds of no B, and so it is contingent that A does not hold of some C. For it was also this way when the minor was assumed to be universal affirmative necessary;273 to prove that in the case of this 5 combination the conclusion is contingent he used274 the fact that it is not possible for an unqualified conclusion to be proved by reductio ad impossibile. 275For it also is not possible in the case of the present combination for it to be proved by reductio ad impossibile that the conclusion is unqualified negative particular, since either there will be a non-syllogistic combination in the second figure, or in the third the combination implies nothing impossible, as was also the case when the 10 minor was necessary universal affirmative and the major contingent negative, as we showed.276 277 As I said previously,278 it is worth asking why the conclusion will not be necessary negative when the major is taken to be necessary universal negative. For if we assert that, when AB is necessary universal negative and the minor is contingent universal affirmative, a 15 necessary universal negative conclusion follows, this will be proved to be so by reductio ad impossibile. For if it is not the case that A holds of no C by necessity, it will then be contingent that it holds of some. But it is also contingent that B holds of all C. Therefore,279 it will be contingent that A holds of some B, which is impossible, since it was assumed that it holds of none by necessity. But even if BC is particular affirmative contingent, it will still be 20 possible to say that the conclusion will be particular negative necessary. For A does not hold of some C by necessity. For otherwise it is contingent that it holds of all. But it is also contingent that B holds of some C. Therefore,280 it will be contingent that A holds of some B, which is impossible since it was assumed that it holds of none by necessity. Therefore it is true that A holds of no C by necessity if BC is 25 contingent universal, or that A does not hold of some C by necessity if BC is taken to be particular contingent. But, as I have already said,281 we have investigated these matters in our book on mixtures. 282 Again, he says that if both premisses are affirmative, the major universal and necessary, the minor particular contingent (this is the 30 meaning of ‘or a universal premiss AB in an affirmative syllogism’,283 the conclusion will be particular contingent affirmative, just as it was when the major was universal affirmative necessary, the minor universal contingent. For it was proved284 by reductio ad impossibile that the conclusion is contingent universal affirmative. But in fact, this ought to be proved in the case of 35 the combination under consideration as well. Let it be assumed that 214,1 with the given assumptions it is contingent that A holds of some C. For, if not, the opposite, ‘It is not contingent that A holds of some C’, i.e., ‘A

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holds of no C by necessity’ . But it is also contingent that B holds of some C. Let this be transformed into B holds of some C. Then – in the third figure – A will not hold of some B, or it is contingent that A does not hold of some B. (For it makes no difference which of these one takes, 285 contingent or one trans- 5 forms it into an unqualified proposition, the argument goes through.) But it is impossible that A does not hold of some B or that it is contingent that it does not hold of some B – that is impossible –, since it was hypothesized that it holds of all B by necessity. Therefore the opposite of the hypothesis from which the impossibility followed holds; that is ‘It is contingent that A holds of some C’, because ‘It is not contingent that A holds of some C’ is equivalent to ‘A holds of no C by 10 necessity’ and was transformed into it. Therefore the proper thing which follows is ‘It is not the case that A holds of no C by necessity’, the opposite of the hypothesis. 286 But suppose the conclusion is taken to be not ‘It is contingent that A holds of some C’ but ‘A holds of some C’. If the opposite of this, ‘A holds of no C’ is hypothesized and ‘It is contingent that B holds of some C’ is added, it will follow that A does not hold of some B or that it is contingent that A does not hold of some B. Both of these are impossible 15 if A holds of all B by necessity. So, the hypothesis being destroyed, it would follow that A holds of some C. However, he says that in the combination under consideration there will not be a conclusion of holding. [36b3287 But if the universal premiss is posited in relation to the minor extreme and is contingent and either affirmative or privative, and the particular premiss is necessary there will not be a syllogism. Terms for holding by necessity: animal, white, human; terms for not being contingent: animal, white, cloak.] But if the minor and not the major is universal, then no matter how the minor is taken, whether affirmative contingent or negative contingent, 20 there will be no syllogism when the major is particular necessary. Again he shows this by setting down terms. Terms for holding of all by necessity are animal, white, human. For animal holds of something white, e.g., swan, by necessity, but it also does not hold of something white, e.g., snow, ; and it is contingent that white holds of every human but also contingent that it holds of none; and animal 25 holds of every human by necessity. Terms for holding of none are animal, white, cloak. For, again, both premisses are true in the same way, and animal holds of no cloak by necessity.

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[36b7288 But when the universal premiss is necessary, the particular contingent, then if the universal is privative, terms for holding are animal, white, crow, terms for not holding animal, white, pitch; and if the universal is affirmative, terms for holding are animal, white, swan, and for not being contingent: animal, white snow.] Nor will there be any syllogism if the universal premiss, i.e., the minor (for this was the universal), is necessary, the particu30 lar contingent. For if the minor premiss is privative universal and necessary, terms for holding are animal, white, crow. For it is contingent that animal holds of something white and contingent that it does not hold of something white, and white holds of no crow by necessity, and animal holds of every crow by necessity. Terms for holding of none are animal, white, pitch; for the premisses are related similarly, and animal holds of no pitch . But if the minor is universal affirmative necessary, terms for holding 35 are animal, white, swan. For again, it is contingent that animal holds 215,1 of something white and contingent that it does not hold of something white, and white holds of every swan by necessity, and animal holds of every swan by necessity. Terms for not holding are animal, white, snow; for the premisses are related similarly, and animal holds of no snow by necessity. [36b12 when the premisses are taken as indeterminate or both particular. Common terms for holding: animal, white, human; for not holding: animal, white soulless. 289For animal holds of something white and white of something soulless, and there is holding by necessity and it not being contingent that something holds. And the situation is the same in the case of contingency, so that the terms are serviceable in all cases.] Nor will there be any syllogism if both premisses are either indetermi5 nate or particular, however they are related in quality.290 Common terms which show holding for all ways whatsoever of taking the premisses are animal, white, human. 291For animal holds of something white, e.g., swan, by necessity, and does not hold of something white, e.g., snow, by necessity. 292And it is contingent that it holds of something white and does not hold of something white, e.g., of human; for it is contingent that some white thing is a human and it is contingent that some white thing is not a human, and, if this is so, it would also be contingent that some white thing is an animal and contingent that some 10 white thing is not an animal. 293And white does not hold of some humans, e.g., Ethiopians, by necessity, and it holds of some, e.g., of Celts, by

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necessity. But it is contingent that it holds of some and contingent that it does not hold of some, as in most cases. And animal holds of every human by necessity. Terms for not holding are animal, white, soulless; for again, if the premisses are taken in the same way, animal holds of nothing soulless by necessity. 294 He sets down material terms for holding of all and of none by 15 necessity for each proof, and, taking one proposition, he shows using the terms that all the possible transformations, relating to both necessity and contingency are indefinite. He leaves to us the task of also seeing the same thing in the case of the other premisses. 36b15 For animal holds of something white and white of something soulless,295 and there is holding by necessity and it not being contingent that something holds. [And the situation is the same in the case of contingency, so that the terms are serviceable in all cases.]

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Since the combinations are composed of two particular premisses, one necessary, one contingent, he shows that in the case of all combinations of such premisses it is possible to use the material terms which have been taken to show them non-syllogistic; he shows that each of the particular premisses in the case of the material terms assumed can be taken as necessary affirmative and necessary negative and, again, as 25 contingent affirmative and negative. Consequently, if the premisses are taken in alternation, one necessary and one contingent, they will be taken truly and it will be established by reference to them that those combinations are non-syllogistic.

36b19296 It is clear297 from what has been said that when the terms are related in the same way [there is or is not a syllogism] with an unqualified premiss and with necessary ones [except that if the privative premiss is posited as unqualified the conclusion of the syllogism is contingent, but if it is posited as necessary the conclusion is both of contingency and of not holding.]

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Just as combinations of the same kind were syllogistic in the case of combinations with both premisses unqualified and of combinations with both necessary, so it is shown to be the same way in the case of mixtures of an unqualified and a contingent premiss and of a necessary . For combinations in which the unqualified or the necessary premiss is assumed similarly are syllogistic in both cases or 216,1 again the similar combinations are non-syllogistic. The difference between them, as he now says, is that in the case of combinations having

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an unqualified premiss with a contingent one, when the major is privative unqualified, the conclusion is contingent; but in the case of combinations of a necessary and a contingent premiss, when the necessary premiss is privative, the conclusion is unqualified and contingent because it is unqualified, but it is not contingent in the way specified. But how can this assertion be sound? For he also showed298 that in the combinations in which the negative premiss is unqualified the conclusion is not contingent in the way specified but is ‘A holds by necessity of no C’ or ‘A does not hold by necessity of all C’. Or because the conclusion is still contingent in a way even if it isn’t straightforwardly contingent in the way specified? ‘A holds by necessity of no C’ is contingent in this sense. 299But in the case in which the premiss is necessary negative, it was proved that the conclusion is straightforwardly unqualified, since he posited to begin with that the conclusion is unqualified, hypothesized the opposite of this, and did a reductio ad impossibile. 300 But this does not happen if we use301 a similar reductio in the case in which the major is unqualified universal negative and the minor contingent universal affirmative; for in this case it is impossible for anything to be proved by reductio ad impossibile in the second figure if the conclusion is taken to be ‘A holds of no C’. For the opposite of the conclusion taken, ‘A holds of no C’, is ‘A holds of some C’, and with ‘A holds of no B’ it implies in the second figure that B does not hold of some C, which is not impossible if it is contingent that B holds of all C. And if the reductio ad impossibile takes place in the third figure, nothing impossible is proved; for what follows is ‘It is contingent that A holds of some B’ when it was assumed that it holds of none, which is not impossible. Or should we say that if the contingent premiss is transformed into an unqualified one which is false but not impossible, what follows does become impossible? For it follows that A holds of some B, but it was assumed to hold of none. 302 When the major premiss was taken to be necessary universal negative, it was proved that B does not hold of some C by necessity, which was impossible. This resulted because of the assumption that when the major in a mixture of a necessary and an unqualified premiss is necessary, the conclusion is necessary. But if it is posited that the conclusion in such a mixture is unqualified, nothing impossible will follow in the second figure whether the negative premiss is necessary or unqualified. For either B does not hold of some C or that B holds of no C; but it was assumed that it is contingent that it holds of all, which is not impossible. 303 Nevertheless an impossibility does result through the third figure. For if one adds to the hypothesis taken, namely that A holds of some C,

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the premiss BC, which is (as was assumed) universal affirmative contingent, or if this premiss is transformed into an unqualified one, something impossible will follow in either case. For A will hold of some B although it is assumed either not to hold of all of it (as in the first mixture) or not to hold of all of it by necessity (in the second mixture). Alternatively if in the third figure it is inferred from ‘A holds of some C’ and ‘It is contingent that B holds of all C’ that it is contingent that A holds of some B, nothing impossible will be inferred in the case of the mixture of an unqualified and a contingent premiss, but an impossibility will be inferred in the case of the mixture of a necessary and a contingent premiss. For if A holds of no B by necessity, it is impossible that it is contingent that A holds of some B. 304 Here it would be reasonable for someone to ask how it is, in a mixture of an unqualified universal negative major and a contingent universal affirmative minor in the first figure, that the impossible result does not come from the transformation of the contingent universal affirmative premiss into an unqualified one, if nothing impossible follows when it is kept fixed and conjoined with the hypothesis , but what follows is impossible if the contingent premiss is transformed into an unqualified one and taken together with the same hypothesis. For the conclusion drawn from ‘A holds of some C’ and ‘It is contingent that B holds of all C’, namely ‘It is contingent that A holds of some B’ is not impossible when it was assumed to hold of none. But from ‘A holds of some C’ and ‘B holds of all C’ it follows that A holds of some B, which is impossible since A was assumed to hold of none of it. For a universal negative proposition having a truth which is temporally restricted results if it no longer remains true when the contingent premiss is transformed into an unqualified affirmative one; nor would what follows be impossible either.305 For since the transformation is false but it is not impossible, nothing impossible would be a consequence of this transformation.

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36b24 It is also clear that all these syllogisms are incomplete [and that they are completed through the figures previously mentioned.]306 Not absolutely all of these , but only those in which 25 the major was taken to be unqualified or necessary. For he proved that such combinations yield a conclusion by reductio ad impossibile. ‘They are completed through figures previously mentioned’ because the reductio ad impossibile is through one of the assumed figures.

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1.17-19 The second figure 1.17 Combinations with two contingent premisses307

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36b26 In the second figure when both premisses are taken308 as contingent, there will be no syllogism, [whether the premisses are posited as affirmative or privative, universal or particular. 36b29309 But when one signifies holding and the other contingency, there will never be a syllogism if the affirmative signifies holding, but there will always be if the privative universal does. The situation is also the same when one premiss is taken as necessary and the other as contingent. And in these cases too it is necessary to take the contingency in the conclusion as it was taken in the preceding.] Having discussed all the mixtures in the first figure, he turns to the second figure; he says that in this figure there is no syllogism from two contingent premisses whether they are taken as both affirmative or both negative or one as affirmative and the other as negative or if both are universal or one particular. The reason is that it was shown310 that in the second figure it is impossible for there to be a syllogism from two affirmative premisses. Therefore there can’t be one from two contingent premisses either. For if the premisses are taken as negative, they will be equivalent to affirmative ones because affirmative contingent propositions convert with negative ones. Someone might ask why, when a contingent negative proposition, which is equivalent to and converts with a contingent affirmative one, is transformed into that affirmative proposition and taken, it has the same use as the affirmative in making the combination syllogistic or non-syllogistic (as in the present case), but the affirmative is not transformed into that negative nor does it make non-syllogistic the combinations which need this affirmative proposition in order to be syllogistic. In the first place it was said311 that if the affirmative premiss makes a combination syllogistic, it is absurd to seek to transform it into something which will make the combination no longer syllogistic. But the transformation is reasonable in the case of a negative premiss because it requires help in order to be syllogistic. For we also do this in conversions of premisses. In those cases in which combinations are proved to be syllogistic by conversion we make use of conversions, but we do not convert those premisses which when converted will not be syllogistic. Furthermore a contingent negative proposition is not in itself or as contingent negative, since what negates is not attached to the modality.312 For a negative contingent proposition potentially affirms the same thing as it denies, since this is what it means to say that

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if it is contingent that X does not hold then it is also contingent that it holds. Thus, in general, a contingent negative proposition is not a negation; and because it is an affirmation, since when it is expressed in this way it has a negative form, it is transformed into that affirmative proposition as being more evident and agreed to.313 Because of this and also because most of the combinations in this figure were shown to yield a conclusion by conversion of the universal negative premiss – only one of them was proved by reductio ad impossibile314 – there is no syllogism in the second figure from two contingent premisses. But he will show that a contingent negative proposition does not convert with itself.315 (31b29)316 If the combinations are mixed of an unqualified and a contingent premiss or of a necessary and a contingent one, then, if only the contingent premiss in them is negative, the combination will not be syllogistic because then both premisses are potentially affirmative since a contingent negative premiss converts with an affirmative one. 317 But something will always follow if the unqualified or necessary premiss is negative and universal. 318 For if the unqualified or necessary premiss is particular negative, the combinations will not be syllogistic. For in the second figure it is necessary that the major premiss be universal; consequently when the major premiss is unqualified particular, the combinations will be non-syllogistic. 319 And even if this minor is universal and the major contingent particular, the combinations will be non-syllogistic. 320 However if the minor is universal and the major contingent universal, the combinations are syllogistic, but they would be proved by reductio ad impossibile, not by conversion; for the universal affirmative contingent proposition does not convert with itself, as has been shown,321 nor does the universal negative, as he will show.322 Furthermore the minor premiss in the first figure becomes negative, if the major is also converted, but a combination of this kind is non-syllogistic. Furthermore conversion is required in combinations which are syllogistic in this way,323 but neither premiss can be converted, not the negative one because it is particular and not the affirmative because it is contingent and universal. 324 But there will be a conclusion if both the unqualified and the contingent premiss are negative and the minor is contingent, because a contingent negative premiss is transformed into a contingent affirmative one. This, then, was the reason why, after discussing contingent combinations in the first figure,325 he then discussed mixtures of an unqualified and a contingent premiss and of a necessary and a contingent one in the first figure326 before discussing combinations of contingent premisses in the second and third figure:327 there is no syllogism from two contingent

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premisses in the second figure, but there is from mixed premisses. 20 However, it was necessary to understand first mixtures in the first figure, since combinations of mixed premisses in the second figure are shown to be syllogistic by reduction to the first figure, just as the simple ones are.328 Therefore it was necessary to understand first which of the mixtures in the first figure yield a conclusion, how many do, and how they are produced. 25

36b33 And in these cases too it is necessary to take the contingency in the conclusions as it was taken in the preceding.

Just as it was the case with mixtures in the first figure with an unqualified or necessary negative major premiss that the conclusion was not contingent in the way specified, so too, he says, it will be the case in the second figure. For in syllogistic combinations which mix an 30 unqualified and a contingent premiss or a necessary and a contingent one in the second figure, the universal negative major is always unqualified or necessary.329

36b35 It should first be shown that a privative contingent331 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. 36b37332 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.] 35 When he discussed propositional conversions and showed which propositions convert with which,333 he said that a universal negative 220,1 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

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require conversions. Since he is going to show that there is no syllogism from contingent premisses in the second figure, and since 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.334 However, as we mentioned at the beginning,335 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.336 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.337 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.338 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 contingent that A holds of no B, it will also be contingent

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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 5 contingency in the way specified is not the only contingency.339 340 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.341 Or perhaps he first showed this by contingent negative propositions being transformed into affirmative ones, affirmative con10 tingent 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 20 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 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 342

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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’343 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. 344 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 possible345 for their opposites to be true with respect to the same thing’.346 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? 347 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,348 since it does not hold of that by necessity. 349 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.

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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 25 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 30 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 35 contingent, it is not assumed to convert with respect to it. 223,1

37a9350 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 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’.

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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 associates351 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 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:352 to

(i) B does not hold of some A by necessity

(ii) It is contingent that B holds of all A and (iii) B holds of some A by necessity

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(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 30 (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) 35 and thereby also does away with (iv)). But, if this is so, the negation of 224,1 (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 5 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)353 –, nothing impossible will follow.354 For it is not 10 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 15 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 possible355 that (i) is true. 356 And also it seems that only when (i) holds does the contingent negative proposition not convert. For although it is contin20 gent 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 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 25 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). 37a17 For if B does not hold of some A357 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

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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’,358 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,359 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. 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,360 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

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25 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 30 – 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, 35 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 226,1 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 5 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 10 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 specified361 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 15 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) 20 was not true362 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’ 25 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 propo-

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sition 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.363

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37a32364 This having been shown, let it be assumed that it is contingent that A holds of no B and contingent that it holds of all C. [Then there will be no syllogism by conversion, since it has been said that a premiss of this kind does not convert. But there won’t be one by means of the impossible either. For if it is posited that it is contingent that B holds of all C, nothing false results; for it would be contingent that A holds of all C and that it holds of none.]

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20 converted, but it has been shown that a contingent negative proposition does not convert with itself. Nor can it be proved by reductio ad impossibile, the method by which combinations which cannot be proved by conversion are proved to yield a conclusion. For if it is contingent that A holds of no B and contingent that it holds of all C, then if someone wishing to prove that it follows 25 that it is contingent that B holds of no C takes the opposite of this (namely ‘It is not contingent that B holds of no C’, which he transforms into a universal affirmative necessary proposition), nothing false results. He has said ‘if it is posited that it is contingent that B holds of all C’ instead of ‘if it is posited and hypothesized that B holds of all C by necessity’. For ‘It is contingent that B holds of all C’, which is taken, 30 cannot be hypothesized as the opposite of ‘It is contingent that B holds of no C’, since the one proposition is the same as the other, not its opposite. But a universal affirmative necessary proposition, which he seems to have taken, is not the opposite of a contingent universal negative one either. Consequently, if one transforms the contingent proposition into a necessary universal one, as we have done, one would 35 not in this way be taking and hypothesizing the opposite, which is what one should do in reductio ad impossibile arguments. It seems then that he goes beyond what is required when he takes the universal affirmative necessary proposition as the opposite of the universal contingent negative one and shows that nothing impossible 228,1 is a consequence; for if nothing impossible follows when the minor premiss367 is posited as universal affirmative necessary, then so much the more will nothing impossible follow if it is particular affirmative necessary. For if the particular affirmative necessary proposition, which is the opposite of the hypothesized conclusion, is hypothesized it 5 is shown similarly that nothing impossible follows. For if B holds of all or some C by necessity, and it is also assumed that it is contingent that A holds of no B, the result is a combination in the first figure having a contingent negative major premiss and a necessary minor.368 The conclusion in such combinations is contingent. So it will follow that it is 10 contingent either that A holds of no C or that it does not hold of some. For if BC has been taken as necessary universal affirmative – as he hypothesizes – the conclusion will be universal negative contingent, ‘It is contingent that A holds of no C’, which is not impossible, since it was assumed that it is contingent that A holds of all C. And if BC is 15 particular negative necessary, the conclusion will again be particular negative contingent; and in this case too what follows is not impossible since it was assumed to be contingent that A holds of all C. But if it is contingent that A holds of all C, it is contingent that it holds of none or does not hold of some. 369 But if ‘It is not contingent that B holds of all C’ were transformed into ‘B does not hold of some C by necessity’ (and ‘It is not contingent

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that B holds of all C’ is also true370 when ‘B does not hold of some C by necessity’ is), the combination will also be non-syllogistic. For the minor in the first figure is negative particular necessary. And it is possible that he has said ‘if it is posited that it is contingent that B holds of all C’ with reference to the conclusion of the combination under consideration, the conclusion which must be proved by reductio ad impossibile – if it were possible to do so. (But then either (a) the text would be defective and take the affirmative instead of the negative proposition, since what should have been written is ‘if it is posited that it is contingent that B does not hold of any C’ since this would be the resulting conclusion; or (b) he has taken the affirmative as equivalent to the negative; and if it is posited that this is the resulting conclusion nothing impossible will be proved to follow from this by a reductio ad impossibile. (He says ‘false’ instead of ‘impossible’.)) But if he has taken ‘It is contingent that B holds of all C’ as the conclusion, he has not posited the method of reductio ad impossibile as it is understood. For if the opposite of ‘It is contingent that B holds of no C’, i.e., ‘B holds of some C by necessity’, is taken, it follows371 that it is contingent that A does not hold of some C’, which is not impossible, since it is also contingent that it holds of all of it (this was assumed) and that it is contingent that A holds of no C (because the contingent affirmative proposition converts). But if it is contingent that A holds of no C, it is clear that ‘It is contingent that A does not hold of some C’ is not impossible. 37a38372 In general, if there is a syllogism, it is clear that the conclusion will be contingent373 [because neither of the premisses has been taken as unqualified. And it would be either affirmative or privative. But it cannot be either. For if it is posited to be affirmative, it will be shown using terms that it is not contingent that B holds of C; and if it is taken to be privative it will be shown that the conclusion is not contingent but necessary. 37b3 For let A be white, B human, C horse. Then it is contingent that A, white, holds of all of one and of none of the other. But it is not contingent that B holds of C and not contingent that it does not hold. It is evident that it is not possible that it holds since no horse is human. But it is also not contingent that it does not hold; for it is necessary that no horse is human, and what is necessary is not contingent. Therefore there is no syllogism. 37b10 It will be proved similarly if the privative premiss is posited conversely, and if both premisses are taken affirmatively or privatively – for the proof will be through the same terms; and also when one is universal, one particular, or both are particular or indeterminate, or however else it is possible374 to transform the

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premisses: for the demonstration will always be through the same terms. It is evident, then, that there is no syllogism if both premisses are posited as contingent.]

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Having shown that it is not possible for the combination under consideration to be proved syllogistic by either conversion or reductio ad impossibile, he now also proves this same thing, going beyond what is required. For he says that it is clear that, in general, if the combination under consideration is syllogistic, the conclusion will be contingent because both premises are contingent. For it has been shown that things that follow by necessity syllogistically from premisses are similar to what they follow from: if the premisses are possible and contingent,375 so is what follows from them, and if the premisses are necessary so is what follows from them necessary. He obtained this result when he showed that an impossibility does not follow from a possibility.376 (It is necessary to understand the words ‘or necessary’ to be added to the phrase ‘because neither of the premisses has been taken as unqualified’.) So, if the combination under consideration is syllogistic, the conclusion will be contingent and either affirmative or negative. But it can’t be either; so there will not be a syllogism. The following considerations make it clear that the conclusion can be neither affirmative or negative: if we say that the conclusion is affirmative it is possible for it to be shown using certain terms that ‘It is not contingent that B holds of C’ follows, and this is not a contingent negative proposition equivalent to an affirmation, but the negation of a contingent proposition; but if we say that the conclusion is negative contingent, it is possible for it to be shown again using the same terms that the conclusion is indeed negative but necessary and not contingent, and the necessary negative conclusion which is proved will do away with both the contingent affirmative and the contingent negative conclusion. (37b3) He proves that this is how things are by taking the terms white for A, human for B, and horse for C. For it is contingent that white holds of no human and of every horse, and human holds of no horse by necessity; and if it holds of none by necessity, it is not contingent that it holds of horse and it is not contingent that it does not hold of horse. He shows this as follows. Just as the contingent affirmation is false because it is not possible that human holds of horse, so too the contingent negative proposition in these terms ‘It is contingent that human holds of no horse’ will be false because human does not hold of horse by necessity. It is not the case that it is contingent that human holds of no horse; rather it holds of none by necessity (37b10) He says that it will be proved in the same way that the combination in which the privative premiss has been taken conversely (the combination which has the major universal affirmative contingent,

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the minor universal negative contingent) is also non-syllogistic; for it cannot be proved to be syllogistic either by conversion or by reductio ad impossibile. And the refutation is through the same terms. And if both premisses are taken as universal affirmative, it will be proved through the same terms that a contingent conclusion cannot follow, whether the conclusion be universal affirmative or universal negative or particular. For if B is shown to hold of no C by necessity, it does away with all cases of contingency. And if both premisses are negative. And also if one premiss is universal and the other particular or both are particular, whether they are both affirmative or both negative or in alternation,377 the same terms when set down will establish that these combinations are non-syllogistic by verifying the premisses under consideration and the necessity of the conclusion. 378 However, one should not suppose that a universal negative necessary conclusion always follows in the combinations under consideration; nor should we ever think that these combinations are syllogistic and yield a conclusion which is not contingent but necessary negative. For it is possible to show in the case of other terms that B also holds of all C by necessity. Aristotle does not mention this on the grounds that he showed sufficiently that the combinations under consideration are not syllogistic because in the case of contingent premisses it is necessary that the conclusion be contingent, but it is found to be necessary negative universal. But the conclusion is sometimes necessary universal affirmative. The following are terms which show that B holds of all C by necessity. Let A be white, B human, C literate. For it is contingent that white holds of every human, and it is contingent that it holds of nothing literate, and human holds of everything literate by necessity. Similarly if it is taken to be contingent that moving holds of every animal and contingent that it holds of no human, or conversely. Terms for the contingent universal affirmative are white, moving, human. For it is contingent that white holds of all that moves and that it holds of no human, and it is contingent that moving holds of every human or of no human. In setting down only for the necessary negative conclusion he was not contented with a deficient number of terms; rather he on the grounds that it has been satisfactorily proved by setting down terms for a necessary conclusion that all combinations of two contingent premisses in the second figure are non-syllogistic.

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37b19379 If one premiss signifies380 unqualified holding and the other contingency, if the affirmative is posited as unqualified [and

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the privative as contingent, there will never be a syllogism, whether the terms are taken as universal or particular; the demonstration is the same and uses the same terms.]

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Having shown that there is no syllogism from two contingent premisses in the second figure, he turns to mixtures, and he first discusses mixtures of an unqualified and a contingent premiss. He says that there will be no syllogism if the unqualified premiss is affirmative and the contingent one negative, however the premisses are taken to be , it being evidently clear that nothing will follow syllogistically if both premisses are affirmative.381 382 He says that the demonstration that nothing follows syllogistically if the unqualified premiss is affirmative and the contingent one negative ‘is the same and uses the same terms’. A combination of this kind is shown to be non-syllogistic because neither a universal negative contingent proposition nor an unqualified affirmative universal one converts. (For if the latter is converted, the result is a particular affirmative major premiss in the first figure.) 383 Moreover, because nothing is proved by reductio ad impossibile either. For if the opposite of either a contingent conclusion or an unqualified one – if someone were to say that the conclusion can be unqualified – is hypothesized , it follows in the first figure that A holds of some C when it was assumed that it is contingent that A holds of no C, which is not impossible. 384 But by setting down the same terms. For if white, human, and horse are taken, they show that the conclusion is neither contingent (affirmative or negative) nor unqualified, but necessary negative; but if the conclusion was inferred syllogistically, it must be either contingent or unqualified since the premisses are of this kind. But it is also possible to provide terms for ‘B holds of all C by necessity’ and ‘B holds of no C by necessity’. He himself used white, human, horse for ‘B holds of no C by necessity’. For, as has been said before,385 one can take it that white holds of all of one of human and horse and that it is contingent that it holds of none of the other, and human holds of no horse by necessity. Terms for ‘B holds of all C by necessity’ are white, human, literate. For let it be contingent that white holds of no human and let it hold of everything literate; human holds of everything literate by necessity. The situation is the same if the extremes are taken to be animal and human. [37b23386 But there will be a syllogism when the affirmative premiss signifies contingency and the privative unqualified holding. For let it be taken that A holds of no B and that it is contingent

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that it holds of all C. If the privative premiss is converted, B will hold of no A. But it was contingent that A holds of all C. The result is a syllogism through the first figure that it is contingent that B holds of no C. The situation is the same if the privative is posited in relation to C.] If the contingent premiss is taken to be universal affirmative and the unqualified one universal negative, there will be a syllogism if the universal negative unqualified premiss is converted, since in this way there will be the first figure, and the conclusion will be that it is contingent that B holds of no C. This is so no matter which term the unqualified privative is posited in relation to; for he says that if the minor premiss is taken as universal negative unqualified and the major as universal affirmative contingent, there will be the same syllogism if the negative unqualified premiss is converted. Thus, let it be contingent that A holds of all B and let A hold of no C. If AC is converted, the result is that C holds of no A, and it is contingent that A holds of all B. Therefore, it is contingent that C holds of no B. But since the conclusion of this syllogism is contingent and a contingent negative proposition does not convert, how will there be an inference to the proposed conclusion? For it is proved through the premisses assumed that it is contingent that C holds of no B, but not that it is contingent that B holds of no C, because a contingent negative proposition does not convert. But it is necessary that B be predicated in the conclusion, since it is assumed as the major term. So what follows is not contingent in the way specified, but in the sense in which contingency is true of what holds.387 For it was proved388 that this is what follows if the major premiss is taken as negative unqualified in the first figure. If this is so, then the conclusion also converts since it is universal negative unqualified,389 but otherwise it doesn’t. He will also make it clear in what follows that the conclusion is of this kind and not contingent in the way specified.390 And he has already said at the beginning of his discussion of the second figure, ‘And in these cases too it is necessary to take the contingency in the conclusion as it was taken in the preceding.’391 392 a contingent negative proposition is equivalent to a contingent affirmative one and converts with respect to it, so that the contingent negative would convert in the same way as affirmative propositions. But particular affirmative propositions convert from universal affirmative ones. So if the conclusion CB is converted in this way the result will be that it is contingent that B does not hold of some C. 393 Someone might ask how anything follows syllogistically in combinations of this kind. For it is possible to show by setting down the same terms that neither a contingent negative conclusion nor an unqualified one follows. For if the premisses are related in this way there is an

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inference involving terms to both ‘B holds of no C by necessity’ and ‘B holds of all C by necessity’. For consider that white holds unqualifiedly of nothing human, and it is contingent that white holds of every horse, and human holds of no horse by necessity, but it is necessary that the conclusion be either contingent or unqualified. And it is also possible to take terms for ‘B holds of all C by necessity’. For again white holds of no human, and it is contingent that white holds of everything literate, and human holds of everything literate by necessity. Therefore there is an inference to a universal affirmative and a universal negative conclusion. But perhaps the proposition that white holds of no human is temporally restricted because it is assumed that it is contingent that white holds of everything literate. For when it is contingent that white already holds of everything literate, at that time it is false that white holds of no human.394 But its being proved that395 ‘B holds of no C by necessity’ is sufficient for rejecting the combination. But also if we take the premisses conversely – i.e., as ‘It is contingent that white holds of every human’ and ‘White holds of no horse’ – the premisses are again true, but the conclusion is necessary negative since human holds of no horse by necessity. For it is not possible to censure the universal negative unqualified premiss as false; for every negative unqualified universal proposition is of this kind, just as every affirmative one is. He himself makes it clear that this is so by using such universal unqualified propositions throughout. Consequently it seems that he posits these combinations as syllogistic by only paying attention to the conversion of the unqualified negative premiss and by not investigating them using terms. 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. 396 Furthermore, it is necessary to make use of the fact that the conclusion is not contingent in the way specified, as he showed in the case of mixtures in the first figure in which the major premiss is necessary universal negative and the minor is contingent. For he said that the negative conclusion will not be contingent in the way specified. 397And the conclusion in the case of a mixture of the present kind must also be of this kind if only the negative and not the affirmative conclusion follows because in their case the negative unqualified premiss is temporally restricted. Consequently, even if what follows is necessary negative, one can say that the conclusion of the syllogism is unqualified negative because not holding is also true when not holding by necessity is. For it is true that something doesn’t hold if it doesn’t hold by necessity. But this was also said before:398 that it is not possible

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for a conclusion to be necessary if neither of the premisses is necessary. And he said about such mixtures, ‘But there will not be a syllogism of 10 by necessity not holding’.399 Or did he say that there will not always be a syllogism of by necessity not holding? 37b29400 But if both premisses are privative and one signifies not holding, [the other contingency, nothing results through the premisses taken themselves. But if the contingent premiss is converted there is a syllogism that it is contingent that B holds of no C, just as in the preceding. For again there will be the first figure.]

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He says that if both of the premisses are privative there will be a syllogism if the contingent negative premiss is converted into a contingent affirmative one. And in the case of this combination the conclusion ‘It is contingent that B holds of no C’ will not be contingent in the way specified, as he proved. For conversion produces a negative unqualified major in the first figure.401 He himself also indicates this when he adds, 20 ‘just as in the preceding. For again there will be the first figure’. For with these words he indicates the conversion of the unqualified negative premiss and the quality of the conclusion. [37b35402 But if both premisses are posited as affirmative, there will not be a syllogism. Terms for holding: health, animal, human. Terms for not holding: health, horse, human.] He shows that in this figure there is no syllogism if both premisses, the unqualified one and the contingent one, are taken as universal affirm- 25 ative by setting down terms in the way that is customary for him to do. Terms for holding are health, animal, human. For let it be contingent that health holds of every animal, and let it hold of every human; and animal holds of every human by necessity. Terms for not holding are health, horse, human. For again horse holds of no human by necessity. 403 [37b39404 The situation will be the same in the case of the particular syllogisms. For when the affirmative premiss is unqualified whether it is taken as universal or particular, there will be no syllogism; this is proved in the same way and through the same terms as in the preceding. 38a3 But when the privative premiss is unqualified, there will be a syllogism by conversion, just as in the preceding.]

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30 The situation is the same in the case of particular syllogisms as it was in the case of the universal ones, where syllogistic combinations resulted when the unqualified premiss was taken as negative. For if the unqualified premiss is taken as negative universal, the contingent as particular, there will be particular syllogisms, whether the major or the minor is taken as unqualified. However, if the major is taken as unqualified, 35 what follows is known. But if the minor is taken as unqualified and it 234,1 is contingent that A holds of some B and A holds of no C, it is clear that C holds of no A. But it is contingent that A holds of some B. The result is that it is contingent that C does not hold of some B. If this proposition is contingent in the way specified, it will also be contingent that B does not hold of some C because both particular contingent propositions convert.405 But if it is not contingent in the way specified, as he showed 5 before,406 the combination will be syllogistic but the conclusion will not be the one proposed. For the minor term will remain predicated in the conclusion because a particular negative does not convert if it is either unqualified or necessary. However, he himself does not clearly say that the combination is also syllogistic when the minor is universal unqualified negative and the major is taken as contingent particular. For a 10 combination in the second figure having a particular major is not syllogistic at all. 407 But there is no syllogism if the unqualified premiss is affirmative. The proof is through the same terms – health, animal, human, and health, horse, human – since it is possible to take the universal contingent and the particular contingent using the same terms. Nor will 15 reductio ad impossibile go through in this case. [38a4408 Again, if both intervals are taken as privative and the not holding is universal, necessity will not result from the premisses themselves. But if the contingent premiss is converted just as in the preceding, there will be a syllogism.] But if both premisses are taken as negative and again the unqualified one is universal, there will be a syllogistic combination, not through the premisses assumed, but when, again, the contingent negative particular premiss is transformed into its equivalent, a particular affirmative contingent proposition. [38a8409 But if the privative premiss is unqualified and it is taken as particular, there will not be a syllogism, no matter whether the other premiss is affirmative or privative.] Nor will there be a syllogism if the unqualified particular premiss is taken as negative and the other contingent premiss is universal and

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either affirmative or negative, because it is clear that the particular negative unqualified premiss does not convert. [38a10 Nor will there be a syllogism when both premisses are taken as indeterminate – whether affirmative or negative – or as particular. The demonstration is the same and through the same terms.] And there will be no syllogism if both premisses are taken as particular or indeterminate, whatever they are in quality. And he says that this is proved through the same terms, that is, again, health, animal, human for holding and health, horse, human for not holding. For the premisses 25 are also true in the case of these terms when they are taken as particular. 1.19 Combinations with a necessary and a contingent premiss410

38a13411 If one premiss signifies necessity and the other contingency, [then if the privative premiss is necessary, there will be a syllogism of which the conclusion is not just that it is contingent that B does not hold of C but also that B does not hold of C; but there will not be a syllogism if the affirmative premiss is necessary.] He turns to the mixture of a necessary and a contingent premiss in the second figure and shows which combinations with this kind of mixture 30 are syllogistic. He says that there will be a syllogism when the contingent premiss is universal affirmative and the necessary premiss is taken as negative universal. The conclusion will be contingent, but not in the way specified, but rather it will be of not holding. He already proved this, because he has said earlier412 that in the first figure, if the negative 35 major premiss is either necessary or unqualified, the conclusion is not contingent in the way specified because when the necessary negative premiss in combinations of this kind in the second figure is converted the result is the first figure having the major premiss necessary nega- 235,1 tive. He sets down how this comes about: 38a16413 Let it, then,414 be assumed that A holds of no B by necessity, and that it is contingent that A holds of all C. [If the privative premiss is converted, B will also hold of no A; but it was contingent that A holds of all C. Again there results through the first figure a syllogism that it is contingent that B holds of no C. At the same time it is clear that B does not hold415 of any C. For let

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it be assumed to hold of C. Then if it is not contingent that A holds of any B, but B holds of some C, it is not contingent that A holds of some C, but it was assumed to be contingent that it holds of all.] 5 If the negative necessary premiss is converted, B holds of no A by necessity. But it is contingent that A holds of all C. The result is the first figure having as conclusion that it is contingent that B holds of no C, but this is not contingency in the way specified, as he indicates when he says ‘At the same time it is clear that B does not hold of any C’. And 10 he proves by reductio ad impossibile that the conclusion is universal negative unqualified and not contingent in the way specified. For if it is not true that B holds of no C, let the opposite of this be taken and let it be assumed that B holds of some C. But it was assumed that A holds of no B by necessity – for this is what the words ‘it is not contingent that A holds of any B’ signify.416 But B holds of some C. The result is that A 15 does not hold of some C by necessity, since he proved417 that in the first figure when the major is necessary and the minor unqualified the conclusion is necessary. Therefore, as he says, A does not hold of some C by necessity, which is impossible, since it was hypothesized that it is contingent that A holds of all C. Therefore the hypothesis from which this conclusion follows is impossible. Therefore its opposite . But this is ‘B holds of no C’. 418 It is necessary to understand that this proof and the impossibility which it infers cohere because he thinks that it is true that in the first figure the conclusion from a necessary major and an unqualified minor is necessary. A proof of this kind does not go through according to those who say that the conclusion of such mixtures is unqualified because nothing impossible follows. For what follows from the hypothesis which says that B holds of some C and from the necessary assumption ‘A holds of no B by necessity’ is ‘A does not hold of some C’, which is not at all impossible when it is assumed that it is contingent that A holds of all C; for nothing prevents it being at the same time true 30 that it is contingent that A holds of all C and that it does not hold of some C. 38a25419 This will be proved in the same way if420 the privative is posited in relation to C. According to Aristotle, if the minor is taken to be universal negative necessary and the major contingent universal affirmative, the conclu35 sion will be the proposed one because, according to him, the conclusion 236,1 is unqualified negative, so that it will also convert. For if it is contingent that A holds of all B and it holds of no C by necessity, C also holds of no A by necessity. But it is contingent that A holds of all B. Therefore it is contingent that C holds of no B – and now contingency is predicated of

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holding. Since, then, a universal negative unqualified proposition con- 5 verts, B also holds of no C; and this is what was required to be inferred. 421 But if the conclusion were not unqualified but contingent in the way specified, it would not be contingent that B holds of no C because a universal negative contingent proposition does not convert. So the proof will not be of the proposed conclusion; but, if there is a proof, a particular contingent would be inferred because, according to him, a 10 particular contingent proposition converts from a contingent universal negative one since it is affirmative. 422 However, according to those for whom the conclusion in the proof under consideration is contingent but not unqualified, the proposed universal contingent negative conclusion will follow because they maintain that a universal contingent negative proposition also converts with itself. [38a26423 Again let the affirmative premiss be necessary and the other be contingent, and let it be contingent that A holds of no B and let A hold of all C by necessity. There will be no syllogism when the terms are related in this way, since it results that424 B does not hold of C by necessity. For let A be white, B human, C swan. White holds of swan by necessity, and it is contingent that it holds of no human; and human holds of no swan by necessity. It is evident then that there is no syllogism with a contingent conclusion, since what is necessary is not contingent. 38a36 And there is no syllogism with a necessary conclusion either. For what is necessary results either from two necessary premisses or from a privative necessary premiss. 38a38 Moreover with these premisses assumed it is also possible that B holds of C. For nothing prevents C being under B and it being contingent that A holds of all B and that A holds of all C by necessity, for example, if C is being awake, B animal, A change. For change holds of what is awake by necessity, and it is contingent that it holds of every animal, and everything awake is an animal. 38b2 So it is evident that there is no syllogism that B does not hold of C either, since when the premisses are related in this way it is necessary that B holds of C. 38b3 Nor is there a syllogism of the opposite affirmations, so that there will be no syllogism.] He shows by setting down terms that there is no syllogism in this figure 15 when the necessary premiss is posited as affirmative and the contingent premiss as negative. He shows first using terms that in such a combination the conclusion is ‘B holds of no C by necessity’; he takes white, human, swan. For it is contingent that white holds of no human, it holds

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20 of every swan by necessity, and human holds of no swan by necessity. Having shown with the assumed terms a necessary negative conclusion in the case of such a combination, he says that the finding of a necessary conclusion in the case of certain terms is a sufficient indication that the conclusion is not contingent, and he says that the contingent is different from the necessary. (38a36) He next adds that the conclusion will not be necessary 25 negative either. (This is what he proved using the terms he set down.) For if things turned out this way in the case of all material terms, someone could say that the combination is syllogistic, and what follows is necessary, not contingent. He first takes it that this is not possible on the basis of what has already been proved and is assumed. For it is assumed that in the second figure the conclusion is 30 necessary if either both premisses are necessary or the negative one is;425 for if the negative premiss is necessary and universal, there results by conversion a necessary major premiss in the first figure. But he thought that when this is so, the conclusion is necessary if the minor is unqualified. But since the other premiss is contingent, he says that the 35 conclusion is not necessary at all. Because in the second figure the conclusion from one assumed necessary premiss is necessary only if the premiss is negative universal necessary, but in the combination under consideration the necessary premiss is not negative but affirmative, the conclusion is not necessary. (But in fact, as I said,426 the conclusion does not turn out to be necessary either because of the contingent premiss.) 237,1 (38a38) However, he also shows that the conclusion cannot be necessary negative by setting down terms. He shows that the conclusion is also ‘B holds of all C by necessity’, and he specifies the sort of material terms with respect to which this can come about. For if the last term C is under the major extreme B and it is contingent that the middle term 5 A holds of all and also of none of the major B, and it holds of all C by necessity, B will hold of all C by necessity. He takes as terms related in this way to one another change for A, animal for B, being awake for C. For it is contingent that change holds of every animal, and if this is so, it is also contingent that it holds of none. (For he is assuming a contingent negative premiss, but he takes an affirmative contingent as 10 being the same as it.) And changing holds of everything awake by necessity, at least if being awake is acting and changing in the sense of perceiving. And animal holds of everything awake by necessity. The proof would be clearer if walking were taken instead of being awake. For it is prima facie clearer that change holds of what walks by necessity than that it does of what is awake; and animal holds similarly of all that walks by necessity . 15 (38b2) And having shown that in the combination under consideration both ‘B holds of all C by necessity’ and ‘B holds of no C by necessity’

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follow, he says that it is evident that the conclusion will not be negative unqualified either; for it has been shown that what follows is affirmative necessary in the case of the terms set out. For it is not just the affirmative unqualified universal proposition which is destructive of the negative unqualified one; rather, the necessary affirmative one is even more destructive of it. At any rate both of the negative propositions 20 have been destroyed by the setting down of ‘B holds of all C by necessity’. 38b3427 Nor is there a syllogism of the opposite affirmations, [so that there will be no syllogism.] It is clear that he means by ‘opposites’ the opposites of the negative propositions, both the universal necessary one and the universal unqualified one. It was shown428 that a universal affirmative necessary proposition does away with these two negative propositions. A particular contingent affirmative proposition is the contradictory opposite of a necessary universal and a particular unqualified affirmative proposition is the contradictory opposite of an unqualified universal . So it is not possible to say that any of these follow. He has shown that neither of these negative propositions follows by using material terms to set down the universal affirmative necessary proposition, which does away with both of them. And he shows again that none of the affirmative propositions opposite to them, whether contradictory or contrary, follows by also inferring the universal negative necessary proposition, which does away with every affirmative proposition and which has been previously shown using material terms. Consequently none of the affirmative propositions which are opposite to these universal negative propositions, not only the contradictory opposites but also the contraries, follows. For again the necessary universal negative proposition does away with all affirmatives. But if there was going to be a syllogism it would have been necessary that one of these things which have been done away with follow. But none of these follow; so nothing follows syllogistically. It is also possible that ‘Nor is there a syllogism of the opposite affirmations’ is meant to be equivalent to ‘Nor will there be a syllogism of the opposite assertions’,429 so that he would be saying not affirmations but assertions, i.e., propositions – since it is also his custom to predicate the name ‘assertion’ of propositions.430 The propositions in question would be the contradictory opposites of the ones which have been proved, namely the universal negative necessary proposition and the universal affirmative necessary one. The contradictory opposites of these are particular contingent propositions, in one case an affirmative proposition and in the other a negative. For he means that nothing

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particular will follow because the universal negative necessary proposition does away with all affirmative propositions and again the universal affirmative proposition does away with all nega10 tive ones. So, if nothing universal or particular follows, nothing will follow at all. [38b4431 It will also be proved similarly if the affirmative premiss is posited conversely.]

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Similarly, if, conversely, the major is taken to be necessary affirmative and the minor to be contingent negative universal. For the refutation is through the same terms. For if white is assigned to A, swan to B, and human to C, white holds of every swan by necessity, it is contingent that it holds of no human, and swan holds of no human by necessity. But the terms which he set down for ‘B holds of all C by necessity’ when he took the minor as necessary affirmative do not show that B holds of all C by necessity if they are taken conversely. For it is not the case that if change holds of everything awake by necessity and it is contingent that it holds of no animal, thereby also being awake holds of every animal by necessity. But the universal negative is sufficient to reject the combination as non-syllogistic. For, as he said,432 a necessary proposition follows when both premisses are necessary or the major premiss is necessary negative. 433 I was perplexed about why these combinations having one premiss universal affirmative necessary and the other contingent universal negative are not syllogistic. For it seems possible for a contingent universal negative conclusion to be proved by reductio ad impossibile. For if it is assumed that A holds of all B by necessity and that it is contingent that it holds of no C, I say that it is contingent that B holds of no C. For if not, it is not contingent that it holds of no C, i.e., it holds of some C by necessity. But A also holds of all B by necessity. Therefore, A holds of some C by necessity, which is impossible, since it was assumed that it is contingent that A holds of none of it. It will be proved similarly if the minor is necessary and the major contingent. For it follows by reductio ad impossibile in the first figure that it is contingent that A does not hold of some C, when it is assumed to hold of all C by necessity. And it follows in the third figure that A holds of some B by necessity when it was assumed that it is contingent that it holds of none. If this is the way things are, then either reductio ad impossibile should be rejected as insufficient to show that a combination is syllogistic, or, if this cannot be rejected, it would seem that material terms are not sufficient to reject a combination as non-syllogistic. I have also said what the solution of this difficulty is in my book on mixtures.

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[38b6434 But if the premisses are similar in form, if they are privative there will always be a syllogism if the contingent premiss is converted, just as in the preceding. For let it be taken that A does not hold of B by necessity and that it is contingent that it does not hold of C. Then, if the premisses are converted,435 B holds of no A and it is contingent that A holds of all C. The result is the first figure. Likewise, too, if the privative is posited in relation to C.] He shows how there is a syllogism and what follows if both premisses are privative (and obviously universal – for this is missing from the text), the contingent negative premiss again being transformed into the affirmative, which converts with it. For the result of converting the necessary premiss is the first figure having the major premiss universal negative necessary and the minor universal affirmative contingent. It was proved436 that a combination of this kind is syllogistic and that here too the conclusion is not contingent in the way specified. The combination is syllogistic whichever extreme the necessary negative is posited in relation to, but if the minor is necessary, through three437 conversions, since it is also necessary to convert the conclusion; the conclusion does convert since it is not contingent in the way specified (as he thinks has been proved); rather it is universal negative unqualified. He says ‘if the premisses are converted’ with reference to the combination having a necessary negative major. For it is necessary that both premisses be converted, but not in the same way; the contingent negative premiss is converted into an affirmative one, but the universal negative necessary premiss is converted by terms into a negative necessary one. In this way the first figure results. But when the universal necessary negative premiss is the minor, there are three conversions: the two premisses are converted, as already described, and in addition the conclusion is converted. [38b13438 But if the premisses are posited as affirmative, there will not be a syllogism. For it is evident that there will not be a syllogism that B does not hold of C or does not hold by necessity, because a privative premiss, unqualified or necessary, has not been taken. But neither will there be a syllogism which concludes that it is contingent that B does not hold of C. For if the premisses are this way, B will not hold of C by necessity, for example if A is posited as white, B is swan, C human. Nor will there be syllogisms of the opposite affirmations since it has been shown that B does not hold of C by necessity. Therefore there is no syllogism at all.]

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20 if both premisses are taken affirmatively, ‘there will not be a syllogism’, and this is reasonable because nothing affirmative follows in the second figure at all. He says that there will not be a negative conclusion because no negative premiss, whether unqualified or necessary, is assumed. (For if an unqualified or necessary premiss is negative, the conclusion is negative; for a contingent and negative 25 premiss which is taken is equivalent to an affirmative.) Having said that there is not a negative conclusion, he says ‘But neither will there be a syllogism which concludes that it is contingent that B does not hold of C’, taking ‘It is contingent that B does not hold of C’ as either not negative or only negative in its verbal expression because it converts with the affirmative. He shows that there is not a syllogism with the conclusion that it is contingent that B does not hold of C by setting down terms for ‘B holds of no C by necessity’, terms which he has already 30 used.439 For white holds of every swan by necessity and it is contingent that it holds of every human, and swan holds of no human by necessity. He says that from the setting down of the terms for ‘B holds of no C by necessity’ it is also clear that in this combination nothing affirmative follows syllogistically. For ‘B holds of no C by necessity’ does away with every affirmative proposition. Having said that a negative proposition 35 does not follow because a negative premiss, whether unqualified or necessary, was not taken, and having shown using terms that ‘B holds of no C by necessity’ – which does away with the affirmations opposite to the negative propositions –, he has shown that nothing follows. (The words ‘Nor will there be syllogisms of the opposite affirmations’ are equivalent to ‘Nor will there be syllogisms of the affirmatives which are opposite to the negative propositions’.) 240,1 However, it is also possible to show by setting down terms for B holds of all C by necessity, that that a negative conclusion follows is done away with. For let it be contingent that change holds of every animal and let it hold of all that walks by necessity. The result is that animal holds of all that walks by necessity. 440 In the case of this combination too someone might ask why it 5 cannot be proved by reductio ad impossibile that what follows syllogistically is that it is contingent that B holds of no C. For if it is assumed that A holds of all B by necessity and that it is contingent that it holds of all C, one could say that something follows – ‘It will be contingent that B holds of no C’. For if not, it is not contingent that B holds of no C, i.e., B holds of some C by necessity. But also A holds of all B by necessity. Therefore, A holds of some C by necessity, which is impossi10 ble, since it was assumed that it is contingent that A holds of all C and also of no C.

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38b24441 The case will be similar with particular syllogisms.442 [For when the privative premiss is universal and necessary, there will always be a syllogism of which the conclusion is both contingent and unqualified; the demonstration is by conversion. 38b27443 But there will never be when the affirmative premiss is universal and necessary; for it will be shown in the same way as in the case of the universal syllogisms and with the same terms.] He has turned to particular syllogisms, and he shows that in their case too if the privative premiss is universal and necessary and the contingent premiss is particular there will be a syllogism with a particular conclusion, but otherwise there will not. And it is clear that in the second figure the combination of a necessary affirmative major and a contingent negative minor is not syllogistic, just as the combination of an unqualified affirmative major and a contingent negative minor is not (although the combination of a necessary and an unqualified premiss taken in this way is syllogistic444). He says that the conclusion will be both ‘It is contingent that B does not hold of C’ – it is necessary to understand this445 – and ‘B does not hold of C’ because it has been proved that in the first figure when the major premiss is negative necessary the conclusion is not contingent in the way specified. But when the negative necessary premiss is converted it becomes a necessary major premiss in the first figure. (38b27) He says that when the necessary premiss is affirmative, there will be no syllogism, just as it was shown446 that there is none when both premisses were universal and the necessary premiss was affirmative. For this was shown because ‘B holds of no C by necessity’ followed in the case of certain terms; he showed this in the case of white, human, and swan. And a necessary conclusion was not possible in the second figure unless either both premisses or just the privative one was necessary. And he also set down terms for ‘B holds of all C by necessity’ in the case of the other combination, namely change, animal, being awake. For animal holds of everything awake by necessity. However, it should be noted that this combination too can be proved by reductio ad impossibile to imply syllogistically that it is contingent that B does not hold of some C. For, if this is not so, the opposite , ‘It is not contingent that B does not hold of some C’, i.e., ‘B holds of all C by necessity’. But it was assumed that A holds of all B by necessity. Therefore A holds of all C by necessity, which is impossible since it was assumed that it is contingent that A holds of some C and that it does not hold of some C.447

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[38b29448 Nor will there be a syllogism when both premisses are taken as affirmative; for the demonstration of this is the same as before.] 241,1 And there will not be a syllogism when both premisses are affirmative, but one is universal, the other particular. For there wasn’t when both were universal and taken in this way. For it was shown using terms that when both premisses are affirmative what follows is negative necessary. The terms set down were white, human, swan. 5 Nevertheless, it is also possible to prove by reductio ad impossibile that this combination syllogistically implies that it is contingent that B does not hold of some C. For if not, it is not contingent that B does not hold of some C, i.e., B holds of all C by necessity. But it is assumed that AB is necessary. It follows that A holds of all C by necessity, but it was assumed that it is contingent that it holds of some and does not hold of some. 10

38b31449 But when both premisses are privative and the one signifying not holding is universal and necessary, [there will be no necessity through the premisses taken themselves, but if the contingent premiss is converted there will be a syllogism, just as in the preceding.]

He has proved that there will be no syllogism when both premisses are affirmative, whether the necessary or the contingent premiss is universal. He now discusses combinations of two negative premisses and says that there will be a syllogism if the necessary premiss is negative 15 universal. (This is what he means by ‘the one signifying not holding is universal and necessary’.) In a combination of this kind if the particular negative contingent premiss is transformed into a particular affirmative and the universal necessary negative premiss is converted, the result is the same as in the other cases in which the combination was syllogistic and the necessary premiss was negative universal. [38b35 If both premisses are posited as indeterminate or particular, there will not be a syllogism. The demonstration is the same and is through the same terms.] 20 He says that if both premisses are taken as indeterminate or particular, whatever quality they have, they will not be syllogistic. He says that the demonstration ‘is through the same terms’. He means it is through the same terms through which he proved in the first figure that combinations of two particular premisses, one necessary and one contingent, 25 are non-syllogistic.450 The terms for holding of all by necessity are white, animal, human. It is clear that white is the middle term in this figure.

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It was also the middle term in the first figure, but not in the same ordering. For in that case animal held of something white by necessity and white held contingently of some human. 451Now white holds of some animal, e.g., swan, by necessity and again does not hold of some animal, e.g., raven, by necessity. And again it is contingent that it holds of some 30 animals and that it does not hold of some animals, e.g., humans. And animal holds of everything human by necessity. 452 Terms for holding of none were white, animal, and soulless. Again white is the middle. Again it holds of some animals by necessity and does not hold of some by necessity, and it is contingent that it holds of some soulless things and contingent that it does not hold of some. And animal holds of nothing soulless by necessity. It is also possible that ‘through the same terms’ is said of the terms 35 he used a short while ago when the premisses were universal and affirmative.453 These were white, swan, human for holding of none by necessity, and, for holding of all by necessity, change, animal, being 242,1 awake. For if with these terms we do not take the premisses as universal as we did then, but as particular, the inference to both ‘B holds of all C by necessity’ and ‘B holds of no C by necessity’ can be proved with the premisses being true.

38b38454 It is evident from what has been said that if the privative premiss is posited as universal [necessary, there is always a syllogism which concludes not only that it is contingent that B does not hold of C, but also that B does not hold of C; but there is never a syllogism if the affirmative premiss is posited as universal necessary. And it is evident that there is or is not a syllogism in the case of necessary premisses and in the case of unqualified ones when the premisses are related in the same way. It is also clear that all the syllogisms are incomplete and that they are completed through the figures previously discussed.]

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It is assumed that a combination in the second figure is syllogistic if one premiss is necessary negative universal and the other is contingent and either universal (whether affirmative or negative) or again particular (whether affirmative or negative). And it has also been shown that the 10 conclusion, which is negative, is not contingent in the way specified. (This is what ‘not only that it is contingent that B does not hold of C, but also that B does not hold of C’ means; for it is thought that the conclusion is both because contingency is also predicated of what is unqualified.) And it has been shown that a combination is not syllogistic if the affirmative premiss is taken to be necessary. And it has also been shown that in mixtures of unqualified and 15

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contingent premisses and of necessary and contingent ones, if the necessary and the unqualified premiss are posited in the same way, the combinations are syllogistic; for when the unqualified premiss was universal negative, the necessary premiss was taken in the same way in syllogistic combinations; and when the necessary premiss is not taken as universal negative, the combinations again are non-syllogistic in the same way. 20 And it has also been shown that all the syllogisms from mixed premisses in the second figure are incomplete, since universally the syllogisms in the second figure are incomplete because they are completed by reduction to the first figure. He may have said ‘the figures previously discussed’ instead of ‘the 25 first figure’; for the completion of the syllogisms under consideration is through the first figure – and it is one figure; or he may have said ‘figures’ instead of ‘combinations’;455 for the syllogisms under consideration are completed through the combinations in the first figure through which he proved them to be syllogistic and to which they are reduced by conversion and thereby completed; and he might call them figures. (The syllogisms were proved by means of the second and fourth ).456 1.20-2 The third figure 1.20 Combinations with two contingent premisses

39a4457 In the last figure if one or both premisses are contingent, there will be a syllogism. [When the premisses signify contingency, the conclusion will also be contingent (and also when one signifies contingency and the other holding). 39a8 But when one premiss is posited as necessary, if it is affirmative, the conclusion will not be necessary or unqualified; but if it is privative there will be a syllogism with a conclusion of not holding, as in the preceding; and in these cases too the contingency in the conclusions should be taken similarly.] 30 He has turned to the third figure. He says that there is also a syllogism in this figure if both premisses are contingent. This was not the case in the second figure, since he showed458 that in the second figure nothing follows syllogistically from two contingent premisses. And he says that if both premisses are contingent, the conclusion will also be contingent, 35 and that the conclusion of a contingent and an unqualified premiss will also be contingent. (39a8) But if one premiss is necessary and one is contingent and the necessary premiss is universal negative, the conclusion will not be

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contingent in the way specified, but it will also be of not holding, as it 243,1 also was in the case of the previous figures. When he was discussing the first figure he said459 that if the unqualified premiss is negative universal, the conclusion will not be contingent in the way specified, but he did not mention this subsequently. Having said that the conclusion of 5 those combinations in which the necessary universal premiss is negative will be of not holding, he adds ‘and in these cases too the contingency in the conclusions should be taken similarly’. Clearly that one should call the conclusion contingent negative and 10 take contingency as holding and not in the way specified.

39a14460 First let the premisses be contingent, [and let it be contingent that A and also B hold of all C. Then since an affirmative proposition converts partially and it is contingent that B holds of all C, it would also be contingent that C holds of some B. So, if it is contingent that A holds of all C and that C holds of some B, it is necessary that it also be contingent that A holds of some B. For the first figure results. 39a19461 And if it is contingent that A holds of no C and that B holds of all C, it is necessary that it is contingent that A does not hold of some B. For again there will be the first figure by conversion. 39a23462 But if both premisses are posited as privative, there will be no necessity based on the premisses taken themselves, but there will be a syllogism if the premisses are converted, as in the preceding. For if it is contingent that A and also B do not hold of C, then if there is a transformation into ‘It is contingent that A and also B hold of C’, there will be again the first figure by conversion.] He speaks about the first combinations of two contingent premisses and first about combinations of two affirmative universal ones. He proves that the combination is syllogistic by conversion of the minor universal affirmative contingent premiss, reducing the combination to the first 15 figure. For the particular proposition converts with this. (39a19) But if the major premiss is taken as universal contingent negative, the minor as universal contingent affirmative, if the universal affirmative contingent minor is converted, the result is again the first figure in which a particular negative contingent conclusion follows from a universal negative contingent major and a particular affirmative 20 contingent minor. (39a23) But if both premisses are taken as negative and contingent universal, nothing will be proved syllogistically from the assumed premisses, but if the minor premiss is transformed into an affirmative

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contingent one and converted, the result will be the same combination 25 in the first figure as the one which has just been treated. And both premisses can be transformed into affirmatives, as he himself also says.

39a28463 But if one of the terms is universal and the other particular, [there will also be or not be a syllogism when the terms are related in the same way as in the case of unqualified premisses.] He says that when one of the premisses is particular, then, whichever combinations in the third figure when taken from unqualified premisses 30 made syllogistic combinations, exactly the same ones will be syllogistic in the case of contingent premisses when they are taken similarly; and the relation of the non-syllogistic combinations with contingent premisses to those with unqualified ones will be the same. For the premisses which are transformed from negative propositions into affirmative ones are transformed as being equivalent to the affirmative ones. [39a31464 For let it be contingent that A holds of all C and that B holds of some C. If the particular premiss is converted there will again be the first figure, since if A holds of all C and that C holds of some B, it is contingent that A holds of some B.] For if the major premiss is taken as universal affirmative contingent, 35 the minor as particular and also affirmative contingent, the combina244,1 tion is syllogistic; the proof comes when the particular contingent affirmative minor premiss is converted; for in this way the first figure again results. [39a35465 Likewise if universality is posited in relation to BC.] And if the major is particular affirmative contingent, the minor univer5 sal affirmative contingent, the combination is also syllogistic. For if the particular affirmative contingent major is converted, the result is again the first figure. But it will be necessary in the case of this combination to convert the conclusion as well, since in this way the syllogism will be of the proposed conclusion. [39a36466 Similarly if AC is privative and BC affirmative, since there will again be the first figure by conversion.] And if the major is taken as universal negative contingent, the minor

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as particular affirmative, the combination is syllogistic. For if the minor, which is particular contingent affirmative, is converted, the result is 10 again the first figure, having the major premiss universal negative contingent and the minor contingent particular affirmative. It was proved467 that the conclusion from these premisses is particular negative contingent. [39a38468 But if both premisses, one universal and one particular, are posited as privative, there will not be a syllogism based on the premisses taken themselves, but there will be if the premisses are converted, as in the preceding.] If both premisses are negative contingent, and one is universal, the other particular, if the particular negative premiss is transformed into a particular affirmative one and converted, the result is the previously mentioned combination in the first figure. For if the negative premiss is transformed into an affirmative one it will make the premisses of the same quality as the ones which were syllogistic in the case of unqualified premisses. So, when the minor premiss is taken as particular negative, we will need one conversion, and when the major is, we will need two, since it will be necessary to convert the conclusion, which is particular negative contingent; but a contingent negative proposition converts in the same way as an affirmative. This is not possible in the case of unqualified premisses in the third figure when the minor premiss is negative universal, the major particular negative, because a particular negative unqualified proposition does not convert, so that there is no syllogism of the proposed conclusion. Consequently someone might ask in what sense it is true that ‘if one of the terms is universal and the other particular, there will also be or not be a syllogism when the terms are related in the same way as in the case of unqualified premisses’. Perhaps it is necessary also in the case of contingent premisses to transform the contingent universal negative premiss into an affirmative. But, because of what has just been said, it will not be true that what holds with contingent premisses will also hold of unqualified ones. [39b2 But when both premisses are taken as indeterminate or particular, there will not be a syllogism. For it is necessary that A holds of all B and that it holds of none. Terms for holding: animal, human, white; for not holding: horse, human, white; the middle is white.] If both contingent premisses are taken as either particular or indeterminate, whether both are affirmative or negative or there is one of each kind, the combinations are non-syllogistic. And again he shows this by setting down terms and showing both holding of all and holding of none,

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35 holding of all with animal, human, and white; for it is contingent that each of animal and human holds of something white and it is contingent that each does not hold of something white – and it is contingent that one holds of white and it is contingent that one does not hold –, and animal holds of every human by necessity. Terms for not holding: horse, human, white. For the premisses hold in a similar way, but horse holds 40 of no human by necessity. 1.21 Combinations with an unqualified and a contingent premiss

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39b7469 If one premiss signifies holding and the other contingency, [the conclusion will be that it is contingent that A holds of B, but not that A holds of B; and there will be a syllogism when the terms are related in the same way as in the preceding.]

He has already said470 that in the case of syllogistic combinations of an unqualified and a contingent premiss in the third figure the conclusion 5 will be contingent in the way specified. So he repeats this here and also says that the combinations which have their terms assumed in the same way as those in syllogistic combinations of two unqualified or two contingent premisses will be syllogistic – for this is what is meant by ‘when the terms are related in the same way as in the preceding’.

[39b10471 For first let the premisses be affirmative, and let A hold of all C, and let it be contingent that B holds of all C. Then, if BC is converted, there will be the first figure, and the conclusion will be that it is contingent that A holds of some B, since in the first figure when one of the premisses signified contingency, the conclusion was also contingent.] 10 He first discusses the combination having the major universal affirmative unqualified and the minor universal affirmative contingent. If the universal affirmative contingent minor premiss is converted into a particular one the result is the first figure having the major unqualified affirmative and the minor particular contingent affirmative. The con15 clusion of this was proved to be particular affirmative contingent. He says ‘in the first figure when one of the premisses signified contingency, the conclusion was also contingent’ with reference to combinations having one premiss unqualified. But how could what has been said hold universally ? He did posit this in the case of affirmative premisses, but in the case of

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combinations of a negative unqualified major and a contingent affirm- 20 ative minor, he said the following: If one premiss is taken to be unqualified and the other contingent, when the premiss relating to the major extreme signifies contingency, all syllogisms will be complete and their conclusion will be contingent in the way specified which has been described; but when the premiss relating to the minor extreme signifies contingency, all of them are incomplete, and the conclusions of the privative syllogisms will not be contingent in the way specified but rather their conclusions will be that something holds of none by necessity or does not hold of all by necessity. For if something holds of none by necessity or does not hold of all by necessity, we say that it is contingent that it holds of none or not of all.472

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So perhaps what he says in the present passage concerns affirmative premisses . Or perhaps he was being extremely precise when he said that the conclusion, which he said was of none by necessity, is not contingent, since ‘of none by necessity’ is different from ‘by necessity of none’, as he showed then. Nevertheless, thereafter he also classifies a 30 negative proposition of this kind among contingent negative propositions because it is not directly unqualified. For in the case of a mixture in the second figure of a negative unqualified major and a contingent minor he already said473 that the conclusion is contingent in the way specified. [39b16474 Similarly too, if B holds of C and it is contingent that A holds of C, and if AC is privative and BC affirmative, whichever is unqualified, in both cases the conclusion will be contingent. For again the first figure results, and it has been shown that if in the first figure one premiss signifies contingency, the conclusion will also be contingent.] But if the major is taken as contingent universal affirmative, the minor 35 as unqualified universal affirmative, the combination is syllogistic. For the unqualified premiss can be converted. And if the major is taken as privative universal and either contingent or unqualified, and the minor is taken as affirmative universal and either contingent (if the major is 246,1 unqualified) or unqualified (if the major is contingent), the combinations are syllogistic. For if the minor premiss is converted there is a reduction to the first figure. 475 39b22476 But if the privative is posited in relation to the minor extreme, or if both premisses are taken as privative, [there will not

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be a syllogism through the assumed premisses themselves, but if they are converted, there will be, as in the preceding.] 246,4 Another syllogistic combination will be the one having the minor privative universal and either unqualified or contingent. 477For if the major premiss is converted there will be the first figure, and the conclusion is contingent particular negative with the minor extreme as predicate. This conclusion converts since it is contingent – as he now says; and if it is converted, the proposed conclusion would be proved to follow. For again it was indicated that,478 however the unqualified premiss is taken, 10 whether as affirmative or negative, the conclusion is contingent. 479 He himself says without qualification that if the minor premiss is taken as universal negative there will not be through the premisses assumed, but if the privative premiss is 15 transformed into an affirmative one. But it is necessary to understand the word ‘contingent’ as added when the minor premiss is privative. For if the minor negative is contingent it will be possible for the negative premiss to be transformed into an affirmative one. But if the minor premiss is unqualified negative, it is no longer possible for there to be a transformation, nor is there a syllogistic combination proving the proposed conclusion, unless we keep the minor premiss BC negative 20 unqualified and convert the contingent affirmative AC. For then it will be the case that B holds of no C and it is contingent that C holds of some A, and the conclusion will be that it is contingent that B does not hold of some A. And if this conclusion is, as he now says, contingent in the way specified, it converts and it is contingent that A does not hold of some B. So it will also follow that this is the conclusion even if no 25 transformation occurs. 480 Again, if both premisses are universal negative and one is unqualified and the other contingent, nothing will follow from the premisses assumed, but if the contingent negative premiss is transformed into an affirmative and converted, the combination is syllogistic. If the minor is contingent negative, the transformation and the proof are evident. 30 But if the major is contingent, the combination will not be syllogistic when the negative minor is unqualified universal. Or it will be necessary to transform the major, since it is contingent negative, into an affirmative contingent proposition and convert it and make it the minor, and then again also convert the conclusion itself, if it is contin35 gent in the way specified, as he now says.

[39b26481 If one of the premisses is universal and the other particular, then, if both are affirmative or the universal is privative and the particular affirmative, the situation concerning syllogisms

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will be the same. For all will yield a conclusion through the first figure, so that it is evident that the syllogism will be of contingency and not of holding.] ‘If one of the premisses is universal and the other particular’, whether both are affirmative or the particular premiss is affirmative and the universal premiss is negative, then, no matter which of them is unquali- 247,1 fied, the combination is syllogistic. For if the affirmative premiss is particular, whether it is contingent or unqualified, if it is converted, the result is a syllogistic combination through the first figure. The words ‘the situation concerning syllogisms will be the same’ indicate that the proof will be by conversion. It is again482 indicated that the conclusion 5 will be contingent on the grounds that mixture of an unqualified and a contingent premiss in the first figure the conclusion is always of this kind. [39b31483 But if the affirmative premiss is universal and the privative particular, the demonstration will be by means of the impossible. For let B hold of all C and let it be contingent that A does not hold of some C. Then it is necessary that it is contingent that A does not hold of some B. For if A holds of all B by necessity, but B is assumed to hold of all C, A will hold of all C by necessity. For this was shown before. But it was hypothesized that it is contingent that A does not hold of some C.] He says that if the minor is affirmative universal and the major negative particular, the combination will be syllogistic. But it will not be proved by a conversion in which it is reduced to the first figure but by reductio ad impossibile. For, let it be assumed that B holds of all C, and let it be contingent that A does not hold of some C. Then it will be contingent that A does not hold of some B. For if not, the opposite and A holds of all B by necessity. But B also holds of all C. Therefore, A will hold of all C by necessity. For the result is a combination of a necessary universal affirmative major and an unqualified universal affirmative minor in the first figure, and he proved484 that the conclusion of this is universal affirmative necessary. But it is impossible that A holds of all C by necessity, since it was assumed that it is contingent that it does not hold of some C. 485 Someone might inquire why according to him it is not also possible to prove a combination of this kind syllogistic by conversion, at least if a particular negative contingent proposition converts with itself. For if it is contingent that A does not hold of some C, it will also be contingent that C does not hold of some A. Rather, transforming the negative particular contingent premiss into an affirmative (for in this way the combination will be syllogistic)

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it is possible to convert it, and it will be contingent that C holds of some A. But it is assumed that B holds of all C. The result is that it is contingent that B holds of some A, and if so, it is also contingent that A holds of some B; for a contingent particular proposition, affirmative or negative, converts. And this proof was thought more acceptable. For the reductio ad impossibile is objectionable.486 487 However, this can work if the major is particular contingent. But if this major were unqualified particular negative, the minor affirmative contingent, there would be no place for a proof by conversion, if these premisses are taken. Nor is anything proved by reductio ad impossibile because, according to him too, in the first figure a universal affirmative necessary conclusion does not result from a necessary universal affirmative major and a contingent universal affirmative minor. The conclusion is not of this kind, but it is contingent universal affirmative, and there is nothing impossible in A not holding of some C and it being contingent that A holds of all C. 488 The proof by impossibility in the case of the combination under consideration goes through on his view because, according to him, the conclusion of a necessary major and an unqualified minor is necessary. But nothing impossible will follow according to his associates489 who say that the conclusion in such cases is unqualified, since it is not impossible that A holds of all C if it is contingent that it does not hold of some C. This is the combination which some people use to try to show that the conclusion of the combination of a necessary universal major and a unqualified minor in the first figure is necessary. For those who carry out a proof by reductio ad impossibile for this combination try to show that what follows is impossible.490 As I said,491 if the major AC is unqualified particular negative, the minor universal affirmative contingent, then, even according to him, no impossibility will be encountered. For if someone were not to agree that it is contingent that A does not hold of some B, the hypothesis would be that A holds of all B by necessity. But it is assumed that it is contingent that B holds of all C. According to him the conclusion of this mixture or combination is contingent; but if one takes it that it is contingent that A does not hold of C or that it is contingent that A does hold of C, nothing impossible results when it is assumed that A does not hold of some C. Therefore, a combination of this kind is absolutely non-syllogistic. 492 Theophrastus does not prove the combination previously described493 by a simple reductio ad impossibile. Rather he first transforms ‘It is contingent that A does not hold of some C’ into ‘A does not hold of some C’ – which is not impossible – and produces two unqualified premisses, a particular negative one resulting from the transformation and the assumed universal affirmative. He says that the conclusion will be that it is contingent that A does not hold of some

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B. For if not, the opposite , ‘ A holds of all B by necessity’. 25 Then, by reductio ad impossibile, he finds that something impossible is a consequence, since it follows that A holds of all C when it was assumed not to hold of some. But he has not encountered the impossibility because of the hypothesis – for it was not impossible for the hypothesis to be taken –, but from positing that A holds of all B by necessity. Therefore, the opposite of ‘A holds of all B by necessity’ , i.e., ‘it is contingent that A does not hold of 30 some B’. [40a1 When both premisses are taken as indeterminate or particular, there will not be a syllogism. The demonstration is the same as in the universals494 and by means of the same terms.] If both premisses are taken as indeterminate or both particular, whether both are affirmative or both negative or there is one of each kind, there will not be a syllogistic combination. He shows that this is so through terms. He says that this will be shown ‘by means of the same terms’ as in the case of the universals, and now he means the consisting of two contingent particular premisses. It is as if he 35 said ‘by means of the terms with which the wholly contingent combinations, that is the contingent ones consisting of two particular premisses, were shown to be non-syllogistic.’ The terms for holding of all were animal, human, white, and for holding of none horse, human, white; and white is the middle. But perhaps the text is wrong, and ‘The demonstration is the same 249,1 as in the universals’ has been written instead of ‘The demonstration is the same as in the case of two contingent premisses.’495 1.22 Combinations with a necessary and a contingent premiss

40a4 If one of the premisses is necessary, the other contingent, [then, if the terms are affirmative, there will always be a syllogism with a contingent conclusion; but when one is affirmative and the other privative, if the affirmative is necessary, there will be a syllogism that it is contingent that A does not hold of B, but if the privative is necessary, there will be a syllogism that it is contingent that A does not hold of B and that it does not hold.]497

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10 that A does not hold of B’, and that if the negative is necessary, it will be ‘that it is contingent that A does not hold of B and that it does not hold’, which is equivalent to saying that the conclusion will not be contingent in the way specified but will involve the contingency which is predicated of what holds. [40a9 And there will not be a syllogism that A does not hold of B by necessity, just as there wasn’t in the other figures.] But the conclusion will not be ‘that A does not hold of B by necessity’; 15 for in the other figures, both in the first and in the second, a necessary negative conclusion did not result from a necessary negative and a contingent affirmative premiss. 498 However, in both the first and the second figure, if the major premiss is necessary universal negative and the minor is contingent, it is possible to prove by reductio ad impossibile that the conclusion is 20 universal necessary negative. For let it be assumed that A holds of no B by necessity, and let it be contingent that B holds of all C. I say that A holds of no C by necessity. For, if not, it is contingent that it holds of some. But it is also contingent that B holds of all C. It follows in the third figure that it is contingent that A holds of some B. But this is impossible, since it was assumed that it holds of no B by necessity. Therefore the hypothesis that it is contingent that A holds of some C is false. Therefore its opposite ‘A holds of no C by necessity’ . 25 Again, in the second figure, let499 A hold of no B by necessity and let it be contingent that A holds of all C. I say that B holds of no C by necessity. For if it is contingent that it holds of some and it is contingent that A holds of all C, it will follow in the third figure again that it is contingent that A holds of some B, when it was assumed that it holds 30 of none by necessity. So, since the conclusion is impossible, the hypothesis that it is contingent that B holds of some C will be destroyed and its opposite ‘B holds of no C by necessity’ posited. It is worth asking why he says that in a mixture of a necessary negative universal premiss and a contingent affirmative one there is no necessary negative conclusion that X holds of no Y by necessity in any 35 of the figures. For either (i) it is necessary that reductio ad impossibile be rejected; or (ii) the combinations in the third figure through which I produced the reductio ad impossibile must be non-syllogistic; or (iii) it follows that X holds of no Y by necessity. As I have said already,500 I have investigated this and spoken about it at greater length 250,1 in On the disagreement of Aristotle and his associates concerning mixtures. And I have discussed it at greater length in my notes on logic.

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[40a12501 First let the terms be affirmative, and let A hold of all C by necessity, and let it be contingent that B holds of all C. Then, since it is necessary that A holds of all C, but it is contingent that C holds of some B, it will also be contingent that A holds of some B, but there will not be an unqualified conclusion, since this is the way it turned out in the case of the first figure.] If both premisses are affirmative universal and the major is necessary, if the minor is converted and becomes particular affirmative, the result 5 is a combination in the first figure having a contingent particular affirmative conclusion – since the major is universal affirmative necessary, and just like the others in which the major was affirmative unqualified and the minor contingent. [40a16502 The proof will be similar if BC is posited as necessary and AC as contingent.] But also, if the minor is necessary universal affirmative and the major contingent universal and affirmative, if, again, the necessary universal 10 affirmative premiss is converted, the result is a combination in the first figure yielding a contingent particular affirmative conclusion, since the major is contingent universal affirmative. This syllogism in the first figure was complete, as were the others having a contingent major. [40a18503 Again, let one premiss be affirmative, one privative, and let the affirmative be necessary. And let it be contingent that A holds of no C and let B hold of all C by necessity. Again there will be the first figure; for the privative premiss signifies contingency. So it is evident that the conclusion will be contingent, since when the premisses were related this way in the first figure, the conclusion was also contingent.] If one of the premisses is taken as negative, the other as affirmative, and the minor is affirmative necessary universal, the major negative 15 contingent universal, if the universal affirmative necessary premiss is converted, there will be a particular negative contingent conclusion in the first figure, since the major is contingent universal negative, and this was a complete syllogism in the first figure. He says ‘Again there will be the first figure’ and adds ‘for the 20 privative premiss signifies contingency’, which shows that the negative conclusion is contingent in the way specified. For the conclusion is of not holding rather than being contingent in the way specified only when the major premiss is privative and necessary.504 The text here is in a way rather incongruous. A congruous formulation of it would be ‘And if 25

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the privative premiss signifies contingency, it is evident that the conclusion will be contingent’.505 506 There can also be a syllogism if we keep the universal affirmative necessary premiss fixed and convert the major, which is contingent universal negative, to a particular proposition and transform it to an 30 affirmative. But the proof is more cumbersome both because of the transformation of the contingent premiss to an affirmative and because it is also necessary to convert the conclusion. And in this way the conclusion becomes particular contingent affirmative, not negative. [40a25507 But if the privative premiss is necessary, the conclusion will be both that it is contingent that A does not hold of some B and that A does not hold of some B. For let it be assumed that A does not hold of C by necessity and that it is contingent that B holds of all C. Then, if the affirmative BC is converted, there will be the first figure, and the privative premiss will be necessary. But when the premisses were related in this way, it resulted both that it is contingent that A does not hold of some C508 and that A does not hold of some C, so that also it is necessary that A does not hold of some B.] Furthermore if the major is privative and necessary, the minor contingent affirmative universal, then, if the contingent premiss is converted, 35 there results a combination in the first figure having the major universal negative necessary and the minor contingent particular affirmative, and a conclusion that A does not hold of some B, which is in this sense contingent particular; for it is not contingent in the way specified. (The words ‘so that also it is necessary that A does not hold of some B’ do not 251,1 say that the conclusion is necessary negative particular; rather they are equivalent to ‘it is necessary that the conclusion be particular negative unqualified’.) [40a33509 But when the privative is posited in relation to the minor extreme, if it is contingent, there will be a syllogism if the premiss is transformed, as in the preceding. If the minor premiss is negative, the major affirmative, then, if the negative premiss is the minor and is contingent, there will not be a 5 syllogism from the assumed premisses; but if the contingent universal negative is transformed into a universal affirmative contingent and it is converted, there will again be the first figure having a particular contingent affirmative conclusion – the major premiss will be universal affirmative necessary, the minor particular affirmative contingent. For 10 nothing will follow if the minor premiss remains negative.510

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[40a35511 But if it is necessary, there will not be a syllogism. For it is necessary that it holds of all and not contingent that it holds of any.512 Terms for holding of all: being asleep, sleeping-horse, human; for holding of none: being asleep, horse-that-is-awake, human.] But if the minor is universal negative necessary, there will not be a syllogistic combination. Again he shows this by setting down terms and showing that the major or first term can hold of all or of none of the last . Terms for holding of all are being asleep, sleeping-horse, human, since it is contingent that sleep holds of every human, sleepinghorse holds of no human by necessity, and being asleep holds of every sleeping-horse by necessity. Terms for holding of none are being asleep, horse-that-is-awake, human, since, again, it is contingent that being asleep holds of every human, horse-that-is-awake holds of no human by necessity, and being asleep holds of no horse-that-is-awake . It is necessary to understand that the conclusions which have been proved here are not necessary without qualification, but on the condition ‘which are in fact so and so’. Things of this sort do away with there being a necessary or unqualified conclusion. For ‘X holds of all Y’ does away with necessary and unqualified negative propositions; similarly again ‘X holds of no Y’ does away with all necessary and unqualified affirmative propositions. However, unqualified propositions do not do away with opposite513 contingent ones, since, if X holds of all Y, it is contingent that it does 514 hold of some or holds of none, and if X holds of no Y, it is contingent that it holds of all or of some of it. 515 And the combination in the third figure having the minor universal negative necessary and the major universal affirmative contingent is syllogistic. For, if the contingent premiss is converted, the result is that B holds of no C by necessity, and it is contingent that C holds of some A. Therefore, B does not hold of some A. For this conclusion was thought516 to result when the necessary premiss was negative. But it is not possible for the proposed conclusion to be inferred – since it is necessary that A be predicated of B. And this is not proved for the mixture under consideration because a particular negative unqualified proposition does not convert. However, if not ‘B does not hold of some A’ but ‘It is contingent that B does 517 hold of some A’ were to follow in the case of the mixture under consideration, a syllogism would seem to result because a particular contingent negative proposition converts. Again, the reason that this combination seems to be non-syllogistic is that the minor in the first figure remains negative.518 519However, it is not true that the minor becomes necessary negative if both premisses do not convert; for BC becomes the major if one does not convert both premisses.

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520 [40a39 The situation will be similar if one of the terms is universal, the other particular, in relation to the middle. For if both are affirmative, there will be a syllogism with a contingent but not an unqualified conclusion. 40b2 And also when one premiss is taken as privative, the other as affirmative, and the affirmative one is necessary. 40b3 But when the privative premiss is necessary the conclusion will also be of not holding. 40b4 For the method of proof will be the same whether the terms are universal or not universal, since it is necessary that the syllogisms be completed through the first figure, so that it necessarily turns out in these cases as it did in those. 40b8 But when the privative universal premiss is posited in relation to the minor extreme, if it is contingent, there will be a syllogism by conversion, but if it is necessary, there will not be one. This will be proved in the same way as with the universal syllogisms and through the same terms.]

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He says that if one of the premisses is universal and one particular and the combination is in the third figure, the syllogisms will be in a similar way to what they were when both premisses were universal. (This is what ‘if one of the terms is universal, the other particular, in relation to the middle’521 means, since he speaks about premisses by mentioning their terms. At the same time he also gives a description of the third figure, in which two terms are predicated of one middle, just as he again described the first, in which there are again three terms of which two are predicated but not of the same thing, but as he said concerning the first figure, ‘when one of the terms is universal and the other is particular in relation to the third’.522 For when both terms are not predicated of the same thing  , but is predicated of a third term, the combination belongs to the first figure.523) For in those cases too524 when the minor premiss was necessary negative universal, no resulted, but there was a syllogistic combination when the minor was contingent negative.525 And similarly when the minor was necessary affirmative.526 For we converted the universal affirmative and made it particular and reduced the combination to the first figure. Similarly, too, in the present cases,527 if we convert the particular premiss, we will produce the same combination in the first figure as when there were two universal premisses. 528 Turning to the other combinations, in the case of the one having the major particular negative contingent and the minor universal affirmative necessary, by reductio ad

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impossibile, as in the case in which the minor premiss was unqualified universal affirmative and the major particular negative contingent.529 For, when he says ‘The situation will be similar’, he is not referring to the proofs but to the fact that the conclusion is contingent in the same sense as it was when both premisses were universal. It is also possible for it to be proved that what follows is contingent particular affirmative if the particular negative contingent premiss is transformed into an affirmative one and converted – but it is also necessary to convert the conclusion. So this kind of combination is not always by reductio ad impossibile. 530 If both premisses are affirmative, the conclusion will be that it is contingent that A holds of some B but not that it does hold of some. It does not matter which premiss is taken as universal, except that if the major is taken as particular, it will be necessary to convert both it and the conclusion, since in this way the conclusion will be the one proposed. 531 (40b2) But if one premiss is taken as negative and the necessary premiss is affirmative, the conclusion is contingent particular negative; †and the necessary premiss must be particular if it is affirmative. For if it were universal, it would be necessary† but it is necessary to transform the particular contingent negative premiss into an affirmative one, and in this way the conclusion will also be affirmative. But it is possible to prove by reductio ad impossibile that the conclusion is contingent particular negative if the minor premiss is taken as universal affirmative necessary. 532 (40b3) If the negative premiss is universal and necessary and it is the major – for then the combination is syllogistic – the conclusion will be that A does not hold of some B and not that it is contingent that A does not hold of some B. This will be proved in the same way as it was proved when both premisses were universal and the negative one was necessary.533 He says that in a combination of a universal negative necessary premiss and a particular affirmative contingent one in the third figure the conclusion will be that A does not hold of some B. And he sets down in a general way the reason why the conclusions are of this sort when he adds ‘since it is necessary that the syllogisms be completed through the first figure, so that it necessarily turns out in these cases as it did in those’, that is, ‘in the case of combinations of this sort, whatever the nature of the conclusions in that figure so they must be in this figure’. 534 Someone might appropriately ask why he did not say that in mixtures of an unqualified and a contingent premiss in this figure are of the same kind they were proved to be in the first figure. Perhaps what he says now is universal, and should be followed: as the conclusion is in the first figure so will it be in the other figures in cases which are proved by conversion to involve the same kind of combination. 535 (40b8) If the minor premiss is privative universal, then, if it is

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contingent and the particular affirmative necessary premiss is converted, the combination will be syllogistic and the conclusion will be that it is contingent that B does not hold of some A, and it will again be also necessary to convert the conclusion. (A particular contingent negative proposition which is contingent in the way specified converts.) But if the minor is privative universal and necessary, the combination will not be syllogistic. This will be shown using the same terms as were taken a short while ago536 when both premisses were universal and the minor was necessary negative. The reason is again that, when the major premiss, which is contingent particular affirmative, is converted, the conclusion is unqualified particular negative with the minor extreme as predicate, but a particular negative unqualified proposition does not convert. The terms for holding were being asleep, sleepinghorse, human, and for not holding being asleep, horse-that-is-awake, human. For it was then taken that it is contingent that being asleep holds of every human, and now it will be taken that it is contingent that it holds of some. Nor is the combination syllogistic when the minor premiss is particular necessary negative, as the same terms when set down will show.537 One will take being awake, sleeping-horse, human. For it is contingent that being awake holds of every human, sleeping-horse does not hold of some human by necessity, and being awake holds of nothing which is sleeping by necessity.538 And again, being awake, horse-that-is-awake, human. For it is contingent that being awake holds of every human, and horse-that-is-awake holds of no human by necessity,539 and being awake holds of every horse-that-is-awake by necessity. He does not mention the other non-syllogistic combinations, those with two particular or indeterminate premisses, whether the two are affirmative or negative or dissimilar in form; for it is already known because of what has been said that these are non-syllogistic.

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This is also what was said in the case of the second figure: ‘they are completed through the figures previously discussed’.540 He said ‘figures’ instead of ‘through the figure previously discussed’. 15 And it is clear that the syllogisms in this figure are also incomplete and are completed through the first figure, since some were shown to yield a conclusion by conversion of premisses and others – those in

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which the minor premiss is universal affirmative and either unqualified or necessary and the major is particular negative contingent541 – to yield one by reductio ad impossibile. He mentioned reductio proof only in the case of the mixture of a contingent and an unqualified premiss;542 and only in the case of a mixture of a necessary and a contingent premiss did he say that the conclusion is contingent in the way specified.543 The words ‘and also when one premiss is taken as privative, the other as affirmative, and the affirmative one is necessary’544 did not set down the proof. For the combination of a universal affirmative necessary minor and a particular negative contingent major in the third figure is proved to be syllogistic by both reductio ad impossibile and conversion. For if we transform the particular contingent negative AC into an affirmative and we convert it, it will follow that it is contingent that B holds of some A; and the previously mentioned conclusion545 will be proved if the conclusion is converted. But what is proved by reductio ad impossibile is particular negative contingent, whereas what is proved by conversion is particular affirmative contingent. Consequently, he does not use the proof by conversion in connection with a combination of this kind because doing so does not keep the conclusion negative when a negative premiss is assumed. 546And in this way, if the minor premiss is necessary universal affirmative, the major contingent particular negative, both propositions are converted .

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Notes 1. As indicated in section III.E.1 of the introduction, there are strong grounds for holding that no combination of contingent premisses is syllogistic. Aristotle recognizes eight CCC syllogisms, four complete analogues of the first-figure UUU syllogisms and four which are reducible to them by EA- or OI-transformationc. 2. Aristotle announces the completeness of Barbara1(CCC) and then of Celarent1(CCC). 3. On the text here see 167,27 with note. 4. On the text here see the note on 167,25. 5. At the end of the previous chapter. Aristotle’s formulation here is perfectly compatible with taking the second premiss to be CON(BaC). 6. Alexander here quotes his paraphrase of 1.1, 24b29-30. See 24,27-30. However, it seems clear that Aristotle is not referring to the definition of ‘said of all’ but to the definition of ‘it is contingent that A holds of all B’ which he has more or less given at 1.13, 32b25-32. Alexander’s reading is presumably due to his desire to make completeness always turn on the so-called dictum de omni et nullo, but he undoubtedly thinks that what Aristotle said at the end of the previous chapter is just a development of the dictum. 7. cf. 25,13-23. 8. Alexander’s difficulty in this paragraph is due to the fact that at 33a5 his text says ‘A holds’ where most manuscripts say ‘A does not hold.’ Hence in his text Aristotle is citing an account of CON(AaB) when one of CON(AeB) seems to be required. 9. The mss of Alexander have legomen where the main ones of Aristotle have elegomen. 10. Barbara1(CCC). 11. Aristotle now uses EE-conversions to establish AEA1(CCC) and EEA1(CCC). At 168,21-4 Alexander supplies a justification of EEE1(CCC), perhaps because he wants to preserve the idea that negative premisses do not imply an affirmative conclusion. 12. 1.13, 32a29-b1. 13. Alexander supplies a justification of EEE1(CCC); cf. his remark at 168,2830. 14. 1.13, 32b18-22. One should read Alexander’s comments on that passage in connection with what he says here. 15. sumplokê; cf. 164,27. 16. And does not consider the result of transforming negative premisses into affirmative ones. 17. Aristotle asserts that all four first-figure combinations with a universal major and a particular minor are syllogistic. 18. For a minor difference between the lemma and our text of Aristotle see Appendix 6 (on 33a21). 19. Aristotle affirms Darii1(CCC). 20. Aristotle affirms Ferio1(CCC).

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21. Aristotle establishes AOI1(CCC), using OI-transformationc. He does not specifically mention EO_1(CCC), which could be reduced to either Darii1(CCC) or Ferio1(CCC). 22. See 167,17-18 with note. Alexander first proposes interpreting Aristotle’s reference to the definition of CON as a reference to the dictum de omni et nullo. 23. Alexander considers as an alternative interpretation that Aristotle is referring to the diorismos of contingency. He argues that the conclusion of Darii1(CCC), CON(AiC) satisfies the diorismos because nothing impossible results if CON(AiC) is assumed along with the premisses. One would have expected him to argue (as he could have) that nothing impossible results if AiC is assumed along with the premisses. The argument is a version of what we called the incompatibility acceptance method in connection with Aristotle’s attempt to show that certain N+U combinations yield an unqualified and not a necessary conclusion; see section II.C of the introduction. But, as is pointed out in the note on 130,9, by itself that method cannot establish that a pair of premisses yields a conclusion. 24. Alexander says (correctly) that nothing impossible results when the ‘opposite’ of the conclusion of Darii1(CCC), ‘i.e.’, NEC(AeC), is conjoined with its premisses. It is not clear whether what he says is intended to be a third attempt to interpret Aristotle’s reference to the definition of contingency or a development of the second. See the note on 170,14. 25. Camestres2(CNCu). There will, of course, be no contradiction even if the conclusion is taken to be BeC rather than CON(BeC). 26. Alexander has given three interpretations of the words ‘This is evident from the definition of “It is contingent” ’. The consideration he now adduces shows a flaw in the third, but gives no grounds for preferring the first or the second. 27. Alexander takes for granted that Aristotle acknowledges EOO1(CCC) as well as AOI1(CCC). 28. Aristotle rejects all cases in which at most the minor premiss is universal. 29. Alexander shows that Aristotle’s general remark about the remaining cases is true by showing that it is true when the major premiss is CON(AiB) and the minor is either CON(BaC) or (equivalently) CON(BeC). The case is sufficient since CON(AiB) is equivalent to CON(AoB) and indefinite propositions are no stronger than definite ones. His interpretation of Aristotle’s argument is the following. Aristotle takes for granted that contingent premisses can only yield contingent conclusions. We want terms A and B such that CON(AiB), but we can satisfy this condition while having a C of which A holds of none by necessity and of all or none of which it is contingent that B holds (e.g. if A is taken as literate, B as being asleep, and C as horse). But then any pair of contingent first-figure premisses with a particular major and a universal minor is compatible with a conclusion of the form NEC(AeC), but NEC(AeC) is incompatible with any genuinely contingent relation between A and C. 30. Following Aristotle, Alexander omits to say that what is taken must be non-A by necessity. 31. If we take into account only combinations with a particular major and a universal minor, Aristotle here takes as true: (i) CON(Animal i White) CON(Animal o White) (ii) CON(White a Human) CON(White a Cloak) CON(White e Human) CON(White e Cloak) (iii) NEC(Animal a Human) NEC(Animal e Cloak) Alexander objects that the propositions numbered (i) are false since NEC(Animal i Swan). He therefore substitutes:

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CON(White i Walking) CON(White o Walking) (ii) CON(Walking a Swan) CON(Walking a Crow) CON(Walking e Swan) CON(Walking e Crow) (iii) NEC(White a Swan) NEC(White e Crow). Aristotle uses these same terms to reject the corresponding U+C combinations at 1.15, 35b11-19 and IA_1(NC_), OA_1(NC_), IE_1(NC_), and OE_1(NC_) at 1.16, 36b3-7, thereby treating ‘Some white things are animals’ and ‘Some white things are not animals’ as unqualified, contingent, and necessary. Alexander does not raise questions about the other two uses of this set of terms. But see his discussion at 215,3-14. 32. endekhetai, here used informally and corresponding to Aristotle’s ouden kôluei at 33a38. 33. For a minor divergence between this lemma and our text of Aristotle see Appendix 6 (on 33b8). 34. Alexander has asserted this at 171,22-4, but we have not found a place where he shows it. It seems likely that he is referring to another work of his own. Cf. 36,5-6 with the note in Barnes et al. (1991). 35. cf. 154,23-155,2. 36. cf. 136,23-9. 37. At 1.13, 32a18-20. 38. Aristotle’s remark seems to apply only to Barbara1(CCC) and EE_1(CCC). Alexander tacitly extends it to Celarent1(CCC) and AE_1(CCC) at 173,1-3. 39. That is, e.g.,  NEC  (AaB) is not equivalent to  NEC  (AeB). 40. That is, when  NEC(AiC), i.e.,  NEC  (AeC), we say that it is contingent that no C are A even if, in fact, NEC(AoC), so that  CON(AaC), and so  CON(AeC). On the expression ‘none by necessity’ (mêdeni ex anankês) see Appendix 1 on the expression ‘by necessity’. 41. See 171,27-172,4. 42. At 1.15, 35a2. There is, however, no trace of these words at 33b3-8 in our texts. Perhaps we have here a logically motivated textual conjecture. 43. This chapter is perhaps the most difficult in Aristotle’s presentation of modal logic. The reader may wish to consult section III.E.2.a of the introduction before proceeding. 44. For the formulation see Appendix 1 on the expression ‘by necessity’. Alexander quotes this whole passage at 245,21-7. 45. i.e. he has not yet argued that EE-conversionc fails, as he will do in chapter 17 at 36b35-37a31. 46. In chapter 15. 47. In chapter 16. 48. Aristotle argues in chapter 17 that there are no second-figure CC syllogisms. In the present passage Alexander formulates the reasoning rather unsatisfactorily, relying on the idea that in the second figure there are no syllogisms with two affirmative unqualified premisses, hence there are none with two affirmative contingent ones; but if there were second-figure syllogisms with one or two contingent negative premisses, the transformation rules for contingent propositions would allow those premisses to be changed into affirmative ones. 49. Aristotle treats these cases in chapter 18. Alexander is unable to provide any conclusive reason for Aristotle’s not doing all the CC figures before the U+C and N+C ones, and there does not seem to be one available. Alexander takes up this question again briefly at 219,14-24. 50. Reading huparkhousês te with the Aldine in place of the anankaias printed

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by Wallies, of which we can make no good sense. But there may be a lacuna after eipein in which Alexander said that Aristotle could have taken up the first-figure N+U cases first, but instead chose to give precedence to the U+C cases. 51. 173,32-174,6, and 174,9-13.17-19 constitute Theophrastus 107A FHSG; 174,1-4 are Eudemus fragment 19 Wehrli. 52. Dropping the to in 174,5. 53. to oudeni ex anankês, meaning  NEC(AiC); see Appendix 1 on the expression ‘by necessity’. 54. to oudeni panti ex anankês, meaning  NEC(AaC); see Appendix 1 on the expression ‘by necessity’. 55. Alexander’s point in the present passage is that the conclusion of Celarent1(UC  N  ) is (equivalent to)  NEC(AiC) and that the conclusion of Ferio1(UC  N ) is  NEC(AaC), and these conclusions are compatible with AiC and AaC, respectively. But here Alexander apparently adopts the position that CON(AeC) means that AeC does not hold now. For difficulties with this position see 161,3-26. 56. That is,  NEC( P) is compatible with P, whereas – on Alexander’s reading – CON(P) rules out P.  NEC( P) is also compatible with NEC(P), which is ruled out by CON(P). 57. hoi peri Theophraston. 58. Alexander attempts to explain why the first-figure UC syllogisms are not complete in terms of the definition of ‘said of all’. He takes Barbara1(UC_) as his example; his argument seems to presuppose that the conclusion of this is not CON(AaC), but something verging on, if not identical with, AaC. He apparently argues as follows. Suppose we know AaB and CON(BaC). Then we know that A holds of everything of which B holds, but because we know only that CON(BaC), we do not know that B holds of any C. Therefore, we cannot infer AaC. Unfortunately he does not discuss a possible justification of Barbara1(UCC) which relies on the intuition that if it is now true that AaB and now true that it is contingent that every C is a B it is now true that it is contingent that every C is an A. 59. For a minor divergence between this lemma and our text of Aristotle see Appendix 6 (on 33b34). 60. For a minor divergence between this lemma and our text of Aristotle see Appendix 6 (on 34a2), and, for a further discrepancy, the note on 34a4. 61. A parenthetical reference to the complete Darii1(CUC) and Ferio1(CUC), which Aristotle takes up at 35a30-5. 62. dunaton. The word occurs 17 times between 34a6 and 32. It and dunatai do not occur again until chapter 24, and have occurred previously only 5 times (in our chapters only at 1.11, 31b9 and 1.13, 32b11). Forms of endekhomai occur only once in 34a6-16 (at 34a11). For a minor difference between this lemma and Ross’ text of Aristotle see Appendix 6 (on 34a6-7). 63. For understanding what Alexander says under this lemma it will be useful to read section III.E.2.a of the introduction. 64. anankaia akolouthia. Alexander does not distinguish clearly between an implication and a conditional (our translation of sunêmmenon). He also uses the words ‘sound’ (hugiês) and ‘true’ rather loosely by today’s standards. 65. We conjecture hepomenon for the first eilêmmenon in the aei to eilêmmenon hepesthai esti tôi to eilêmmenon hôs hêgoumenon einai printed by Wallies. The parallel passage in [Themistius] (in An. Pr. 25,20-1) is equally opaque to us: eilêmmenon aei hepesthai esti dia to eilêmmenon hôs hepomenon hôs hêgoumenon einai. 66. Alexander raises questions about whether in the reductio one derives a falsehood rather than an impossibility. His main line of interpretation commits

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him to the idea that one derives an impossibility. For if all that were at stake in the argument for Barbara1(UC‘C’) were derivation of a falsehood from a falsehood, there would be no way to deflect ‘blame’ from BaC, since it is false on Alexander’s understanding of CON(BaC). 67. Aristotle’s argument that if it is necessary that if P then Q and it is possible that P, then it is possible that Q is most easily understood in terms of the standard temporal interpretation of the modal operators. Suppose that if P then Q at all times, P at some time t and Q at no time (i.e. that it is not possible that Q). But then ‘if P then Q’ is true at t, so Q is true at t, contradicting Q at no time. (For possible minor differences between Alexander’s text of this passage and our text of Aristotle see Appendix 6, on 34a8-10 and 34a10-11.) 68. Ross prints eiê here where Alexander and the manuscripts of Aristotle have ei. 69. Wallies prints keimenou toutôi to A hepesthai. We read keimenou tou to A hepesthai to B with M and the Aldine. But note that this appears to be said again in line 6. In general we find the syntax of this paragraph hard to follow. 70. Reading adunaton at 177,16, where Wallies prints dunaton, a correction in B. 71. 177,19-182,8 is fragment 994 Hülser (1987-88). 177,25-178,8 is SVF II.202a. 72. The Stoic term lêgon; see the Greek-English lexicon. 73. Alexander turns to what he takes as an explicit attempt by Chrysippus to produce counterexamples to the claim that ‘if P then Q’ excludes it being the case that it is possible that P and not possible that Q. It seems likely that Chrysippus’ concern was to refute the so-called Master Argument of Diodorus by refuting this premiss of it. See Long and Sedley, vol. I, pp. 230-6. Chrysippus’ counterexamples are difficult because they depend on the use of pronouns and time of utterance. Alexander mentions two alleged counterexamples: (i) If Dion has died, he has died; (ii) If it is night, this is not day. Since Alexander says little about (ii) (see 181,34-182,8) we concentrate on (i). Chrysippus seems to take it that ‘Dion’ names a unique quality possessed only by Dion and therefore has a reference independently of whether Dion is alive, but ‘he’ can refer only to a living person so that an attempt to refer to a dead person as ‘he’ fails. Thus ‘He has died’ is impossible although it is possible that Dion has died. But why did Chrysippus think that ‘If Dion has died, he has died’ is a sound conditional? Perhaps he argued as follows. ‘He has died’ is significant only when ‘he’ has a referent, a living being; so ‘He has died’ is impossible. On the other hand, since ‘Dion’ picks out a unique qualification, ‘Dion has died’ is significant even when Dion has died. So ‘Dion has died’ is possible. Thus ‘If Dion has died, he has died’ has a possible antecedent and an impossible consequent. Presumably it also has significance only when Dion is alive, but then it is true because, e.g., it has a false antecedent and a false consequent. 74. axiôma. 75. Alexander here touches on the Stoic doctrine of eternal recurrence; see below 180,28ff. 76. Alexander now gives a series of arguments against Chrysippus’ alleged counterexamples. He first argues that if ‘Dion has died’ can be said in circumstances in which ‘He has died’ cannot, the conditional ‘If Dion has died, he has died’ cannot be necessary. 77. Alexander expresses the antecedent with and without the word ‘if’, apparently indifferently. 78. ho Diôn, not to Diôn. It is not clear that Alexander (or the scribes) maintain consistency in distinguishing between the name ‘Dion’ and the person Dion. 79. Just above at 178,13-14. 80. Apparently Chrysippus claimed that ‘If Dion has died, he has died’ remains

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true even if ‘he has died’ loses its significance because of the death of Dion. One might compare this with a view that ‘If Zeus is strong, Zeus is strong’ is true even though ‘Zeus is strong’ is ‘meaningless’ because Zeus does not exist. Alexander insists that Chrysippus’ conditional is false because – according to Chrysippus – ‘he has died’ may fail to hold when ‘Dion has died’ holds. 81. ho Diôn, not to Diôn. 82. i.e. in such a way that ‘Dion has died’ can have application when ‘he has died’ does not. What follows suggests that someone attempted to defend Chrysippus by making a (far-fetched) comparison between the relation of ‘he’ to Dion and that of species to genus. In the example it seems to be imagined that there are no scalene triangles, but that it is true that all triangles have their angles equal to two right angles and also true that if all triangles have their angles equal to two right angles, all scalene triangles do, even though ‘all scalene triangles have their angles equal to two right angles’ is false (or meaningless). Alexander’s rejection of the comparison seems sound, but he does not really argue for it. 83. idiôs poios. On this Stoic notion see Long and Sedley (1987), vol. I, pp. 166-79. 84. Presumably the Stoics. 85. In this difficult paragraph Alexander apparently insists that Q follows from P if and only if Q follows from the hypothesis that P. Perhaps what he means is that Q holds when P holds if and only if P  Q is sound. One way of understanding Chrysippus’ position is as saying that although ‘if Dion has died, he has died’ is sound, when ‘Dion has died’ holds, ‘he has died’ is destroyed and so does not hold. 86. epikheirêsis. 87. Apparently Alexander is referring to 34a5-12 and 34a25-33. 88. logikôteron. In what follows Alexander argues in terms of certain Stoic assumptions, e.g., that ‘He has died’ is destroyed when Dion has died, and the belief in cosmic conflagration. 89. Alexander now invokes the Stoic doctrine of eternal recurrence. For discussion see Long and Sedley (1987), vol. I, pp. 308-13. On their interpretation Alexander’s uncertainty about the numerical identity of the recurring Dions reflects Stoic alterations of an original doctrine according to which the Dions are numerically identical. (180,31-6, 181,13-17, and 181,25-31 are SVF II.624.) 90. According to Nemesius (2.81 Morani = SVF II.790), Chrysippus characterized death as the separation of the soul from the body. 91. In the only other extant reference to this work Stobaeus (I.79.1 = SVF II.913) says that in its second book Chrysippus called the substance of fate a pneumatic power controlling the universe in an orderly way. 92. We have translated Wallies’ text as a negated conjunction equivalent to ‘It is not the case that the following two things are true: (i) the recycled Dion is the same as the earlier one; (ii) the word ‘he’ said of the recycled Dion does not refer to the earlier one,’ that is, ‘If the recycled Dion is the same as the earlier one, then the word “he” refers to them both.’ An anonymous reader has commended to us the rendering of Feliciano (Alexandri Aphrodisiensis Super Priora resolutoria Aristotelis subtillissima Explanatio: a Ioanne Bernardo Feliciano in latinum conuersa, Venice: H. Scottus, 1560, p. 147), who takes the contents to be two rhetorical questions: isn’t the later Dion the same as the earlier one? Isn’t the word ‘he’ predicated of the same thing? (Feliciano’s Greek exemplar actually differs from Wallies’ text in a way irrelevant to the point raised here.) 93. The material in parentheses seems to be an aside from the main argument, but obviously relates to the question of the relation between the Dion(s) in different kosmoi.

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94. One might have expected Alexander to conclude that ‘If Dion has died, he has died’ is a true conditional with a contingent antecedent and consequent. But he seems to prefer – at least for the sake of this argument – the view that ‘He has died’ is not a self-refuting proposition and can be false when ‘Dion has died’ is true. cf. 178,8-22. 95. metaballousa gar ouk estai. We take the point to be that ‘It is night and ‘It is not day’ are both contingent because there is day and night in alteration. Hülser ((1987-1988), p. 1279) has ‘[der Sinn dieses Ausdrucks] wird ja nicht umschlagen können.’ 96. Our text of Aristotle has ‘impossibility and possibility’. 97. to dunaton kai endekhomenon. The point of Aristotle’s remark is not clear, but it would seem that all he wants to say is that in whatever way X is possible (e.g., possibly true, possibly holding), then if Y follows from X, Y will be possible in the same way. Alexander extends the remark to cover the ‘senses’ of contingency he understands Aristotle to have mentioned at 1.3, 25a37-9. 98. At 1.13, 32a18-20, the diorismos of contingency. 99. 34a8-9. 100. cf. 182,9-12. 101. i.e. just as one of the ‘senses’ of possibility is truth, so one of the senses of impossibility is falsehood. 102. At 182,23-183,7. 103. We read ei tethnêke Diôn, tethnêke anthrôpos with the Aldine. Wallies prints to tethnêke anthrôpos. 104. 1.13, 32b4-23. 105. See 182,33-183,7. 106. The next two paragraphs constitute testimony 135 Döring (1972); fragment 992 Hülser (1987-1988) includes the next two lines as well. For other ancient material on the subject of Diodorean and Philonian possibility see Döring (1972), testimonies 130-9 and Sharples (1982), pp. 91-6. 107. to akhuron to en têi atomôi. Long and Sedley (1987) have ‘chaff in atomic dissolution’, Hülser (1987-1988) ‘nicht weiter teilbar Spreu’, Muller (1985) ‘le chaume sur pied dans un champ non fauchée’. At 184,16 below it is tentatively assumed that ho atomos is impassive. The example recurs in Alexander, Quaest. 31,9-10; and apparently he used it in his commentary on De Caelo (Simplicius, in Cael. 316,25-9, where there is only talk of the chaff’s being destroyed). 108. Because, according to Alexander, Aristotle, unlike Diodorus, thinks that some possibilities will never be realized, but, unlike Philo, he does not think that it is possible for chaff at the bottom of the sea to be burnt. If Alexander intends what he says to apply to contingency, what he says about possibilities which are never realized would be incompatible with any standard temporal interpretation of the modalities, including the one which Alexander seems to flirt with. 109. Alexander paraphrases Aristotle’s hôsper oun ei tis theiê to men A tas protaseis to de B to sumperasma as kai ei tis to men A anti tôn protaseôn laboi to de B sumperasma. He is presumably worried about the brachylogy of the formulation, on which see Ross ad loc. 110. Reading hôs en tôi deuterôi deixei. The reference is to chapters 2-4 of book 2. Alexander here corrects Aristotle’s careless statement that ‘if A is false but not impossible and if when A is, B is, B will also be false ’. 111. At 34a16-19. Alexander and Aristotle overlook the case in which two possible premisses are together impossible. 112. For a minor difference between Alexander’s citation and our text of Aristotle see Appendix 6 (on 34a32).

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113. Aristotle turns to arguing for Barbara1(UC‘C’). See section III.E.2.a of the introduction; see also Mignucci (1972) for an attempt to defend Aristotle’s argument. The phrases in the lemma signalled as (i), (ii), and (iii) all cause difficulty. Ross emends (i) to say ‘it is not contingent that A holds of all C’. Alexander (186,17-19) says that Aristotle’s formulation is deficient (endeesteron) in leaving out the ‘all’, (ii), which is sometimes translated as if it said ‘It is not contingent that A holds of all B’ should say (ii’) ‘A does not hold of all B’ (i.e. AoB), and (iii) should say (iii’) ‘it was hypothesized that A holds of all B’. Alexander says (186,19-21) that Aristotle says (ii) instead of (anti) (ii’); he apparently intends to say that this is what Aristotle means (cf. 186,25-7, where Alexander says that Aristotle says (ii) as equivalent to (hôs ison) (ii’)). Alexander also says (186,23-5) that Aristotle says (iii) instead of (iii’). He points out (186,31-4) that if we take (ii) and (iii) straightforwardly, they do not contradict one another. He then suggests that Aristotle’s formulations are all right because in one sense the unqualified is said to be contingent. Alexander’s discussion of Aristotle’s treatment of Barbara1(UC‘C’) lasts until 193,21 (omitting 187,9-188,7), and ends with the somewhat pathetic words ‘These things should be investigated in a better way’ (episkepteon de peri toutôn beltion). Alexander seems committed to defending Aristotle’s position on Barbara1(UC‘C’), but the position is untenable, as his discussion of 34b7ff. shows. There is further discussion related to Barbara1(UC‘C’) at 198,5-199,15, where Alexander makes clear that the conclusion is not contingent in the way specified. 114. See 34a1-5. Alexander proceeds to give a version of Aristotle’s argument before discussing the difficulties in Aristotle’s formulation of it. 115. Guided by 34a40-1 (iii), Alexander cites Bocardo3(NU C ), formulating the conclusion as  CON(AaB), i.e.,  CON (AoB), rather than as NEC(AoB). In fact, as Alexander goes on to point out, Aristotle denies the validity of Bocardo3(NUN) at 1.11, 32a4-5, which Alexander discusses at 150,25-151,30. Of course, as Alexander also realizes, Aristotle needs only Bocardo3(NUU) to get an inconsistency. 116. Ross accepts Becker’s view that this passage is ‘the work of a rather stupid glossator’. In it the original premisses of Barbara1(UC‘C’), AaB and CON(BaC), are transformed into CON(AaB) and BaC, from which it is correctly inferred (Barbara1(CUC)) that CON(AaC), which is then said to be incompatible with  CON(AaC)). It is very hard to know how to interpret this argument; indeed, if the difficulties in Aristotle’s formulation in the preceding passage carry over to this one, it is not even clear what argument is being offered. Alexander does not provide any real elucidation here, and what he says at 188,7ff. is quite obscure. 117. The lemma has thenta where our texts of Aristotle have thentas. 118. In Aristotle’s affirmation of the completeness of Barbara1(CUC) at 33b33-6. 119. Alexander now points out that the transformation of AaB into CON(AaB) is unnecessary, since AaB and BaC imply (Barbara1UUU)) AaC, which is incompatible with  CON(AaC). He suggests that Aristotle chose to transform AaB into CON(AaB) and infer CON(AaC) because the incompatibility of AaC and  CON(AaC) is not as ‘clear and indisputable’ as that of CON(AaC) and  CON(AaC). 120. Alexander points out that – by analogy with his understanding of the preceding argument for Barbara1(UC‘C’) – one could understand ‘It is contingent that A holds of all B’ as AaB and ‘It would be contingent that A holds of all C’ as AaC. This reading would give the argument ruled out in the previous paragraph. 121. In the preceding argument for Barbara1(UC‘C’). 122. We are not able to construe this difficult paragraph in a fully satisfactory

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way, but we think it might be an argument against indirect proofs depending on the transformation of a contingent premiss into an unqualified one (what we call U-for-C substitution). It would then be aimed, at least partly, against the first argument for Barbara1(UC‘C’), but it is possible that Alexander is only pointing out that the second argument for Barbara1(UC‘C’) really infers the desired conclusion directly and so is not a reductio. We formulate our notes in terms of the first argument. 123. Reading kataskeuazetai with the Aldine. 124. This is apparently AaB, the major premiss of Barbara1(UC‘C’). 125. BaC, the transformed minor premiss of Barbara1(UC‘C’). 126. We suspect a lacuna here in which the hypothesizing of the opposite of the conclusion and the derivation of an impossibity were described. 127. We have excised the words ‘for this is impossible’: adunaton gar touto. 128. See the note on 207,36. 129. On this lemma and Alexander’s discussion see section III.E.2.a of the introduction. Alexander’s discussion is made less cogent than it need be by his misunderstanding of rejection by setting down terms. (For a minor divergence between the lemma and our text of Aristotle see Appendix 6, on 34b7.) 130. Alexander says that ‘All that moves is human’ is true only at a specific time because moving has a greater extension than human, so that there may be C, e.g., horse, of which moving can hold but of which human cannot possibly hold; consequently, all that moves cannot be human at the time when all horses are moving. 131. This remark doesn’t seem to be required for Alexander to make his claim. Alexander’s point is (again) that if B has a larger extension than A, and C is such that  CON(AaC), and there is a time at which BaC, one cannot have AaB at that time. 132. i.e. the universal unqualified major premiss ceases to be true. 133. endekhetai. 134. Alexander gives no argument for this unfortunate assertion. We have already seen that the combination is not syllogistic in the sense that there are counter-examples to it, that the method mentioned by Alexander is illegitimate, and that simpler methods of the same illegitimate kind are available. 135. Alexander slightly changes Aristotle’s example, taking as the interpretation of the premisses: (i) All that moves is an animal. (ii) It is contingent that all stones are moving. Again he points out that (i) cannot be true at a time when even one stone is moving. 136. skhêma. It is difficult to see how this claim can be defended, since Aristotle and Alexander have just been saying that only some instances of Barbara1(UC‘C’) are valid. 137. i.e. when the minor contingent premiss is taken to hold actually, the universal affirmative major becomes false. 138. This is the first hint that Alexander realizes that the conclusion of Barbara1(UC_) is not contingent in the way specified. See also 198,5-13. Alexander goes on to discuss the fact that of Aristotle’s two interpretations only the one which renders NEC(AeC) true produces the temporal conflict between the two premisses which Alexander wants to emphasize. 139. Alexander’s point is that ‘All that moves is an animal’ and ‘All humans are moving’ can be true at the same time. 140. See the note on 207,36 below.

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141. At 34a34-b2. Although it is, indeed, worthwhile to attack the method of that alleged reductio, the issue raised here is a red herring. In a reductio one derives an inconsistency, that is, one derives a proposition P from assumptions which include or entail a proposition incompatible with P; it follows that one of the assumptions is false, whether P is ‘false’ or ‘impossible’. The argument considered by Alexander concedes that Aristotle derives  (AaB), i.e., AoB, when one of the original premisses is AaB, but argues that, if AaB is unqualified and not necessary, then AoB will be false but not impossible. Alexander’s response is to say that although CON(AoB) is true at the time when AaB is (namely now, when it is being taken as a premiss), at that time one cannot change CON(AoB) into AoB, i.e., presumably, AoB is impossible at a time when AaB. 142. Using Bocardo3(NUU). 143. CON(AoB). 144. The response shifts attention to the real issue, the change of the second premiss CON(BaC) into BaC, and argues that if Alexander is right about the impossibility of transforming CON(AoB) into AoB when AaB is assumed, it is equally impossible for Aristotle to transform CON(BaC) into BaC, when it is true (as it might be) that BeC. 145. Alexander’s resolution of the problem is not very satisfying; the end of the discussion shows his own dissatisfaction. His remarks are directed at the opponent’s transformation of CON(BaC) into BaC on the assumption that BeC, but apply also to his transformation of CON(AoB) into AoB, given the premiss AaB. Both transformations are impossible because the new unqualified propositions, BaC and AoB, are incompatible with the other unqualified assumptions, BeC and AaB. On the other hand, Aristotle’s transformation of CON(BaC) into BaC is all right because he is simply assuming what will at some future time hold without denying that at the present BeC (possibly) and CON(BaC). 146. BaC. We take Alexander’s claim to be that Aristotle’s hypothesizing of BaC could be said to be false only if it did away with the premiss CON(BaC) as the assumption of AoB does away with the premiss AaB. For our punctuation of this sentence see the textual emendation for 192,32-193,1. 147. Of CON(BaC) into BaC. 148. Reading dokei for the dokein printed by Wallies. 149. Aristotle’s argument for Celarent1(UC  N  ) is completely analogous to his argument for Barbara(UC‘C’). He assumes  CON(AeC), ‘i.e.’, NEC(AiC), changes CON(BaC) to BaC, and infers (Disamis3(NUU)) that AiB, contradicting AeB. Alexander spells out the argument and insists that it works. There is a more critical, but inconclusive, discussion of Celarent1(UC_) at 216,7-217,20, where Alexander justifies Celarent1(UCU) with a reductio argument in which the contingent premiss is changed into an unqualified one. 150. The endekhetai of M seems to us more likely than the endexetai printed by Wallies. 151. At 34a5-33. 152. mêdeni ex anankês, i.e.,  NEC(AiC). 153. i.e. in the way specified. 154. On the obeli see the note on 199,16-18 below. 155. mêdeni ex anankês. 156. For this distinction between ‘A holds of no C by necessity’ (NEC(AeC)) and ‘A holds by necessity of no C’ ( NEC(AiC)), see Appendix 1 on the expression ‘by necessity’. 157. We offer the following paraphrase of the first part of this paragraph.  CON(AeC), the negation of the ‘conclusion’ of Celarent1(UC_) was transformed

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into NEC(AiC), and the contradictory of NEC(AiC) is  NEC(AiC) expressed as either ‘A holds by necessity of no C’ or ‘It is not necessary that A holds of some C’ (ouk ex anankês tini). So  NEC(AiC) is what is inferred in the ‘reductio’, and this is not equivalent to CON(AeC) because NEC(AiC) is not equivalent to  CON(AeC); for  CON(AeC) is true when NEC(AiC) is false and NEC(AoC) is true. For example,  CON(Walking e Animal) because NEC(Walking o Animal) ; but also  NEC(Walking i Animal) . On this passage and the sequel see Appendix 5 on weak two-sided Theophrastean contingency. 158. As Aristotle, but not Theophrastus, holds. 159. For the point of this remark see Appendix 5 on weak two-sided Theophrastean contingency. 160. Aristotle gives terms to show that the conclusion of Celarent1(UC_) is not contingent in the way specified. For this purpose it suffices to give an interpretation under which the premisses are true and there is some necessary connection between the major and minor term. His first terms provide such an interpretation: Raven e Reflective CON(Reflective a Human) NEC(Raven e Human) Aristotle decides to add terms showing that Celarent1(UC_) does not yield a necessary conclusion. These terms verify: Moving e Knowledge CON(Knowledge a Human)  NEC(Moving o Human)  NEC(Moving i Human) (For some reason Aristotle mentions the verification of  NEC(Moving e Human) rather than  NEC(Moving o Human).) Aristotle indicates dissatisfaction with these terms, but does not say why. Perhaps his dissatisfaction relates to a possible ambiguity in his middle term ‘knowledge’: the first premiss would seem to mean ‘No knowledge is in motion’, the second ‘It is contingent that no human is knowledgeable’ (but perhaps Aristotle is taking the possible truth ‘Nothing knowledgeable is moving’ as an unqualified truth). Alexander’s dissatisfaction with the terms is buttressed by his view that the specification of terms is a way of confirming that a certain conclusion follows. He finds two failings in the first set of terms: (i) that they make the first premiss necessary not unqualified; (ii) that, as Aristotle indicates, they verify NEC(AeC) and not just the desired conclusion  NEC(AiC). According to Alexander, the second set of terms takes care of (ii), but leaves (i), which shows that he is taking ‘Moving e Knowledge’ to mean ‘No knowledge is in motion’. Alexander proposes a set of ‘better terms’: Angry e Laughing CON(Laughing a Human)  NEC(Angry i Human). Alexander is not comfortable with these terms, perhaps because the major premiss might seem to be necessary. He offers another set which are not clearly better on this score. Walking e Resting CON(Resting a Animal)  NEC(Walking i Animal). These terms might be better than the first set because one might hold that CON(Angry e Human), whereas, since Alexander takes for granted that NEC(Walking o Animal), he holds that  CON(Walking e Animal). Alexander takes up this setting down of terms a second time at 199,19ff.

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161. Alexander’s point is not that  NEC(Raven i Human) is false but that the stronger NEC(Raven e Human) is true. 162. Alexander’s suggestion seems incorrect. Aristotle encountered the alleged difficulty because he took a minor term of which the major does not hold by necessity. He might have taken, e.g., NEC(Raven e Reflective) CON(Reflective a Walking)  NEC(Raven e Walking) and  NEC(Raven i Walking). 163. Here is a striking example of Alexander’s view that a proposition which is contingent in the way specified does not hold. He thinks that CON(Angry e Human), which is equivalent to CON(Angry a Human) is not true because many humans are, in fact, angry;  NEC(Angry i Human) is, however, true. 164. Alexander now raises the Theophrastean question whether ‘It is necessary that AiC’ isn’t the contradictory of ‘It is contingent that AeC’. 165. pseudês. 166. We have paraphrased Alexander’s terse formulation of the two alternatives, which might be rendered: ‘as denying by necessity the assumption (to keimenon) or as denying the necessity.’ (In the immediate sequel we refer to a sense of ‘A holds by necessity of no C’ where Alexander uses one of his formulations to describe the sense.) We might express the two alternatives by saying that ‘A holds by necessity of no C’ may assert either (1) NEC(AeC) or (2)  NEC(AiC). Note that at 194,13-19 Alexander suggested that only (2) corresponded to the words ‘A holds by necessity of no C’, (1) corresponding to ‘A holds of no C by necessity’, and in general he sticks to what he says there. But he is not always careful; see, e.g., 202,22. In our translation we have tried to smooth over these careless formulations. 167. Alexander again makes the point that, unlike CON(AeC),  NEC(AiC) can be true even when NEC(AoC). 168. That is, CON(AeC) converts with CON(AaC) and so is incompatible, as  NEC(AiC) is not, with NEC(AoC) and indeed with any necessary proposition AC. 169. cf., e.g., 156,13-14 and 194,33-195,1. 170. That is,  CON(AeC) follows from NEC(AiC) but does not imply it. Again, we paraphrase the rest of the paragraph.  CON(AeC) is true if either NEC(AiC) or NEC(AoC). Aristotle chooses to use NEC(AiC) rather than NEC(AoC) for his reductio because he is able to get a contradiction using it. The conclusion from NEC(AoC) and the second premiss BaC of Celarent1(UC_) would be (Bocardo3(NUU)) AoB, which is compatible with the first premiss AeB. So Aristotle changes  CON(AeC) into NEC(AiC), gets a contradiction, and quite reasonably infers, not   CON(AeC), i.e., CON(AeC) but,  NEC(AiC). For  NEC(AiC) is the opposite of NEC(AiC), but the opposite of CON(AeC) is  CON(AeC), which is implied by either NEC(AiC) or NEC(AoC), both of which are incompatible with CON(AeC) and compatible with  CON(AeC). 171. The reference is to Aristotle’s rejection of an indirect argument for EEconversionc at 1.17, 37a9-31. 172. cf. 195,6-17. 173. i.e. not CON(AeC). 174. The reference is to 35a1-2, ‘It is clear then that the conclusion is that A holds by necessity of no C,’ but at 34b25-6 Aristotle says ‘But this is impossible, so that it will be contingent that A holds of no C.’ 175. Alexander now raises the question whether the conclusion of Barbara1(UC_), taken up at 34a34ff., is contingent in the way specified. It is not, as

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Alexander makes clear, although his discussion is not as lucid as it might be; see also 207,3-18. 198,5-199,10 constitute Theophrastus 103B FHSG. 176. Celarent1(UC_). Cf. 197,7-32 and Appendix 5 on weak two-sided Theophrastean contingency. 177. On this way of expressing  NEC(AoC) see Appendix 1 on the expression ‘by necessity’. 178. Alexander asks whether the difference between CON(AaC) and  NEC(AoC) is merely verbal. If it is not, the conclusion of Barbara1(UC_) must be taken to be the latter rather than the former. Alexander goes on to look for terms which verify  NEC(AoC) and  CON(AaC). See Appendix 5 on weak two-sided Theophrastean contingency. 179. The rest of this paragraph is difficult. We propose the following interpretation hesitantly. Alexander is looking for terms which will verify  CON(AaC),  NEC(AoC). For this purpose he need only find terms which make  NEC(AoC) and NEC(AiC) true. He first considers  NEC(Breather o Animal) (‘No animal is by necessity not a breather’). He points out that this statement will not do because, even though NEC(Breather i Animal) is true, it is also true that NEC(Breather o Animal), presumably because fish don’t breathe. He then points out in a somewhat long-winded way that both  CON(Breather e Animal) and  CON(Breather a Animal) are true. 180. viz.  CON(Breather a Animal) and CON(Breather e Animal). 181.  NEC(Thinking o Rational). Alexander presumably thinks it is not necessary that any rational being not be thinking. But also NEC(Thinking i Rational) because it is necessary that god be thinking, and, consequently  CON(Thinking a Rational). In the next paragraph he offers ,  NEC(Motion o Body) (because there is no necessarily stationary body – or is the earth necessarily stationary?). Also NEC(Motion i Body) because the cosmos, the rotating body, necessarily rotates), and, consequently,  CON(Motion a Body). 182. Alexander draws the correct inference: the conclusion of Barbara1(UC_) is not contingent in the way specified. 183. hoi peri Theophraston. The natural reading of what Alexander says is that Theophrastus thought  NEC( P) says that P is contingent in the way specified. Alexander points out that such a position involves abandoning EA-transformation for contingent propositions. 184. We here translate the text as printed in CAG. In a note to the lemma Wallies writes ‘non lemma, sed textus verba in M’. He refers to 200,5-9 for the evidence that the obelized words were not known to Alexander. 185. Alexander has discussed Aristotle’s specification of terms at 195,18196,12, a passage which should be read in connection with this one. 186. i.e. it is necessary that nothing reflective is a raven. Alexander thinks that Aristotle’s project is to find terms which make the premisses of Celarent1(UC_) true and  NEC(AiC) true. Aristotle’s terms do this since  NEC(Moving i Human). But what Aristotle points out about these terms is both (i)  NEC(Moving e Human) and (ii)  NEC(Moving i Human). To explain this Alexander here takes Aristotle to be saying (i) that the conclusion is not NEC(AeC) and (ii) that it is  NEC(AiC). 187. ouk anankê mêdena, apparently an alternative formulation of the oudeni ex anankês of 196,28 (which Alexander takes from Aristotle’s mêdeni ex anankês (34b28) and finds necessary to reconcile with Aristotle’s use of ouk anankê tina in the present passage, although Aristotle’s formulation seems like a better rendering of  NEC(AiC)).

198

Notes to pp. 112-115

188. Aristotle reduces AEA1(UC‘C’) to Barbara1(UC‘C’) and EEE1(UC  N  ) to Celarent1(UC  N  ). 189. Our text of Aristotle says ‘and it is taken that it is contingent that B holds of all C’, but Alexander does not have the word endekhesthai, as is made clear by 200,23-8. 190. In the treatment of AEA1(CCC) at 1.14, 33a5-12. 191. That is, Aristotle would seem to be reducing AEA1(UC‘C’) to Barbara1(UC‘C’), but the words (in Alexander’s text) ‘if BC is converted and it is taken that B holds of all C’ suggest that an unqualified minor premiss has come into play. The next sentence, which we have put in parentheses, suggests that Aristotle is not just describing a reduction of AEA1(UC‘C’) to Barbara1(UC‘C’), but the further reductio ‘justification’ of Barbara1(UC‘C’), in which the contingent premiss is converted to an unqualified one. This sentence comes in harshly and lacks a connecting particle; perhaps it is a gloss. 192. Aristotle rejects EE_1(CU_) and AE_1(CU_), taking the following to be true: (i) CON(White a Animal) CON(White e Animal) (ii) Animal e Snow Animal e Pitch (iii) NEC(White a Snow) NEC(White e Pitch) Alexander worries about (i) since NEC(White a Swan) and NEC(White e Crow). He substitutes: (i’) CON(Moving a White) CON(Moving e White) (ii’) White e Walking White e Standing still (iii’) NEC(Moving a Walking) NEC(Moving e Standing still) He points out that the assertions of (iii’) are only necessary on a condition; see Appendix 3 on conditional necessity. He drops the subject insisting that the two combinations are not syllogistic. Aristotle uses the same terms to reject EE_1(CN_) and AE_1(CN_) at 1.16, 36a27-31, where they seem to be more appropriate, since the propositions (ii) seem necessary rather than unqualified, although the problem with (i) remains. In his comment at 212,1-2 Alexander suggests dealing with the problem of (i) by substituting: (i’) CON(White a Horse) CON(White e Horse) (ii’) NEC(Horse e Snow) NEC(Horse e Pitch) (iii’) NEC(White a Snow) NEC(White e Pitch). 193. At 35a2. 194. Aristotle claims completeness for Darii1(CUC) and Ferio1(CUC). 195. Aristotle asserts the validity of Darii1(UC‘C’), Ferio1(UC  N ), AOI1(UC‘C’), and EOO1(UC  N  ). Alexander gives the argument for the first at 202,17-30 and for the second at 202,30-203,1; he briefly describes what is done in connection with the last two at 203,1-9. Neither he nor Aristotle raise any of the doubts about these cases analogous to those treated earlier in connection with their analogues with a universal minor premiss. 196. For apparent minor divergence between Alexander’s text here and our text of Aristotle, see Appendix 6 (on 35b1). 197. Alexander made no mention of this fact in connection with 1.4.26b21 where Aristotle also calls premisses intervals. 198. Alexander insists that Barbara1(UC‘C’) and Celarent1(UC  N  ) are not complete, since they are established by reductio; so when Aristotle says that in their cases the syllogisms are through the premisses themselves, he means only that the premisses don’t have to be transformed in quality to reach the conclusion. 199. Alexander gives a ‘proof’ of: Darii1(UC‘C’) AaB CON(BiC)  NEC  (AiC).

Notes to pp. 115-118

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not remarking that it involves the same difficulties as Aristotle’s proofs of Barbara1(UC‘C’) and Celarent1(UC  N  ). (The same can be said of Alexander’s treatment of the remaining syllogisms in this section.) But see the note on 202,30. Here is the ‘proof’: Assume  CON(AiC), ‘i.e.’, NEC(AeC); assume BiC; then (Ferison3(NUN)), NEC(AoB), contradicting AaB. Alexander points out that it would suffice for the contradiction to take the conclusion of Ferison3(NU_) to be unqualified, thereby circumventing the controversy over the proper conclusion of that combination. 200. Here Alexander moves silently from ex anankês oudeni to oudeni ex anankês. It seems unlikely that he cares about the difference between these two expressions which he invoked in connection with Celarent1(UC  N  ) at 194,14. 201. Alexander’s ‘proof’ for: Ferio1(UC  N  ) AeB CON(BiC)  NEC  (AoC) is analogous: Assume  CON(AoC), ‘i.e.’, NEC(AaC); assume BiC; then (Datisi3(NUN)) NEC(AiB), contradicting AeB. Again Alexander points out that it suffices for present purposes to assume the validity of Datisi3(NUU). Alexander says a little more about this case in the next chapter at 205,26-206,11, and makes clear that the conclusion is  NEC  (AoC). 202. 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.’ 203. Aristotle rejects AO_1(CU_) and EO_1(CU_). 204. The lemma reads Hotan de to mê huparkhein tini lambanêi where Aristotle has hotan de to mê huparkhein lambanêi hê kata meros tetheisa. 205. For a minor textual point see Appendix 6 (on 35b11). 206. Alexander points out that since Darii1(CUC) and Ferio1(CUC) are valid, one cannot show the invalidity of AO_1(CU_) or EO_1(CU_) using terms which verify the particular negative minor premisses but do not verify the corresponding universal negative assertion. For if they do not verify the latter, they will verify the particular minor premisses of the two valid syllogisms and hence verify their conclusions. 207. e.g., at 1.4, 26b10-20, 1.5, 27b9-28, 1.6, 28b24-31. 208. At 35a20-4. 209. Aristotle rejects cases with a particular major premiss and either a universal or a particular minor premiss. (For a minor divergence between the lemma and our text of Aristotle see Appendix 6, on 35b12.) 210. It is difficult to see how else to interpret Aristotle’s ‘contingent or unqualified or in alternation’, even though it makes Aristotle refer to the results of previous chapters. 211. See the note on 171,14. 212. Certainly the text doesn’t give all the relevant alternatives; the omissions may be due to a scribe. 213. For a minor textual divergence from Aristotle see Appendix 6 (on 35b17). 214. Aristotle would seem to be asserting the validity of all of: Darii1(CUC), Ferio1(CUC), Darii1(UC‘C’), Ferio1(UC‘C’), AOI1(UC‘C’), EOO1(UC‘C’), AOI1(CU_), EOO1(CU_), and rejecting the corresponding cases with a particular minor and universal major. Alexander realizes that Aristotle has rejected AOI1(CU_) and EOO1(CU_), so he re-interprets Aristotle to be referring only to the validity of Darii1(UCC). 215. The words tês elattonos dêladê look very much like a gloss.

200

Notes to pp. 118-120

216. Like chapter 15, chapter 16 is very difficult. The reader may wish to consult section III.E.3.a of the introduction. 217. The words ‘or not holding’ in the lemma do not occur in our texts of Aristotle. For two other very minor variations see Appendix 6 (on 35b23; see also the notes on 35b34 and 35). 218. Our texts of Aristotle do not have the words ‘that something does not hold’ (mê huparkhein) quoted by Alexander at 205,21-2. 219. i.e.  NEC is different from NEC  . For possible minor divergences between Alexander’s text of this last sentence and ours see Appendix 6 (on 36b34 and 36b35). 220. In the previous chapter. 221. Despite the correct reservations expressed at 198,5-199,15, Alexander now follows Aristotle and reasserts the false claim that the conclusions of Barbara1(UC‘C’) and Darii1(UC‘C’) are contingent in the way specified. 222. In Alexander’s view when Aristotle argues that a pair of premisses yields the conclusion ‘It is contingent that P’ because it yields the conclusion P, he is not talking about contingency in the way specified since for Alexander CON(P) entails  P. What he says at 216,7-14 suggests that he understands the difference between CONu and  NEC  , but for the most part he lumps them together under the rubric ‘contingent, but not in the way specified.’ See section III.E.3.a of the introduction. 223. Alexander refers to the treatment of Celarent1(UC  N  ) at 1.15, 34b1935a2 and of Ferio1(UC  N  ) at 1.15, 35a35-b8. See 194,9ff. The conclusion of the former is  NEC(AiC) (A holds by necessity of no C), and that of the latter is  NEC(AaC) (A does not hold by necessity of all C). See Appendix 1 on the expression ‘by necessity’. 224. Alexander considers Ferio1(UC  N  ), of which the conclusion is  NEC(AaC). He wants to explain the non-equivalence of this and CON(AoC) or, equivalently, of NEC(AaC) and  CON(AoC). His basic point is that  CON(AoC) is verified not just by NEC(AaC) but by NEC(AeC). See Appendix 5 on weak two-sided Theophrastean contingency. 225. See Appendix 5 on weak two-sided Theophrastean contingency. 226. i.e. NEC(AaC) implies but is not implied by  CON(AoC). A more literal translation of what Alexander says might be: ‘Its negation has not been transformed into its equivalent but into one of the things under it by which it is rendered true’ (tina tôn hup’ autên, kath’ hôn alêtheuetai). 227. This paragraph makes no clear sense to us. We think it is corrupt. Even paraphrasing what Alexander says is difficult because he uses alternative formulations for equivalent propositions. We use the following abbreviations. endekhetai tini mê CON(AoC) (a) ou panti ex anankês  NEC(AaC) (b) ouk ex anankês tini  NEC(AiC) (c) oudeni ex anankês  NEC  (AeC) (c’) panti ex anankês NEC(AaC) ( b) ouk endekhetai tini mê  CON(AoC) ( a) Alexander seems to make the following claims: (i) (a) is true when either (b) or (c) is; (ii) (b) falsifies ( b) and is implied by (c’) because (c’) implies the falsehood of ( b); also ( b) implies ( a); (iii) (c) implies the denial of NEC(AiC); (iv) not only (a), but also the conjunction of (b) and (c) are incompatible with any necessary relation between A and C; (a) and the conjunction of (b) and (c) are equivalent;

Notes to pp. 120-121

201

(v) each of (b) and (c’) are true if the corresponding unqualified propositions are true, i.e., hold unqualifiedly. What is said under (ii) and (v) is correct and does not involve (c). (iii) involves (c) and seems very trivial. (iv) is incorrect since although (a) implies that there is no necessary relation between A and C,  NEC(AiC) implies  NEC(AaC), and they are quite compatible with NEC(AeC). (i) is also incorrect since neither (b) nor (c) implies (a). Both of these difficulties would be removed if (c) could be read as  NEC(AeC) and (i) could understood as saying that (a) is implied by the conjunction of (b) and (c). This passage would then be an assertion of what we have called weak two-sided Theophrastean contingency. The latter change seems not impossible, but we are reluctant to suggest that ouk ex anankês tini might have been altered from ouk ex anankês oudeni at 206,3-4, and we cannot see how to explain an orginal ouk ex anankês oudeni in 206,8. 228. In the way specified. 229. Celarent1(UC  N  ). 230. Aristotle apparently asserts the validity of Barbara1(NCC). Alexander gives a U-for-C proof of: Barbara1(NC‘C’) NEC(AaB) CON(BaC)  NEC  (AaC) Assume  CON(AaC), ‘i.e.’ NEC(AoC). Take for the contingent second premiss BaC. Then (Bocardo3(NUU)) AoB, contradicting NEC(AaB). At 207,3-18 Alexander makes clear that the conclusion is not contingent in the way specified. 231. At 205,4-5. 232. i.e.  NEC(AaC). The point of this last sentence is probably to be understood in terms of Alexander’s remark at 206,22-4, which simply paraphrases Aristotle’s statement at 35b37-8 that the conclusion of Barbara1(NC_) will not be necessary. 233. i.e.  NEC(AoC). See Appendix 1 on the expression ‘by necessity’. At this point Wallies writes ‘nescio, quo vitio periodus turbetur’, but the sense of the text is clear enough. Alexander points out that NEC(AoC) is not equivalent to  CON(AaC), because the latter is entailed by either NEC(AiC) or NEC(AoC). He then points out that one could ‘transform’  CON(AaC) into either of NEC(AiC) or NEC(AoC), but only the latter works for getting a contradiction; see Appendix 5 on weak two-sided Theophrastean contingency. We have omitted from our translation words which could be rendered ‘which, if “A does not hold of some C by necessity” is not equivalent to “It is not contingent that A holds of all C”.’ 234. In connection with U+C combinations at 198,5-199,10. 235. Alexander points out that  NEC(AoC) denies NEC(AoC),  NEC(AiC) denies NEC(AiC), and that  NEC(AoC) is compatible with NEC(AiC). He illustrates the compatibility with  NEC(Moving o Body) (since all bodies might be in motion) and NEC(Moving i Body) (since it is necessary that the heavens move). 236. At 199,2-4. 237. Alexander gives a circle justification of Barbara1(NC‘C’): Assume  CON(AaC), ‘i.e.’, NEC(AoC); but NEC(AoC) and CON(BaC) yield (Bocardo3(NC‘C’)) ‘CON’(AoB), contradicting NEC(AaB). On this last paragraph see section III.E.3.a of the introduction. 238. 35b34-5. 239. On this lost work see Sharples (1987), p.1196. Alexander also refers to it at 125,30-1, 127,15-16, 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.

202

Notes to pp. 122-123

240. Aristotle asserts the completeness of Barbara1(CNC). 241. Note that this is not what Aristotle says. 242. On Alexander’s discussion of Aristotle’s indirect justification of Celarent1(NCCu) via Celarent1(NCU) NEC(AeB) CON(BaC) AeC see III.E.3.a of the introduction. The Greek behind the obelized words is in Ross’ text of Aristotle, but perhaps not in Alexander’s; see 209,22-5; for a more minor divergence see Appendix 6 (textual note on 36a11). We here summarize the argument as it appears in our text of Aristotle (which corresponds to Alexander’s initial representation at 208,8-209,3). Instead of just assuming the contradictory AiC of the conclusion AeC Aristotle considers two alternatives: (i) AaC AiC. He then (ii) takes the first premiss, converts it to NEC(BeA), and says that there follows (Celarent1(NUN) or Ferio1(NUN)): (iii) NEC(BeC) NEC(BoC), either of which is incompatible with CON(BaC). 243. cf. 126,33. 244. That is, the conclusion is not a necessary truth, but it follows necessarily from the premisses. 245. Alexander points to the use of first-figure NUN cases. 246. Alexander does a reductio to establish: Celarent1(NC  N  ) NEC(AeB) CON(BaC)  NEC  (AeC). He assumes NEC  (AeC), i.e., NEC(AiC), converts the major premiss to NEC(BeA) and uses Ferio1(NNN) to get NEC(BoC), contradicting CON(BaC). It is striking that he refers to  NEC  as ‘contingency in the way specified’; cf. the note on 199,7. Philoponus (in An. Pr. 205,13-27, Theophrastus 109A FHSG) ascribes a different reductio to ‘those around Theophrastus’. It involves converting NEC(AiC) to NEC(CiA) and applying Darii1(CNC) to infer CON(BiA), which conflicts with NEC(BeA), the converted form of NEC(AeB). 247. cf. 1.15, 34b27-35a2 with Alexander’s discussion, starting at 194,9. 248. The remainder of this section is more continuous than our breaking it into two paragraphs suggests. We have done this because the sense of our second paragraph is much clearer to us than the sense of our first. We discuss some textual issues in the first paragraph in the next note. Here we point out that the paragraph appears to be directed at Aristotle’s expressing the contradiction of the reductio argument with the words ‘But A is assumed to hold of all or some C, so that it is not contingent that B holds of [any or of] all C. But it was hypothesized to hold of all originally’. Aristotle should have said what Alexander represents him at 208,31-3 as saying: ‘But it was hypothesized originally that it is contingent that it holds of all.’ The author of this paragraph takes ‘It is not contingent that B holds of  all C’ to mean ‘It is contingent that B does not hold of all C’, which he says Aristotle takes to be equivalent to  (BaC), which makes a clear conflict with ‘But it was hypothesized to hold of all originally’, i.e., with BaC. The notion that Aristotle assumed BaC instead of the given premiss CON(BaC) leads to the notion of a U-for-C argument, which is given in the second paragraph. 249. The words ‘so that it is not contingent that B holds of all C’ appear to correspond to 36a14 ‘so that it is not contingent that B holds of [any or of] all C’, which Alexander’s earlier remarks suggest that he too read. We are not sure what to make of this difference, but we note that Becker (1933), p. 44 suggests (without referring to Alexander) that the original text may have been something like what Alexander says here. Nor do we know what to make of the words which follow in

Notes to pp. 123-125

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Wallies’ text (ho hupekeito [de] ex arkhês) and have emended the text to make them a continuation of what Aristotle says. 250. Alexander offers a U-for-C proof for Celarent1(NCU): assume  (AeC), i.e., AiC. But – transforming the minor contingent premiss into an unqualified one – BaC. Therefore (Disamis3(UUU)) AiB, contradicting NEC(AeB). 251. Aristotle asserts that Celarent1(CNC): CON(AeB) NEC(BaC) CON(AeC). is complete and that it is not possible to prove Celarent1(CNU) by reductio. The second claim is false since AiC and NEC(BaC) imply (Disamis3(UNN)) NEC(AiB), contradicting CON(AeB), a point which seems to have escaped Alexander’s notice. (Or perhaps he preferred to avoid invoking a controversial UNN syllogism.) His discussion is made complicated by what he calls the conciseness of the last sentence of this passage. 252. The lemma reads katêgorikê where our text of Aristotle has kataphatikê. 253. Ross emends ‘no’ to ‘some’. We quote his comment: ‘ In a23 tini must be right. The MSS. of [Alexander] record tini mê as a variant (210,32, but Al.’s commentary (ib. 32-4) shows that the variant he recognized was tini. mêdeni, the reading he accepts (210,21-30), is indefensible.’ 254. See, e.g., 205,2-8. 255. Again Alexander invokes the dictum de omni et nullo, saying that if CON(AeB) and NEC(BaC), then CON(AaB) and the C’s are included among the B’s, of all of which it is contingent that A is said; hence CON(AaC) and CON(AeC). 256. Alexander attempts to explain why no impossibility follows from AiC and CON(AeB), a combination (Festino2(CU_)) which is not discussed until chapter 18 at 37b39-38a2, where it is rejected. Presumably his point is that CON(AeC) is equivalent to CON(AaC) and it is ‘known’ that there are no second-figure syllogisms with two affirmative premisses. (The same point can be made about AaC and CON(AeB).) 257. Alexander turns to discuss Aristotle’s problematic sentence ‘For if A is hypothesized to hold of no C and it is also assumed that it is contingent that it holds of no B, nothing impossible results from these things’, which he thinks should say something like ‘For if the conclusion of Celarent1(CN_) is hypothesized to be A holds of no C and one posits the opposite of this AiC and it is also assumed that it is contingent that A holds of no B, nothing impossible results from these things.’ 258. mê bracketed by Wallies. 259. Alexander points out that no contradiction follows if AiC, the denial of the conclusion of Celarent1(CNU), is conjoined with BaC, but the premiss in question is actually NEC(BaC), with which a contradiction can be derived; see the first note on the lemma. 260. Disamis3(UUU) 261. Alexander mentions the following indirect way to establish Celarent1(CN‘C’). Assume  CON(AeC), ‘i.e.’, NEC(AiC). But NEC(BaC), so that (Disamis3(NNN)) NEC(AiB), contradicting CON(AeB). Alexander claims that we do not in this case have to take the conclusion to be  NEC  (AeC), since we can also rule out NEC(AoC), because NEC(AoC) and NEC(BaC) imply (Bocardo3(NNN)) NEC(AoB), which is incompatible with CON(AeB). See Appendix 5 on weak two-sided Theophrastean contingency. In our translation of the first words of this paragraph we have followed the Aldine. 262. Alexander perhaps adds the words in parentheses, which are irrelevant to

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Notes to pp. 125-128

the present argument, because in the next argument he will invoke an incompatibility with NEC(AoB). See the note on 211,14. 263. See 207,3-12 with the notes and Appendix 5 on weak two-sided Theophrastean contingency. 264. Because CON(AaB) is equivalent to the major premiss CON(AeB). 265. Aristotle accepts the waste case AEA1(NC‘C’). He rejects AE_1(CN_) and EE_1(CN_), using the same terms he used for their CU analogues at 1.15, 35a20-4; see 201,1-24 with the note. The next lemma after this one occurs at 215,19. 266. cf. 201,1-24 with the note on 201,1. 267. Aristotle offers the following argument for: Ferio1(NCU) NEC(AeB) CON(BiC) AoC Assume  (AoC), i.e., AaC. But NEC(AeB), so that (Cesare2(NUN)) NEC(BeC), contradicting CON(BiC). Alexander describes the argument, including the reduction of Cesare2(NUN) to Celarent1(NUN), and then points out that it depends upon Aristotle’s position on NUN in the first figure. At 212,22 he offers an ‘uncontroversial’ alternative derivation of Ferio1(NC  N  ). 268. For a minor textual difference between the citation and our text of Aristotle see Appendix 6 (textual note on 36a32). 269. Alexander actually says ‘the hypothesized conclusion’. 270. Alexander gives a reductio to establish: Ferio1(NC  N  ) NEC(AeB) CON(BiC)  NEC  (AoC) Assume  CON(AoC), ‘i.e.’, NEC(AaC). But NEC(AeB), so that (Cesare2(NNN)) NEC(BeC), contradicting CON(BiC). Alexander does not point out, but presumably takes for granted, that the legitimate conclusion here is  NEC  (AoC) rather than CON(AoC). 271. Aristotle affirms Ferio1(CNC) and Darii1(NC‘C’). 272. Alexander turns to Ferio1(CN_) until 213,27. 273. i.e., in Celarent1(CNC); see 209,35-210,8. 274. At 36a17-25; see the note on 209,33. 275. Alexander is looking at: Ferio1(CN_) CON(AeB) NEC(BiC), and asking whether one can show by reductio that the premisses imply AoC, that is whether either of CON(AeB) and NEC(BiC) together with AaC yields an impossibility. The first pair (Cesare2(UC_)) will be rejected by Aristotle in chapter 18 at 37b19-23, and the second pair yields only (Datisi3(UNU)) AiB, which does not contradict CON(AeB). 276. See 210,8-21. 277. Alexander now gives circle arguments validating Celarent1(NCN) and Ferio1(NCN). See III.E.3.a of the introduction. 278. cf. 207,28-33. 279. Disamis3(CCC). 280. Datisi3(CCC) 281. At 207,35-6. 282. Alexander turns to Darii1(NC‘C’). 283. For a minor difference between this citation and Ross’ text of Aristotle see the note on 36b1. 284. Aristotle asserts this about Barbara1(NC_) at 35b37-36a2. Alexander supplied a U-for-C proof at 206,29-207,3. He now gives a confusing argument (or pair of arguments) for: Darii1(NC‘C’) NEC(AaB) CON(BiC) CON(AiC). He assumes  CON(AiC), ‘i.e.’, NEC(AeC) and transforms CON(BiC) into BiC. He then says that these imply either AoB or ‘CON’(AoB). Either of these contradicts

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NEC(AaB), but the only sense we can make of his saying that his two assumptions might imply ‘CON’(AoB) is if he is imagining an alternative proof in which CON(BiC) is not transformed into BiC. In that proof one would use Ferison3(NCCu), which, of course, has not been dealt with yet and depends on Ferio1(NC_). (Ferison3(NCCu) could, of course, be said to yield either AoB and CONu(AoB), but applying it does not require transforming CON(BiC) into BiC.) When Alexander asserts that the consequence of NEC(AeC) and BiC is AoB, he apparently chooses to ignore the fact that Aristotle espouses (1.11, 31b31-3) Ferison3(NUN) rather than the Theophrastean Ferison3(NUU). Alexander also points out that the conclusion is not contingent in the way specified. 285. Accepting a conjecture of Wallies, which he does not print in the text; see Textual Emendations at the beginning of the volume (on 214,5). 286. Alexander now gives an argument for: Darii1(NCU) NEC(AaB) CON(BiC) AiC. Assume  (AiC), i.e., AeC. Alexander says that the consequence of AeC and CON(BiC) is either AoB or ‘CON’(AoB), either of which contradicts NEC(AaB). The second of these conclusions follows from Ferison3(UC‘C’), which Aristotle affirms at 1.21, 39b26-31. The first would follow only if CON(BiC) were transformed into BiC and Ferison3(UUU) applied. 287. Aristotle rejects IA_1(NC_), OA_1(NC_), IE_1(NC_), and OE_1(NC_), using the same terms as he used to reject CC and U+C combinations with a particular major premiss at 1.14, 33b3-8 and 1.15, 35b11-19; see the note on 171,14. He takes as true: NEC(Animal i White) NEC(Animal o White) CON(White a Human) CON(White e Human) CON(White a Cloak) CON(White e Cloak) NEC(Animal a Human) NEC(Animal e Cloak). 288. Aristotle rejects IE_1(CN_), OE_1(CN_), taking as true: CON(Animal i White) CON(Animal o White) NEC(White e Crow) NEC(White e Pitch) NEC(Animal a Crow) NEC(Animal e Pitch) and IA_1(CN_), and OA_1(CN_), taking as true: CON(Animal i White) CON(Animal o White) NEC(White a Swan) NEC(White a Snow) NEC(Animal a Swan) NEC(Animal e Snow). It is of some interest that Aristotle takes ‘Some white things are animals’ and ‘Some white things are not animals’ as contingent here, when he has just taken them as necessary. 289. Since indeterminate premisses are no stronger than particular ones, Aristotle needs that the two ‘conclusions’ are true, viz, NEC(Animal a Human) NEC(Animal e Soulless) and that the following premisses are true: 1. NEC(Animal i White) 2. NEC(Animal o White) 3. CON(Animal i White) 4. CON(Animal o White) 5. NEC(White i Human) 9. NEC(White i Soulless) 6. NEC(White o Human) 10. NEC(White o Soulless)

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7. CON(White i Human) 11. CON(White i Soulless) 8. CON(White o Human) 12. CON(White o Soulless) Aristotle does only the set for NEC(Animal e Soulless). He seems to assert the truth of Animal i White and White i Soulless, and then of 1, 2, 9, 10, and remarks that the case will be the same for 3, 4, 11, 12 290. i.e. whether they are affirmative or negative. 291. Alexander asserts the (de re) truth of 1 and 2. 292. At this point Alexander wishes to explain the truth of 3 and 4, which would seem to be incompatible with 1 and 2. He does so in an obscure way by claiming that CON(Human i White) and CON(Human o White) and ‘inferring’ 3 and 4. It does not seem possible to treat any of these propositions as de re true. 293. Alexander turns to 5-8 and treats them de re, taking it to be necessary that Ethiopians are black and Celts white, and to be contingent for most humans whether or not they are white. 294. We find the text here difficult, but take the sense to be this: Aristotle gives material terms for both NEC(AaC) and NEC(AeC), but he does only the latter case, showing that all of 1-4 and 9-12 hold, so that the connection both between A and B and between B and C is ‘indefinite’. The next lemma reproduces some of the text already discussed. It and the brief discussion, which seems to say falsely that Aristotle has done all the cases, may be a later insertion. 295. For a minor divergence from our text of Aristotle, see Appendix 6 (on 36b16). 296. Aristotle makes a general parallel between the U+C and N+C first-figure syllogisms, but contrasts Celarent1(UC  N  ) and Ferio1(UC  N  ) (and perhaps EEE1(UC  N  ) and EOO1(UC  N  )) with Celarent1(NCCu) and Ferio1(NCCu) (EEE1(NCCu) and EOO1(NCCu)). Alexander again raises the question whether the  N  conclusions should be considered contingent. 297. dêlon; our texts of Aristotle have phaneron. 298. Alexander refers to Aristotle’s treatment of Celarent1(UC  N  ) at 1.15, 34b19-35a2 and Ferio1(UC  N  ) at 1.15, 35a35-b8. Alexander suggests making a distinction between CONu conclusions which are really unqualified and  NEC conclusions which are not. 299. See Aristotle’s treatments of Celarent1(NCCu) at 36a7-17 and Ferio1(NCCu) at 36a32-9 with Alexander’s comments and the notes. 300. Alexander now argues that one cannot establish: Celarent1(UCU): AeB CON(BaC) AeC by an argument analogous to the reductio for Celarent1(NCCu); he points out that: (i) AiC and AeB imply (Festino2(UUU)) BoC, which is consistent with CON(BaC); (ii) AiC and CON(BaC) imply (Disamis3(UCC)) CON(AiB), which is consistent with AeB. He points out that (ii) would obviously be false if one could change CON(BaC) into BaC, as in a U-for-C argument; the same change would produce an inconsistency with (i). Alexander makes this point again immediately below at 216,34-217,7. 301. We read khrômenôn ; cf. 133,32 and 134,16. 302. This paragraph is compressed, but it essentially contains the information that, if one rejects Aristotle’s point of view on first-figure NU combinations and does not allow U-for-C substitution, there is no indirect justification for any of Celarent1(NCCu), Celarent1(UC‘C’), Ferio1(NCCu), Ferio1(UC‘C’) in which the negation of the purported conclusion and the major premiss are used to derive something incompatible with the minor, contingent premiss. Alexander refers to such reductios as using the second figure because the shortest ones would use it,

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but only a straightforward conversion of the major premiss is required for a reduction to the first figure. Alexander first refers to the straightforward second-figure reductio for Celarent1(NC  N  ): assume NEC(AiC) and infer with AeB that NEC(BoC), conflicting with CON(BaC). He points out that this argument won’t go through without Aristotle’s assumption about first-figure NU combinations ‘whether the negative premiss is necessary or unqualified’. He then says that without this assumption the reductios will yield either BoC (for the two Celarent1 cases), which is compatible with the minor premiss CON(BaC) or BeC (for the two Ferio1 cases), which is compatible with the minor premiss CON(BiC). 303. Alexander now looks at ‘third-figure’ reductios of Celarent1(UCU) AeB CON(BaC) AeC and Celarent1(NCU) NEC(AeB) CON(BaC) AeC In these one tries to infer from AiC and CON(BaC) an inconsistency with the major premiss. Given Disamis3(UCC) (or Darii1(CUC) and two II-conversions) one can infer CON(AiB), which is inconsistent with NEC(AeB). As Alexander points out a corresponding argument for Celarent1(UCU) requires applying U-for-C substitution to CON(BaC) and using Disamis3(UUU). 304. Alexander again asks whether the transformation of CON(BaC) into BaC isn’t ‘responsible’ for the contradiction since none results if the transformation isn’t made. He again insists that it is not, but what he says is cryptic. Cf. his discussion of 1.15, 34b7-18 at 188,18-193,21. 305. That is, although AiB is incompatible with the premiss AeB, it is not impossible, if the incompatibility results from the transformation of CON(BaC) into BaC. 306. This sentence is excised by Ross, who cites Maier. Alexander’s comment gives the reason. 307. The main interest in this chapter is the lengthy discussion (1.16, 36b351.17, 37a31) of the failure of EE-conversion for contingent propositions. Aristotle uses the non-convertibility result to argue (37a32-5) against the possibility of reducing Cesare2(CC_) to Celarent1(CCC) in the way Cesare2(UUU) was reduced to Celarent1(UUU). The argument obviously generalizes to Camestres2(CC_) and Festino2(CC_). Aristotle also gives an obscure argument (37a35-7) against the possibility of an indirect reduction. 308. For a minor discrepancy from our text of Aristotle see Appendix 6 (on 36b26). 309. Aristotle discusses U+C second-figure combinations in chapter 18 and N+C ones in chapter 19. The force of what he says here can be gathered by looking at the summaries of U+C and N+C syllogisms. 310. In chapter 5; cf. 28a7-9; cf. 227,16-17. Alexander’s point is that if a CC combination with at least one negative premiss yielded a conclusion, so would the combination with the negative premisses transformed into affirmative ones. But, of course, what is true of UU combinations need not be true of the corresponding CC combinations. The important point is that Aristotle’s transformation rules for contingent propositions make it impossible to reduce second-figure CC combinations to combinations already proved valid. Alexander’s perspective leads him to go on in the next paragraph to raise the following sort of question: why should one assume that AA_2(CC_) is non-syllogistic and infer that Cesare2(CCC) is nonsyllogistic rather than assuming that Cesare2(CCC) is syllogistic and inferring that AA_2(CCC) is syllogistic? 311. Alexander claims some kind of priority of affirmative over negative assertions; one should not undermine syllogistic combinations involving affirmative

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premisses on the basis of combinations involving negative ones, but the latter combinations need to be justified. The imperfect ‘it was said’ suggests an independent discussion. 312. That is, CON(XeY) is not of the form  P; cf. 158, 32-159,4. Alexander goes on to argue that somehow CON(XaY) is a prior formulation of CON(XeY). 313. i.e. the person who accepts CON(XeY) eo ipso accepts CON(XaY). 314. That is in 1.5 Cesare2(UUU), Camestres2(UUU), and Festino2(UUU) were established by conversion of the negative premiss, Baroco2(UUU) by reductio. 315. At 1.17, 36b35-37a31. 316. The discussion in the remainder of this section is not easy to follow. Alexander is trying to give a general explanation of why the valid second-figure U+C and N+C combinations are all and only those having their non-contingent premiss universal negative. Our notes – which deal only with the U+C cases since the N+C cases are completely analogous – indicate difficulties in construing what Alexander says. The easiest access to what he is trying to describe is through the summaries of the U+C and N+C cases. Alexander takes for granted that no second-figure syllogism has two affirmative or particular premisses. In this paragraph he rules out AE_2(UC_), EA_2(CU_), IE_2(UC_), EI_2(CU_), OA_2(CU_), and AO_2(UC_). 317. This description covers all the combinations accepted as syllogistic by Aristotle in chapter 18: Cesare2(UC‘C’), Camestres2(CU‘C’), Festino2(UC‘C’), IEO2(CU_), and the four which reduce to them by EA- or OI-transformationc: EEE2(UC‘C’), EEE2(CU‘C’), EOO2(UC‘C’), OEO2(CU_). 318. This rules out OA_2(UC_), AO_2(CU_), OE_2(UC_), and EO_2(CU_). However, the reasoning which follows is not clear to us. 319. If we assume that Alexander is only dealing with cases he hasn’t already discussed or taken for granted, then he is ruling out IE_2(CU_) and OA_2(CU_) (and not IA_2(CU_), which is a two-affirmative case or OE_2(CU_), which he has previously accepted). 320. This description covers four cases, two of which have already been ruled out: AA_2(CU_) and EA_2(CU_). The remaining two cases have already been accepted: Camestres2(CU‘C’) and EEE2(CU‘C’). When Alexander says these are proved by reductio ad absurdum, he presumably means that they are proved by reduction to Celarent1(UC  N  ), which is itself proved by reductio. He goes on to point out that there is no possibility of reducing them to a first-figure CUC combination because neither AA-conversionc nor EE-conversionc is valid, and, even if it were, the result of applying it to the first premiss would be a first-figure combination with an unqualified universal negative minor. 321. Probably a reference to 1.2, 25a7-10 in the general discussion of conversion in chapter 2. 322. At 36b35-37a31. 323. It is not clear what kind of syllogisms Alexander is talking about. He goes on to refer to two cases which he has already rejected: OA_2(UC) and AO_2(CU_). In these cases there can be no reduction to the first figure because the negative premiss will not convert and the affirmative will only convert to a particular affirmative or a universal negative premiss. 324. Alexander’s description applies to EEE2(UC‘C’), EOO2(UC‘C’), and OEO2(UC_), the first two of which he has already affirmed, the third of which he has rejected. The reduction he describes applies only to the first two cases. 325. In chapter 14. 326. In 1.15 and 16. 327. In 1.17 and 20.

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328. Alexander does not give any clear reason why one has to study first-figure mixtures before studying second-figure CC combinations. One possible reason is that Aristotle argues that there is no indirect justification for second-figure CCC combinations, but an indirect reduction would need to invoke a syllogism with a necessary (i.e. a  C) premiss; cf. 227,36-228,18. 329. That is, all the second-figure U+C and N+C syllogisms reduce ultimately to Celarent1(UC  N  ), Ferio1(UC  N  ), Celarent1(NCCu), or Ferio1(NCCu), so that for Alexander their conclusions are not contingent in the way specified. 330. For discussion of this passage see section III.D.2 of the introduction. 331. For a minor divergence from our text of Aristotle see Appendix 6 (on 36b35). 332. Aristole 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. 333. In chapters 2 and 3. The discussion of universal negatives is at 1.3, 25b3-25. 334. i.e. Aristotle is taking for granted EA-transformationc and arguing against EE-conversionc. Theophrastus accepts EE-conversionc and rejects EA-transformationc. 335. 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. 336. pantôn tôn tou B. On the terminology of disjointedness see the note on 124,20 in volume 1. 337. Aristotle’s argument is not a reductio, but what we have called an incompatibility rejection argument. 338. Theophrastus and Eudemus. 339. That is to say,  NEC  (AaB) is not equivalent to  NEC  (AeB). 340. 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 (on 37a4). 341. We read ek tês proeirêmenês kataskeuês with B where Wallies prints [ek] tês proeirêmenês kataskeuê, following the Aldine. 342. 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

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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. 343. ouk ex anankês panti. 344. 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. 345. endekhetai. 346. 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. 347. Alexander now suggests that Aristotle is accepting AO- and EI-transformationc, but not OA- and IE-transformationc. 348. i.e. the heaven. 349. 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. 350. On this Aristotelian passage see section III.D.2 of the introduction. 351. hoi hetaroi autou, Theophrastus and Eudemus. 223,3-15 are Theophrastus 102B FHSG. Alexander gives a 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. 352. 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. 353. A better formulation would be that neither (i) nor (iii) is a consequence of the denial of (iv). 354. In the alleged reductio justification of EE-conversionc. 355. endekhetai

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356. 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. 357. For a minor difference between the lemma and our text of Aristotle see Appendix 6 (on 37a18). 358. The correct implication is in the opposite direction, and what Alexander says establishes the correct implication. 359. 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. 360. 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 because NEC(CiD). Alexander takes it that Aristotle is assuming CaD and also  CON(CaD) because NEC(CiD). 361. For a minor difference between the lemma and our text of Aristotle see Appendix 6 (on 37a27). 362. See 224,18-27. 363. Perhaps Alexander’s clearest statement of the point that a logical rule must be true in every instance. 364. Aristotle takes up Cesare2(CC_). Cesare2(UUU) was reduced to Celarent1(UUU) at 1.5, 27a5-9 by converting the first premiss. The argument just given shows that CON(AeB) cannot be converted. Aristotle then turns to showing that a reductio argument won’t work either. The premisses of Cesare2(CC_) are: (i) CON(AeB) (ii) CON(AaC) and the potential conclusion presumably should be: (iii) CON(BeC) So the hypothesis for reductio would be: (iv)  CON(BeC) ‘i.e.’, (v) NEC(BiC) With (i) this implies (Ferio1(CNC)): (vi) CON(AoC) which is compatible with (ii), so there is no reductio. (Alexander does not consider the possibility of combining (v) with (ii), but that also yields no inconsistency.) However, the text makes it seem that Aristotle takes instead of (iv): (iva) CON(BaC) which makes no sense at all. Alexander first suggests (227,28-33) reading (iva) as: (ivb) NEC(BaC) He gives no explanation of how the text could possibly be understood this way, but he points out (227,36-228,17) that no impossibility follows from (ivb) and (i), so that certainly no impossibility follows from the weaker (v) and (i). Alexander considers as a second possibility (228,20ff.) that Aristotle is taking (iva) not as the denial of the purported conclusion of (i) and (ii), but as the conclusion itself, which it could be since (iii) and (iva) are equivalent for contingency in the way specified. Alexander more or less admits this point (228,26-30), but ends (228,30-7) by insisting that the conclusion ought to be (iii). In the course of the discussion he mentions

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(228,23-6) the possibility that the text ought to be read as ‘It is contingent that B does not hold of any C’ rather than ‘It is contingent that B holds of all C’; see Appendix 6 (on 37a35-6). 365. At 36b26-9. 366. Alexander presumably has in mind the fact that Aristotle has stated this for second-figure combinations with unqualified premisses at 28a7-9. But the fact that CON(XaY) and CON(XeY) convert alters the situation. See 218,3-7 with the note. 367. That is, the minor premiss in the alleged reductio argument using NEC(BaC) and CON(AeB). 368. Celarent1(CNC) or Ferio1(CNC) 369. As Alexander has made clear in the discussion of the first part of this chapter, he considers both NEC(BiC) and NEC(BoC) legitimate reductio hypotheses for establishing CON(BeC) or CON(BaC). He here points out that if NEC(BoC) is taken one gets the non-syllogistic pair EO_1(CN_) rather than Ferio1(CNC), but he does not make clear whether he is taking the purported conclusion to be CON(BeC) or (a possibility he takes up immediately) CON(BaC). 370. More literally ‘is truly predicated’ (alêthôs katêgoreitai). 371. Ferio1(CNC). 372. Aristotle assumes that the conclusion of Cesare2(CC_) must be contingent and gives terms for showing it is not. It would be sufficient for him to give terms verifying the premisses and some necessary proposition BC. But he chooses to give a set of terms and point out separately that they verify  CON(BaC) and  CON(BeC): CON(White e Human) CON(White a Horse)  CON(Human a Horse)  CON(Human e Horse). At 231,7-10 Alexander, who takes Aristotle to be showing that neither a contingent affirmative nor contingent negative conclusion follows, supplies these same terms to explain Aristotle’s rejection of Cesare2(CU_) and Camestres2(UC_) at 1.18, 37b19-23. But then at 232,10-36 he uses them to undermine Aristotle’s espousal of Cesare2(UC‘C’) and Camestres2(CU‘C’) at 1.18, 37b23-9. At 37b10 Aristotle generalizes his claim about Cesare2(CC_) to all second-figure CC combinations. 373. For a minor difference between the lemma and our text of Aristotle see Appendix 6 (on 37a38). 374. endekhetai. 375. dunata kai endekhomena. 376. At 1.15, 34a5-24. Note especially 23-4. However, Aristotle’s argument is a general theoretical one and does not establish the specific result Alexander invokes here. 377. i.e. if one is affirmative the other negative. 378. Alexander worries that someone might think Aristotle has failed to show that the second-figure CC combinations do not yield a necessary negative conclusion and produces terms to show they do not. He uses the first set, white, human, literate, in connection with U+C combinations in the way which we have described in the note on 229,1. 379. Aristotle rejects: Cesare2(CU_) CON(AeB) AaC Camestres2(UC_) AaB CON(AeC) and also some corresponding combinations with particular premisses which he takes up at 37b39. He indicates that the arguments will be the same as those which

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he used against the second-figure CC combinations. Alexander fills in some details. (The commentary on chapter 18 contains only two lemmas, this one and one at 233,13.) 380. For a minor divergence between the lemma and our text of Aristotle see Appendix 6 on 37b19. 381. cf. 218,3-7 and 227,16-17. In fact, Aristotle gives terms for rejecting AA_2(CU_) and AA_2(UC_) at 37b35-8. 382. Alexander points out that it would be illegitimate to use EE-conversionc or AA-conversionu to justify Camestres2(UC_) and Cesare2(CU_), and that the legitimate conversion of the unqualified premiss produces a non-syllogistic first-figure combination. 383. Alexander claims correctly that there is no indirect argument to show that these two combinations are syllogistic although he does not consider all possible indirect justifications. He focuses on the case of Camestres2(UC_), imagining a reductio in which the conclusion is assumed to be BeC (or CON(BeC)) and it is argued that AaB and BiC (NEC(BiC)) imply (Darii1(UUU) or Darii1(UNU)) AiC, which is perfectly compatible with CON(AeC). 384. The terms for Cesare2(CC_) from 1.17, 37b3-4 yield the following truths: CON(White e Horse) White a Horse White a Human CON(White e Human) NEC(Horse e Human) and so – according to Alexander – rule out any contingent or unqualified conclusion. To rule out the possibility of a necessary negative conclusion, Alexander adds as terms verifying NEC(BaC), the same terms he used for the same purpose in connection with Cesare2(CC_) at 230,14-17. CON(White e Human) White a Human White a Literate CON(White e Literate) NEC(Human a Literate). 385. For a close parallel to this remark see 204,15-19. 386. Aristotle describes a derivation of: Cesare2(UC‘C’) AeB CON(AaC)  NEC  (BeC). Convert AeB to BeA and apply Celarent1(UC  N  ). Unfortunately in this case he does not mention that the conclusion is not contingent in the way specified. For Alexander this problem comes to a head with Aristotle’s remark that the situation is the same for: Camestres2(CU‘C’) CON(AaB) AeC  NEC  (BeC). For in this case after converting AeC and applying Celarent2(UC  N  ) one has ‘CON’(CeB), which cannot be converted to ‘CON’(BeC) if we are dealing with contingency in the way specified. 387. This characterization is more appropriate for the contingency which we represent by CONu, but, as we have indicated, Alexander tends to think of this kind of contingency as the same as the one we represent by  NEC  . Of course, either kind of contingency will permit EE-conversion. 388. At 1.15, 34b19-35a2. 389. Alexander puts this too strongly.  NEC  (BeC) is not unqualified, but it – unlike CON(BeC) on Alexander’s usual construal – is compatible with BeC, and it does convert with  NEC  (CeB). Elsewhere (e.g. at 245,30-2) Alexander recognizes that  NEC(  P) is not ‘directly’ unqualified. 390. Aristotle does not make this clear in the sequel, but Alexander works hard to make him do so. See 233,15-23 and perhaps 232,36-233,12. 391. 1.17, 36b33-4 in Aristotle’s general account of second-figure combinations with a contingent premiss.

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392. Alexander considers the possibility that the conclusion reached applying Celarent1(UC_) is CON(CeB) and that this could be converted to CON(BoC) to make the major term the predicate of the conclusion. He suggests that this conversion is possible because CON(CeB) is equivalent to CON(CaB) and that converts to CON(BiC). But one does not need the analogy to justify the conversion since CON(BiC) converts with CON(BoC). Elsewhere (e.g. at 234,3-4) Alexander straightforwardly asserts OO-conversionc. In any case the point is of little significance since the conclusion of Celarent1(UC_) is not contingent in the way specified. 393. Alexander now makes a decisive objection to Aristotle’s modal syllogistic, although he does not make it as decisively as one might like: using terms which Aristotle first used in connection with Cesare2(CC_) at 1.17, 37b3-4 and apparently again in connection with Cesare2(CU_) and Camestres2(UC_) at 37b19-23 (see the note on 229,1) he offers the following counterinterpretation to Cesare2(UC‘C’): (i) White e Human (ii) CON(White a Horse) (iii) NEC(Human e Horse). Then, taking his own terms used in the same connection at 230,14-17 and 231,1619, he gives a second counterinterpretation: (i) White e Human (ii) CON(White a Literate) (iii) NEC(Human a Literate). Alexander worries about shortcomings of the second terms on the grounds that everything literate can’t be white when no humans are white (cf. the discussion of 1.15, 34b7ff. starting at 188,20). Instead of rejecting this suggestion, he decides that he can do without the second interpretation because the premisses shouldn’t be compatible with a necessary conclusion. Alexander goes on to use the first interpretation against Camestres2(CU‘C’): CON(White a Human) White e Horse NEC(Human e Horse). In this case he considers the objection that ‘No horse is white’ is not a satisfactory interpretation of an unqualified proposition and rightly rejects it. 394. Here again Alexander appears to be imagining the realization of a contingent proposition at a given time. See section III.D.1 of the introduction. 395. Adopting Wallies’ conjecture of deikhthen for lekhthen. 396. As we have seen, Alexander is quite right that the conclusions of Camestres2(CU_) and Cesare2(UC_) are not contingent in the way specified because they are justified using Celarent1(UC  N  ). He now cites Aristotle’s treatment of Celarent1(NCCu) at 1.16, 36a7-17 and perhaps of Ferio1(NCCu) at 36a39-b2 as a confirmation of his view, again not distinguishing CONu and  NEC . He is particularly interested in Aristotle’s words at 35b28-36, which he takes to reaffirm that the conclusion of Celarent1(UC‘C’) is not contingent in the way specified. But he ends this section with another expression of uncertainty. 397. cf. 232,10-23. Alexander remains caught in his misunderstanding of interpretations. He suspects that only his interpretation verifying NEC(BeC) is legitimate and that this shows that the two combinations may imply NEC(BeC). He thinks that this means that the conclusion is contingent but not contingent in the way specified. 398. At 1.12, 32a6-14. 399. 1.16, 35b34-5. 400. Aristotle affirms EEE2(CU‘C’) and EEE2(UC‘C’). In either case one applies EA-conversionc to the contingent premiss yielding the premisses of Cames-

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tres2(CU‘C’) or Cesare2(UC‘C’). Alexander insists correctly that the conclusion is not contingent in the way specified. (For a minor difference between the lemma and our text of Aristotle see Appendix 6, on 37b30.) 401. i.e. the ultimate reduction in both cases is to Celarent1(UC  N  ). 402. Aristotle rejects AA_2(CU_) and AA_2(UC_), taking the same propositions as contingent or unqualified. Alexander does the former case a little more explicitly. 403. Aristotle does not explicitly discuss particular combinations with two affirmative premisses; nor does Alexander. Presumably they are rejected. 404. Aristotle’s words imply rejection of all of: Festino2(CU_) IE_2(UC) OA_2(CU_) Baroco2(UC_) and then acceptance of Festino2(UC‘C’) AeB CON(AiC)  NEC  (BoC) CON(AiB) AeC  NEC  (BoC) IEO2(CU‘C’) OA_2(UC_) Baroco2(CU_) However, he rejects the last two of these at 1.18, 38a8-10. Alexander discusses the two accepted cases first. The situation with Festino2(UC‘C’) is, he says, known. That is, one converts AeB to BeA and invokes Ferio1(UC  N  ) to get  NEC  (BoC). Neither Aristotle nor Alexander mentions anything about the nature of the conclusion. However, IEO2(CU‘C’) raises more serious difficulties for Alexander than Camestres2(CU‘C’) did earlier, since after the conversion of AeC to CeA and the application of Ferio1(UC  N  ) one is left with  NEC  (CoB), which, Alexander points out, cannot be converted unless the conclusion is contingent in the way specified. He ends by suggesting that perhaps Aristotle doesn’t intend IEO2(CU_) to be syllogistic at all since no second-figure UU combination with a particular major premiss was syllogistic. 405. i.e. both II-conversionc and OO-conversionc hold. 406. Alexander presumably means that the reducing syllogism Ferio1(UC N ) was ‘shown’ by Aristotle not to be contingent in the way specified. Aristotle asserted this to be the case at the beginning of chapter 15. 407. Alexander gives the terms used by Aristotle at 1.18, 37b36-8 to reject AA_2(UC_) and AA_2(CU_) to reject all the combinations rejected by Aristotle at 37b39-38a2 and says that none of them could be shown to be syllogistic by reductio. 408. Aristotle apparently accepts EOO2(UC‘C’) and OEO2(CU‘C’), which reduce to Festino2(UC‘C’) and the problematic IEO2(CU‘C’) by OI-transformationc. 409. Aristotle rejects OA_2(UC_), Baroco2(CU_), OE_2(UC_), and EO_2(CU_). Alexander says only that the unqualified o-premiss cannot be converted. 410. For an overview of this chapter see section III.E.3.b of the introduction. 411. Aristotle announces the validity of Cesare2(NCCu) and Camestres2(CNCu), and denies that of Cesare2(CN_) and Camestres2(NC_). 412. Alexander is thinking of the cases Celarent1(UC  N  ) and Celarent1(NCCu). Only the latter case is clearly relevant in the present context, but Alexander is always concerned about the parallels between the U+C and the N+C cases because of Aristotle’s indefiniteness about contingency. Celarent1(NCCu) is treated at 1.16, 36a7-17 and described at 35b30-4. For complications see sections III.E.3.a of the introduction. 413. Aristotle does: Cesare2(NCCu) NEC(AeB) CON(AaC) CONu(BeC).

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Notes to pp. 155-157

This clearly has a straightforward reduction to Celarent1(NCCu) if one converts the major premiss to NEC(BeA). Aristotle expresses this conversion as if it were to BeA, but Alexander takes it for granted that he intends the proper conversion, and that he infers ‘It is contingent (but not in the way specified) that BeC’. Aristotle, however, gives a separate argument that BeC follows. He assumes BeC), i.e., BiC, and NEC(AeB), from which it follows (Ferio1(NUN)) that NEC(AoC), contradicting CON(AaC). Alexander points out that this argument depends upon the non-Theophrastean Ferio1(NUN). 414. The lemma has oun where our text of Aristotle has gar. 415. Reading the oukh huparkhei of Alexander (235,9) and some mss. Ross prints oud’ huparxei, but does not record Alexander’s citation. 416. cf. 1.10, 30b9 with Alexander’s commentary at 136,20-9. 417. In chapter 9. 418. The next paragraph is Theophrastus 108B FHSG. 419. Aristotle briefly affirms: Camestres2(CNCu) CON(AaB) NEC(AeC) CONu(BeC), for which Alexander supplies the following derivation. He converts the minor premiss to NEC(CeA) and invokes Celarent1(NCCu) to get CONu(CeB), which – since it is really unqualified – converts to CONu(BeC). Alexander points out that such a conversion is not possible for CON(CeB), and then that Theophrasteans are able to use Celarent1(CN  N  ) to infer  NEC  (CeB), which does convert to  NEC  (BeC). So the Theophrasteans have Camestres2(CN N ). 420. For a minor divergence between the text of the lemma and our text of Aristotle see Appendix 6 (on 38a25). 421. With this paragraph cf. 232,5-9. Alexander’s point is that if the conclusion reached before the conversion were CON(CeB), then one could only transform it into CON(CaB) and then into CON(BiC) (and perhaps (cf. 234,3-4) into CON(BoC)). 422. 236,11-14 constitute Theophrastus 109C FHSG. 423. With this passage one should look at Aristotle’s rejection of Cesare2(CC_) at 1.17, 37a38-b10 with the note on 229,1. Aristotle here wishes to show that the premisses of: Cesare2(CN_) CON(AeB) NEC(AaC) are not syllogistic. He gives the following interpretation: CON(White e Human) NEC(White a Swan) NEC(Human e Swan) which he takes to show that no contingent conclusion follows from the premisses. He gives an argument that no necessary conclusion does either and then offers what appears to be the following interpretation: CON(Change a Animal) NEC(Change a Awake) Animal a Awake The first premiss here should be: CON(Change e Animal) Alexander (237,9-10) handles this difficulty by saying that Aristotle takes for granted EA-transformationc. He changes Animal a Awake to NEC(Animal a Awake) without comment, but he expresses misgivings about the perspicuousness of this proposition, and substitutes walking for being awake at 237,13-16. If Aristotle’s interpretations are acceptable, they suffice to show that the premisses of Cesare2(CN_) yield no conclusion, so that Aristotle could be finished at this point. He, however, adds the unnecessary remark that he has shown that

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the premisses don’t imply BeC (38b2-3) or the ‘opposite affirmations’ (38b3-4). Alexander takes up the last remark in the next section of the commentary. At 238,29-34 Alexander points out that the Theophrastean Cesare2(CN  N ) can be proved. 424. i.e. ‘there is an interpretation under which the premisses are true and’. 425. This is the upshot of chapters 8 and 10; for a brief formulation for the N+U cases see 1.10, 30b7-9. Alexander interprets Aristotle as saying or assuming that a N+C combination cannot yield a necessary conclusion in cases where the corresponding N+U combination does not. He adds that N+C combinations never yield a necessary conclusion. 426. cf. 154,23-155,2. 427. As we have seen, whatever Aristotle means by this remark (he makes a similar statement at 38b21-2), he doesn’t need it to argue that there will be no syllogism with the premisses: CON(AeB) NEC(AaC). Alexander mentions two accounts of what Aristotle says. According to the first Aristotle has shown that neither NEC(BeC) nor BeC can be inferred and now points out that their opposites, both their ‘contradictories’, CON(BiC) and BiC, and their ‘contraries’ CON(BaC) and BaC, cannot be inferred either. On the second interpretation Aristotle has ruled out a universal conclusion and now says that one will not be able to infer the opposites of NEC(BaC) and NEC(BeC), understood as the CON(BoC) and CON(BiC). An unclear (but acceptable) generalization is made to all particular propositions, and it is concluded that nothing can be inferred from the premisses. A striking feature of this passage is that Alexander treats CON( P) as the contradictory of NEC(P), when he normally insists that they may be false together. 428. It does not seem that Alexander could have any specific passage in mind here. 429. Alexander considers interpreting kataphaseôn as phaseôn (= protaseôn), a reading which has found its way into some mss. Both readings were already known to Philoponus (in An. Pr. 226,5). 430. cf. Bonitz (1870), 813a24-34. 431. Having rejected Cesare2(CN_), Aristotle now says there will be a similar rejection for: Camestres2(NC_) NEC(AaB) CON(AeC). Alexander points out that, with a variation in order, the same terms used to verify the premisses of Cesare2(CN_) and NEC(BeC) can be used to verify the premisses of Camestres2(NC_) and NEC(BeC). But the same is not true for NEC(BaC), since although it may be true that: NEC(Change a Awake) CON (Change e Animal) it is not true that NEC(Awake a Animal) Alexander is clearly right to say this since any interpretation verifying NEC(BaC) and NEC(AaB) will falsify CON(AeC). Alexander first says that the one counterinterpretation is sufficient, but he goes on to give proofs for both Camestres2(NC  N  ) and Cesare2(CN  N  ). He refers the reader to his work on mixtures for a solution to the impasse. 432. See 236,29-31 with note. Alexander’s point would seem to be again that the one interpretation with a necessary conclusion rules out all contingent conclusions. 433. Alexander gives a reductio argument for: Camestres2(NC  N  ) NEC(AaB) CON(AeC)  NEC  (BeC)

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Notes to pp. 160-161

He assumes  CON(BeC), ‘i.e.,’ NEC(BiC), and infers (Darii1(NNN)) NEC(AiC), contradicting CON(AeC). He does not remark that the conclusion is not contingent in the way specified. He then offers two reductio arguments for: Cesare2(CN  N  ) CON(AeB) NEC(AaC)  NEC  (BeC). He assumes NEC  (BeC), i.e., NEC(BiC) and uses either Ferio1(CNC)) to infer CON(AoC), contradicting NEC(AaC) or Disamis3(NNN) to infer NEC(AiB), contradicting CON(AeB). It is not clear to us why Alexander leaves out the alternative demonstration for Camestres2(NC  N  ), converting NEC(BiC) to NEC(CiB) and using Ferio1(CNC) to get CON(AoB), contradicting NEC(AaB). We note also that one could give similar derivations for: Festino2(CN  N  ) CON(AeB) NEC(AiC)  NEC  (BoC), which Aristotle presumably rejects although neither he nor Alexander mentions it specifically. For if one assumes NEC(BaC) one can use Disamis3(NNN) to get NEC(AiB), contradicting CON(AeB) or Celarent1(CNC) to get CON(AeC), contradicting NEC(AiC). 434. Aristotle now asserts that both of: EE_2(NC_) NEC(AeB) CON(AeC) EE_2(CN_) CON(AeB) NEC(AeC) yield a conclusion, presumably CONu(BeC). He says the second pair is treated similarly to the first. For the first he transforms the premisses into NEC(BeA) and CON(AaC), the premisses of Celarent1(NCCu). Alexander points out that the conclusion is not contingent in the way specified. Alexander also points out that in the second case the same two conversions yield CON(AaB) and NEC(CeA), the premisses of Celarent1(NCCu), but in this case the conclusion has C as predicate and so has to be converted, a conversion which is legitimate if the conclusion is either CONu(CeB) or  NEC  (CeB), but not if it is CON(CeB). 435. For a small textual point here see Appendix 6 (on 38b10). 436. 1.16, 36a7-17. 437. Reading triôn for Wallies’ duo. This may be a slip by Alexander; it is corrected eight lines below at 239,18. 438. Aristotle now apparently rejects: AA_2(NC_) NEC(AaB) CON(AaC), AA_2(CN_) CON(AaB) NEC(AaC), although he explicitly does only AA_2(NC_). He first rules out NEC(BeC) and BeC as conclusions on the ground that either would require a negative necessary or unqualified premiss. Alexander understands him to be ruling out all negative conclusions with this remark, contingent negative premisses being irrelevant since they are equivalent to contingent affirmative ones. He is therefore somewhat perplexed by Aristotle’s next remark ruling out CON(BeC), since that is a ‘negative’ conclusion. In any case Aristotle gives terms verifying the premisses of AA_2(NC_) and NEC(BeC). These terms take as true: NEC(White a Swan) CON(White a Human) NEC(Swan e Human). The same terms will work for AA_2(CN_) if Human and Swan are interchanged. So any negative conclusion is ruled out, but so is an affirmative one since NEC(BeC) is incompatible with any affirmative one. At 239,39 Alexander seems to say that he will give terms verifying NEC(BaC) and the premisses of AA_2(NC_), but he actually verifies the premisses of AA_2(CN_), taking as true:

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CON(Change a Animal) NEC(Change a Walking) NEC(Animal a Walking). In fact, he could not give terms verifying NEC(BaC) and the premisses of AA_2(NC_) since if he verifies NEC(BaC) and NEC(AaB), he will verify NEC(AaC) and hence not verify CON(AaC) if contingency is taken in the way specified. At the end of his remarks on the passage Alexander points out – but draws no moral from it – that the premisses of AA_2(NC_) actually yield  NEC  (BeC). 439. cf. 38a30-2. 440. Alexander now derives: AAE2(NC  N  ) NEC(AaB) CON(AaC)  NEC  (BeC) Assume NEC  (BeC), i.e., NEC(BiC). Then Darii1(NNN), NEC(AiC), contradicting CON(AeC), which follows from (AE-transformationc) CON(AaC). Obviously he could also verify: AAE2(CN  N  ) CON(AaB) NEC(AaC)  NEC  (BeC) Assume NEC  (BeC), i.e., NEC(BiC). Then NEC(CiB) and (Darii1(NNN)) NEC(AiB), contradicting CON(AeB), which follows from (AE-transformationc) CON(AaB). 441. Aristotle apparently asserts that both: Festino2(NCCu)NEC(AeB) CON(AiC) CONu(BoC) IE_2(CNCu)CON(AiB) NEC(AeC) CONu(BoC) can be proved syllogistic by conversion. This is clearly correct in the first case since if we convert NEC(AeB) to NEC(BeA), we can apply Ferio1(NCCu). Alexander remarks that for Aristotle the conclusion is CONu(BoC). He does not point out that one could also establish Festino2(NC  N  ), but it may well be that he is taking this point for granted. He does not discuss the second case, which involves the same difficulty as IE_2(CU_): if we convert NEC(AeC) to NEC(CeA), we get ‘CON’(CoB), which can be converted to CON(BoC) only if the conclusion of Ferio1(NC_) is contingent in the way specified, which it is not. See Alexander’s discussion of IE_2(CU_) at 233,34-234,11. 442. For a minor divergence between the lemma and our text of Aristotle see Appendix 6 (on 38b24). 443. Aristotle asserts that neither of: Baroco2(NC_) NEC(AaB) CON(AoC) OA_2(CN_) CON(AoB) NEC(AaC) yields a conclusion on the grounds that he has already shown that they do not yield one when the contingent premiss is made universal, that is, Camestres2(NC_) and Cesare2(CN_) do not yield a conclusion. Alexander gives a reductio argument that the first combination yields  NEC  (BoC). Assume NEC (BoC), i.e., NEC(BaC). Then (Barbara1(NNN)), NEC(AaC), contradicting CON(AoC). (The other combination can be proved syllogistic only if it is assumed that corresponding necessary and contingent particular propositions are incompatible.) 444. Baroco2(NUU), treated at 1.10, 31a10-15. 445. Aristotle says literally that the conclusion will be both of being contingent (tou endekhesthai) and of not holding. Alexander points out that ‘being contingent’ must be understood as contingently not holding (tou endekhesthai mê huparkhein). 446. At 1.19, 38a26-b5, but for complications see above 236,15-238,38. Alexander’s discussion in this paragraph only applies to Camestres2(NC_), not Cesare2(CN_), which Aristotle described as disprovable in a similar way. 447. The premiss in question is CON(AoC). Alexander’s argument does not require CON(AiC) as well. However, he does need this at 241,5-9 below.

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448. Aristotle now asserts that none of: AI_2(NC_) NEC(AaB) CON(AiC) AI_2(CN_) CON(AaB) NEC(AiC) IA_2(NC_) NEC(AiB) CON(AaC) IA_2(CN_) CON(AiB) NEC(AaC) yield a conclusion, presumably on the ground that he showed at 1.19, 38b13-23 that the pairs yield no conclusion when the particular premiss is made universal. But see the discussion at 239,20-240,11. Alexander gives a reductio argument that the premisses of AI_2(NC_) yield  NEC  (BoC). Assume  CON(BoC), ‘i.e.,’ NEC(BaC). Since NEC(AaB), it follows by Barbara1(NNN) that NEC(AaC), contradicting CON(AoC), which follows from CON(AiC) by OI-transformationc. There is an analogous derivation for AI_2(CN_): Assume  CON(BoC), ‘i.e.,’ NEC(BaC). Since NEC(AiC), NEC(CiA) and it follows by Darii1(NNN) that NEC(BiA), i.e., NEC(AiB), contradicting CON(AaB). The two remaining cases can be proved only if it is assumed that NEC(AiB) and CON(AiB) are incompatible. 449. Aristotle apparently asserts that both of: EO_2(NC_) NEC(AeB) CON(AoC) OE_2(CN_) CON(AoB) NEC(AeC) yield a conclusion when the contingent premiss is ‘converted’, but he does not say what the conclusion is. An application of OI-transformationc reduces these to the cases affirmed at 38b2427; see the note on 240,12. Alexander does the ‘good’ case, EO_2(NC_), carrying out the full reduction to Ferio1(NCCu). For a minor textual issue see Appendix 6 (on 38b32). 450. At 1.16, 36b12-18; see Alexander’s discussion of that passage at 215,3-28 with the notes. The propositions to be verified here are the same as those to be verified there except for the interchange of subject and predicate in the major premiss, i.e., the only difference is that Alexander now must verify: 1’. NEC(White i Animal) 2’. NEC(White o Animal) 3’. CON(White i Animal) 4’. CON(White o Animal) 451. Alexander verifies 1’ and 2’ understood de re. For his probable understanding of 3’ and 4’ see his treatment of CON(White i Human) and CON (White o Human) at 215,10-13 with the note. 452. Alexander turns to the NEC(BeC) case, for which there is really nothing new to do. He mentions only four of the eight relevant premisses. 453. Alexander presumably has in mind the treatment of AA_2(NC_) and AA_2(CN_) at 1.19, 38b13-23. There Aristotle only mentions the terms for holding of none by necessity, but Alexander supplies the other terms at 239,39-240,4. Aristotle had given both sets of terms in his treatment of Cesare2(CN_) at 38a26b4. It is not clear why Alexander thinks these terms might do. He can at best verify NEC(White i Swan), and not NEC(White o Swan) or CON(White o Swan). 454. Aristotle reasserts that the only N+C syllogisms in the second figure are Cesare2(NCCu), Camestres(CNCu), EEE2(NCCu), EEE2(CNCu), Festino2(NCCu), IEO2(CNCu), EOO2(NCCu), and OEO2(CNCu), and he reaffirms the parallelism between the U+C and N+C cases. For a minor textual difference between the lemma and our text of Aristotle see Appendix 6 (on 38b39). 455. Alexander here uses the two words which we translate ‘combination’, suzugia and sumplokê. With this brief passage cf. 254,12-14, where Alexander opts for something close to the first alternative. 456. i.e., Celarent1 and Ferio1.

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457. To get a sense of the content of this passage one should consult the summary. If one restricts attention to the standard syllogistic combinations Aristotle appears to be saying that all third-figure pairs with one or two contingent premisses yield a conclusion which is contingent in the way specified except for Felapton3(NC_), Ferison3(NC_), and Bocardo3(NC_), which yield CONu conclusions. 458. In chapter 17. 459. Aristotle used the expression ‘not in the way specified’ in connection with Celarent1(UC  N  ) and Ferio1(UC  N  ) at 1.15, 33b28-33; cf. 34b27-31 and 35a35-b2. In the case of Celarent1(NCCu) and Ferio1(NCCu) he said that the conclusion is of not holding and of contingency (1.16, 35b30-34; cf. 36a7-17 and 32-9). 460. Aristotle first does: Darapti3(CCC) CON(AaC) CON(BaC) CON(AiB) by converting the second premiss and invoking Darii1(CCC). 461. Aristotle affirms: Felapton3(CCC) CON(AeC) CON(BaC) CON(AoB) since Ferio1(CCC) results when the second premiss is converted. 462. Aristotle affirms: EEI3(CCC) CON(AeC) CON(BeC) CON(AiB) essentially reducing the combination to Darapti3(CCC) by transforming both premisses. Alexander points out that Aristotle could have justified: EEO3(CCC) CON(AeC) CON(BeC) CON(AoB) by reducing it to Felapton3(CCC) by just transforming the second premiss. He does not point out that Aristotle could also verify, e.g. AEI3(CCC) CON(AaC) CON(BeC) CON(AiB) with the same kind of reasoning. Of course, he could also give terms for rejecting all third-figure CC syllogisms, e.g., for holding of all by necessity: A Animal, B Human, C Moving; for holding of none by necessity: A Horse, B Human, C Moving. The same terms will work for rejecting all third-figure U+C combinations. 463. Aristotle’s remark here suggests that he will recognize the validity of Datisi3(CCC), Disamis3(CCC), Ferison3(CCC), and Bocardo1(CCC) and of no other particular syllogisms with two contingent premisses, exactly one of which is universal. But at 1.20, 39a38-b2 he claims that EO_3(CC_) and OE_3(CC_) are syllogistic. Here Alexander hints at the solution he will suggest somewhat more explicitly at 244,14-32, but finally take back: the premisses of EOO3(CCC) are ‘really’ those of Ferison3(CCC) and those of OEO3(CCC) are those of Disamis3(CCC). This is the last lemma in our text of Alexander for this chapter. For two possible minor differences between Alexander’s text of this passage and our text of Aristotle see Appendix 6 (textual notes on 39a29 and 30). 464. Aristotle does: Datisi3(CCC) CON(AaC) CON(BiC) CON(AiB) by converting the minor premiss and invoking Darii1(CCC). At 249,25-32 Alexander uses Datisi3(CCC) to establish Cesare2(N‘C’N), a mood rejected by Aristotle. 465. Disamis3(CCC) CON(AiC) CON(BaC) CON(AiB) is reduced to Darii1(CCC) by converting the major premiss. But in this case, as Alexander points out, the resulting conclusion has to be converted. 466. Here Aristotle seems to announce the validity of both: Ferison3(CCC) CON(AeC) CON(BiC) CON(AoB) Bocardo3(CCC) CON(AoC) CON(BaC) CON(AoB) Alexander does only the former case, reducing it to Ferio1(CCC) by converting the minor premiss to CON(CiB). He never mentions the problematic Bocardo3(CCC).

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Ross (ad loc.) proposes a derivation for OAI3(CCC) by converting CON(AoC) to CON(AiC) and then to CON(CiA), using Darii1(CCC) to get CON(BiA) and converting that to CON(AiB). Alexander’s remarks in the next section suggest that he would prefer to ignore OA_3(CCC). 467. See 1.14, 33a25-7. 468. Aristotle now does what Alexander takes to be: EOO3(CCC) CON(AeC) CON(BoC) CON(AoB) OEO3(CCC) CON(AoC) CON(BeC) CON(AoB) Alexander does the first by transforming the minor premiss to CON(BiC), which yields Ferison3(CCC), just reduced to Ferio1(CCC). He says that in the second case if the major premiss is transformed to CON(AiC), Ferison3(CCC) will yield CON(BoA), which can be converted to CON(AoB). Alexander points out that this latter conversion has no analogue for unqualified propositions, so that the Aristotle’s claim of a parallelism between third-figure UUU and CCC particular syllogisms at 39a28-31 breaks down. 469. The only other lemma in this chapter is at 246,11. For a minor textual point see Appendix 6 (on 39b7). 470. 1.20, 39a5-8. There Aristotle says only that the conclusion will be contingent. But what Aristotle says in this chapter leads Alexander to conclude that he has abandoned the distinction between  NEC  and CON, and is treating both as contingency in the way specified. See 245,16-35, and note Alexander’s references back to 1.20, 39b14-16 at 246,8, 9, 22, and 34-5, and 247,6-9. Alexander is sufficiently uncertain or reverential to refrain from judgment when Aristotle blunders at 39b22-5. See 246,13-36. 471. Aristotle does: Darapti3(UC‘C’) AaC CON(BaC)  NEC  (AiB), converting CON(BaC) to CON(CiB) and using Darii1(UC‘C’) to infer the conclusion. 472. 1.15, 33b25-33. 473. At 1.18, 37b28 Aristotle describes the conclusion of Cesare2(UC‘C’) as ‘It is contingent that B holds of no C’, but he doesn’t say anything about the nature of the contingency. 474. Aristotle now does: Darapti3(CUC) CON(AaC) BaC CON(AiB) CON(AeC) BaC CON(AoB) Felapton3(CUC) Felapton3(UC‘C’) AeC CON(BaC)  NEC  (AoB) converting the minor premiss and invoking Darii1(CUC), Ferio1(CUC), and Ferio1(UC‘C’). Again Aristotle marks no distinction between contingency and non-necessity. 475. We have moved 246,4-10 after the next lemma, where they belong. 476. Aristotle apparently announces that the following combinations are syllogistic: (i) AE_3(UC_) AaC CON(BeC) (ii) EE_3(UC_) AeC CON(BeC) (iii) AE_3(CU_) CON(AaC) BeC CON(AeC) BeC (iv) EE_3(CU_) (i) and (ii) are waste cases of Darapti3(UC‘C’) and Felapton3(UC‘C’), and so imply: (ia)  NEC  (AiB) (iia)  NEC  (AoB) (iv) is a waste case of (iii) or vice versa, but, as Alexander indicates, neither can be shown to imply syllogistically a conclusion with A as predicate and B as subject. 477. Alexander first considers cases (i) and (iii). He imagines converting the

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major premiss to CiA or CON(CiA) and applying Ferio1(CUC) or Ferio1(UC  N ) to get CON(BoA) or  NEC  (BoA). The former of these converts to: (ib) CON(AoB), but the latter does not convert, as Alexander’s remarks at 246,17-25 and 30-5 make clear. He is, however, sufficiently confused by Aristotle’s lax treatment of contingency and perhaps sufficiently uncertain of his own logical acumen to leave open the question whether the conversion is possible. 478. 1.20, 39b14-16. 479. Alexander now suggests that Aristotle only intended to do cases (i) and (ii), essentially as waste cases of Darapti3(UC‘C’) and Felapton2(UC‘C’). He points out that an analogous procedure will not work for (iii) and (iv). He repeats the procedure he has already described for (iii) and suggests that Aristotle now holds that the conclusion of Ferio1(UC  N  ) is contingent in the way specified. 480. Alexander reaffirms the feasibility of validating case (ii), but points out that combination (iv) will not be syllogistic unless it is legitimate to transform CON(AeC) into CON(AaC), convert that to CON(CiA), apply Ferio1(UC  N  ) to get  NEC  (BoA) and pretend that that can be converted to CON(AoB). 481. Aristotle apparently asserts the validity of: Datisi3(UC‘C’), Datisi3(CUC), Disamis3(UCC), Disamis3(CU‘C’), Ferison3(UC‘C’), Ferison3(CUC), IE_3(UCC), and IE_3(CU‘C’). Alexander points out that in each case conversion of the particular premiss produces a first-figure syllogism with the desired conclusion. He does not indicate that in the last case there is an OO-conversion‘c’ problem. 482. cf. 1.20, 39b14-16. 483. Aristotle apparently asserts that the following can be shown to yield a conclusion by reductio but not by conversion: Bocardo3(CU_), Bocardo3(UC_), AO_3(CU_), and AO_3(UC_). However, he may only be interested in the case he does: Bocardo3(CU‘C’) CON(AoC) BaC  NEC  (AoB), For a detailed discussion of the important next section of the commentary see section III.E.2.c of the introduction. 484. At 1.9, 30a15-23. 485. Alexander now sketches two arguments for OAI3(CU  N  ). In the first one presumably transforms CON(AoC) into CON(CoA) and invokes AOI1(UC‘C’) (accepted by Aristotle at 1.15, 35a35-b8) to get  NEC  (BiA), which converts to  NEC  (AiB). In the second one changes CON(AoC) into CON(AiC), converts that to CON(CiA), infers (Darii1(UC‘C’))  NEC  (BiA), which converts to  NEC  (AiB). 486. Because of the use of the non-Theophrastean Barbara1(NUN). 487. Alexander denies the possibility of establishing: Bocardo3(UC‘C’) AoC CON(BaC)  NEC  (AoB) on the grounds that transformation of the minor premiss yields a non-syllogistic pair and a reductio gives as premisses NEC(AaB) and CON(BaC), which yield (Barbara1(NC‘C’)  NEC  (AaC), a conclusion compatible with AoC. Alexander repeats this point below at 248,10-19. He does not point out that one gets no inconsistency either, if one combines NEC(AaB) with AoC because – according to Aristotle – Baroco2(NU_) has an unqualified conclusion. 488. This paragraph is Theophrastus 108A FHSG. 489. Theophrastus and Eudemus. 490. cf. 127,3-14. Alexander refers to people who invoke Bocardo3(CU‘C’) to derive from the minor premiss, BaC, and the denial,  NEC(AaC), of the conclusion, NEC(AaC), of Barbara1(NUN) ‘CON’(AoB), which is ‘impossible’, i.e., incompatible with the major premiss, NEC(AaB), of Barbara1(NUN). 491. 247,30-9; see the note there.

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492. This is Theophrastus 107B FHSG. For the significance of what is said here for Theophrastus’ ‘logic’ see section III.E.2.c of the introduction. 493. Bocardo3(CtUCt). 494. Alexander and all the mss of Aristotle read ‘in the universals’ (en tois katholou). Ross (40a1-2) prints ‘in the preceding’ (en tois proteron). But he takes the reference to be the same as Alexander does, viz, 1.20, 39b2-6. 495. Alexander suggests reading epi tôn ex amphoterôn endekhomenôn in place of en tois katholou. 496. Aristotle asserts the validity of: Darapti3(NC‘C’) (accepted at 40a12-16) Darapti3(CNC) (accepted at 40a16-18) Felapton3(CNC) (accepted at 40a18-25) Felapton3(NCCu) (accepted at 40a25-32) His words also imply acceptance of AEO3(NC‘C’) (AEI3(NC‘C’) (apparently accepted at 40a33-5), AEO3(CNCu) (rejected at 40a35-8). Neither Aristotle nor Alexander ever discusses the waste EE cases, but EE_3(CN_) stands or falls with AE_3(CN_) and EE3(NCCu) with Felapton1(NCCu). Alexander considers the possibility of justifying AEO3(CN  N  ). The only other lemma in this chapter is at 254,10. 497. For a divergence between Ross’s text of Aristotle and Alexander’s see Appendix 6 (on 40a8-9). 498. Alexander here argues for Celarent1(NCtN), using Disamis3(CtCtCt); see section III.E.3.a of the introduction. He adds an argument for: Cesare2(NCtN) NEC(AeB)  NEC  (AaC) NEC(BeC) Assume  NEC(BeC), i.e.,  NEC  (BiC). Then (Datisi1(CtCtCt))  NEC  (AiB), contradicting NEC(AeB). 499. Reading estô for Wallies’ estai (omitted in the Aldine). 500. The reference is to 207,35-6 and 213,25-7. We have no other knowledge of Alexander’s notes on logic (scholia logika). The sentence referring to it may be a gloss. 501. Aristotle does: Darapti3(NC‘C’) NEC(AaC) CON(BaC)  NEC  (AiB) Alexander reproduces his argument, converting CON(BaC) to CON(CiB) and invoking Darii1(NC‘C’). Alexander says nothing about the character of the conclusion, but contents himself with remarking that the situation is analogous to the U+C case. 502. Darapti3(CNC) CON(AaC) NEC(BaC) CON(AiB) Alexander fills out Aristotle’s ‘similar’ proof, converting NEC(BaC) to NEC(CiB) and invoking Darii1(CNC). Alexander is no doubt right to call Darii1(CNC) complete (as far as Aristotle is concerned), but neither he nor Aristotle discusses the mood, which probably explains why Alexander says ‘This syllogism in the first figure was complete, as were the others having a contingent major.’ 503. Aristotle does: Felapton3(CNC) CON(AeC) NEC(BaC) CON(AoB) converting NEC(BaC) to NEC(CiB) and invoking Ferio1(CNC). Alexander repeats the proof. 504. Alexander repeats Aristotle’s view of Barbara1(NC‘C’) and Darii1(NC‘C’), which he knows to be false; see section III.E.3.a of the introduction. Nothing turns on the point in the present circumstances. 505. Alexander suggests understanding estai dê palin to prôton skhêma; kai ei hê sterêtikê protasis endekhesthai sêmainei, phaneron hoti to sumperasma estai

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endekhomenon where Aristotle has estai dê palin to prôton skhêma; kai gar hê sterêtikê protasis endekhesthai sêmainei; phaneron oun hoti to sumperasma estai endekhomenon. 506. Alexander mentions but apparently rejects a way of establishing that the premisses of Felapton3(CNC) yield the conclusion CON(AiB): ‘convert’ CON(AeC) to CON(CoA) and then to CON(CiA). By Darii1(NC‘C’), ‘CON’(BiA), which can be converted to ‘CON’(AiB). Alexander may mention this alternative because he is going to invoke changing CON(XeY) into CON(YiX) in connection with AEO3(NCC), which Aristotle says is handled ‘as in the preceding’ (40a35). 507. Aristotle does: Felapton3(NCCu) NEC(AeC) CON(BaC) CONu(AoB) converting CON(BaC) to CON(CiB) and invoking Ferio1(NCCu); Alexander reproduces the proof, and does not mention the possibility of establishing  NEC  (AoB) by reductio. 508. Aristotle uses the letter representing the minor term in the discussion of the first figure rather than the ‘B’ which is appropriate here. 509. Aristotle asserts the validity of AE_3(NC_). Alexander takes him to be accepting AEI3(NC‘C’), a waste case of Darapti3(NC‘C’). See the next note. 510. This remark seems false, since one could establish: AEO3(NCC) NEC(AaC) CON(BeC) CON(AoB) by converting NEC(AaC) to NEC(CiA) and inferring (Ferio1(CNC)) CON(BoA), which can be transformed into CON(AoB). 511. Aristotle is apparently rejecting: AE_3(CN_) CON(AaC) NEC(BeC) For discussion of the next part of the commentary see section III.E.3.c of the introduction. 512. i.e. NEC(AaB) and NEC(AeB) under the interpretations given. 513. enantios. 514. Wallies cites 251,36 to justify not inserting a ‘not’ here; we think the sense in both places requires it. 515. Alexander now considers the possibility of proving: AEO3(CN‘C’) CON(AaC) NEC(BeC) ‘CON’(AoB) He converts CON(AaC) to CON(CiA) and infers (Ferio1(NCCu)) BoA (= CONu(BoA)), but, as Alexander points out, it is not possible to convert this to AoB. But then he expresses doubts about whether the conclusion might not be CON(BoA), which would convert. 516. See 1.16, 36a32-9 with Alexander’s discussion at 209,35-211,17. 517. cf. the note on 251,26. 518. Alexander seems to suppose that the obvious way to try to justify AE_3(CN_) is to convert the minor premiss to NEC(CeB) producing a first-figure combination which yields no conclusion. 519. We retain the mê of 251,39 bracketed by Wallies. It is not clear to us why Alexander chooses to speak of not converting both premisses rather than of just converting the major CON(AaC). In any case it seems to us likely that the text is corrupt since we would expect Alexander to say ‘if both premisses are not converted’ rather than ‘if both premisses do not convert’. 520. In this note we offer an overview of this last part of the commentary on the modal logic. On the basis of 1.22, 40a39-b2 and 40b4-8 we have an alleged parallelism between the proofs of the following pairs: (i) Datisi3(NC‘C’) and Darapti3(NC‘C’) (ii) Datisi3(CNC) and Darapti3(CNC) (iii) Disamis3(NCC) and Darapti3(NC‘C’)

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(iv) Disamis3(CN‘C’) and Darapti3(CNC) On the basis of 40b2-4 we can add parallelism for: (v) Ferison3(CNC) and Felapton3(CNC) (vi) Ferison3(NCCu) and Felapton3(NCCu) Alexander confirms the claims of parallelism at 254,14-21, 31-35, and 253,6-17. However, what Aristotle says at 40b2-4 also implies that there will be parallelisms for: (vii) Bocardo3(CNC) and Felapton3(CNC) (viii) Bocardo3(NCCu) and Felapton3(NCCu) as well as for: (ix) AO_3(CN_), IE_3(CN_) and AE_3(CN_) (x) AO_3 (NC_), IE_3(NC_) and AE_3(NC_) Aristotle accepts AE_3(NC_) at 40a33-5. Alexander takes him to be accepting AEI3(NC‘C’), and apparently mistakenly denies that he could have established AEO3(NCC); see the note on 251,10. At 40b8-12 Aristotle suggests a parallel treatment for IE_3(NC‘C’). Alexander discusses what he says at 253,23-7. In this case, however, he takes Aristotle to be plumping for IEO3(NCC), and doesn’t even consider the possibility that he has IEI3(NCC), a waste case of Disamis3(NCC), in mind. Neither Aristotle nor Alexander says anything explicit about AO_3(NC_), which can be treated as a waste case of Datisi3(NC‘C’) or shown to yield the conclusion CON(AiB) by reduction to Darii1(CNC). Since Aristotle has rejected AE_3(CN_) at 40a35-8 and rejects IE_3(CN_) at 40b8-12, it seems unlikely that he intends to accept IE_3(CN_) at 40b3-4. Alexander explains Aristotle’s rejection of IE_3(CN_) at 253,27-36 and offers terms for rejecting AO_3(CN) at 253,36-254,6. This leaves us with (vii) and (viii). It is clear that we cannot establish either case of Bocardo3 by mimicking the derivations for Felapton3, since conversion of the minor premiss leaves two particular premisses. Alexander never discusses: Bocardo3(NCCu) NEC(AoC) CON(BaC) CONu(AoB) although it admits of an easy indirect derivation. Assume  (AoB), i.e., AaB. Then (Barbara(UC‘C’)) ‘CON’(AaC), contradicting NEC(AoC). However, Alexander does spend a lot of time on OA_3(CN_), for which he mentions two kinds of justification, a reductio for: Bocardo3(CN  N  ) CON(AoC) NEC(BaC)  NEC  (AoB) which he never gives, and a direct proof for: OAI3(CN‘C’) CON(AoC) NEC(BaC)  NEC  (AiB) This is just a waste case of Disamis3(CN‘C’) but Alexander describes a full reduction to Darii1NC‘C’) briefly at 252,27-9 and more fully at 254,26-9. The indirect proof obviously proceeds by assuming NEC(AaB) and using Barbara1(NNN) to infer that NEC(AaC), contradicting CON(AoC). At 252,25-7 Alexander indicates his preference for assigning the reductio to Aristotle by saying that Aristotle’s words ‘the situation will be similar’ refer to the character of the conclusion not the nature of the proof. Of course, the conclusion of Felapton3(CN_) is ‘contingent in the way specified’, that of the indirectly proved Bocardo3(CN_) and the directly proved OAI3(CN_) is  NEC  . Again at 254,21-35 Alexander re-affirms his claim that Aristotle intends the indirect proof this time on the grounds that he wants there to be a negative conclusion when there is a negative premiss. What Alexander says just previously suggests that he is also influenced by the fact that Aristotle offered an indirect justification for Bocardo3(CU‘C’) at 1.12, 31b31-9. 521. For a minor divergence between Alexander’s citation and our text of Aristotle see Appendix 6 (on 40a39). 522. 1.4, 26a17. For minor textual variations see Appendix 6.

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523. Wallies prints hotan gar mê tou autou katêgorôntai amphoteroi, alla mêde hoi duo allêlôn antikatêgorôntai, alla tou heterou, tou prôtou skhêmatos hê sumplokê. The inserted words are from the Aldine. We have not translated the words alla mêde hoi duo allêlôn antikatêgorôntai, which seem to be rendered redundant by the insertion (which is however required). 524. i.e. the universal ones already discussed. Alexander here reports Aristotle’s rejection of AE_3(CN_), although he has just worried about it at 251,19-252,2. 525. AEI3(NC‘C’) (and perhaps the undiscussed EEO3(NCCu)). 526. Darapti3(CNC) and Felapton3(CNC). It is not clear why Alexander leaves out the two corresponding NC cases. 527. Alexander would appear to be thinking about all six N+C cases of Datisi3, Disamis3, and Ferison3. 528. Alexander turns to OA_3(CN_). See the introductory note on 252,3. 529. Bocardo3(CU‘C’), which Aristotle justified at 1.21, 39b31-9. But that case required Barbara1(NUN), whereas the present one needs only Barbara1(NNN). 530. Alexander returns to the Datisi3 and Disamis3 cases, pointing out that the former just require conversion of the minor premiss, but those for Disamis3 require converting the major premiss, invoking the relevant case of Darii1, and then converting the conclusion. 531. It seems to us likely that this paragraph is corrupt. We believe that Alexander is picking up on 40b2 and is thinking about: Ferison3(CNC) CON(AeC) NEC(BiC) CON(AoB) CON(AoC) NEC(BaC) OA_3(CN_) He first says that they both yield a contingent particular negative conclusion, but then (ignoring the obelized words) refers to the two derivations for the second case described in the note on 252,3. We have, in effect, translated the following text of 252,35-253,2: alla kan hê hetera apophatikê lêphthêi hê {for Wallies’ lêphthêi ê} anankaia kataphatikê, endekhomenon epi merous apophatikon to sumperasma. dei de [kai en merei einai tên anankaian ousan kataphatikên; an gar autê katholou, deêsei] tên epi merous endekhomenên apophatikên eis tên kataphatikên metalabein. 532. Alexander takes up Ferison3(NCCu). 533. Felapton3(NCCu) established at 40a25-32. 534. For the point of Alexander’s query see, e.g., the note on 245,3. 535. At 40b8 Aristotle accepts: IEO3(NCC) NEC(AiC) CON(BeC) CON(AoB) and rejects: IE_3(CN_) CON(AiC) NEC(BeC) Alexander gives the proof for the first of these: convert NEC(AiC) to NEC(CiA) and apply Ferio1(CNC) to get CON(BoA), which can be converted to CON(AoB). For the second Alexander first explains why a straightforward reduction won’t work: if one converts CON(AiC) to CON(CiA), one can apply Ferio1(NCCu) to get BoA (or  NEC  (BoA)), but that will not convert. He then gives Aristotle’s terms. He does not repeat his claim (see 252,19-252,2) that the terms are inadequate to reject a contingent conclusion. At 253,36 Alexander uses similar terms to reject: AO_3(CN_) CON(AaC) NEC(BoC) 536. 40a35-8. 537. Alexander speaks loosely, since the terms are not exactly the same. 538. And hence of no sleeping horse.

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539. And therefore a fortiori, does not hold of some human by necessity. 540. See 1.19, 39a2-3 and Alexander’s discussion at 242,22-7. 541. i.e. Bocardo3(CU‘C’) and Bocardo3(CN‘C’). 542. In justifying Bocardo3(CU‘C’) at 1.21, 39b31-9; see Alexander’s discussion at 247,9-248,30. 543. See the note on 245,3. 544. 40b2-3. Alexander claims that those words in their context are compatible with either a proof by conversion of OAI 3(CNC) or a reductio for Bocardo 3 (CN N ). He goes on to describe the proof by conversion for OAI3(CNC) again, and then says that Aristotle would not use it because he expects a combination with a negative premiss to yield a negative conclusion. 545. It is contingent that A holds of some B. We suspect that Alexander wrote (or intended to write) ‘proposed conclusion’ (prokeimenon rather than proeirêmenon). 546. This last sentence is difficult. We have interchanged the words ‘major’ and ‘minor’ in Wallies’ text. If Wallies text is right Alexander is suddenly turning from OA_3(CN_) to AO_3(NC_), and saying that its justification requires that a premiss and the conclusion be converted. But that is simply not true; see the note on 253,23. We suspect that someone – possibly Alexander himself – has interchanged the words ‘minor’ and ‘major’, and that Alexander is simply characterizing again the direct justification of OAI3(CN‘C’).

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, 34b27-28 (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 panti ex anankês, for which we use the translation ‘A does not hold by necessity of all B’. Thus we have the correspondences:

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‘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 e- and 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, 38a1011) 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 minor premiss necessary, and the

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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 counterinterpretations 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, 30b3640) 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 condition; for animal

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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 believe that Alexander invokes this distinction here in defense of AI-conversionn. His general point seems clear: we cannot convert a proposition like ‘Human holds of

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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 207,36 with the note. 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

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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 and section III.E. 3.c of the Introduction. 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 of

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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 on the grounds

242

Appendix 5

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 pp. 73-4 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 32b4

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). de (Ross); dê (lemma, 161,27. Both Ross and Wallies print legômen

244 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 35b12 35b17 35b23 35b32-3

Appendix 6 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, hot’ adunaton, ouk an genoito (Ross); to men [A] dunaton, hote 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). 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).

Appendix 6 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 38b39 39a29 39a30 39b7 40a2 40a8-9 40a39

245

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. 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: 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, 18821909). 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 directly: autothen; euthus disjoin (v.): apozeugnunai dissimilar in form: anomoioskhêmôn

English-Greek Glossary 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

249

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

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

last (term): eskhatos

genus: genos go through (said of a proof) (v.): proeinai; proerkhesthai; prokhôrein

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

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;

major: meizon material terms: hulê mean (v.): sêmainein minor: elattôn mixture: mixis modality: tropos

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 posit (v.): tithenai

250

English-Greek Glossary

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 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 second. 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 Greek-English index of Barnes et al. should be corrected.) At 36a21-2 Aristotle

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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 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

Greek-English Index 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 it is converted (136,220(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;

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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 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)).

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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 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;

Greek-English Index 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’). 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

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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 diorismenos 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; 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

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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 (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

Greek-English Index 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 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;

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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 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;

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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 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

Greek-English Index 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; 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

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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 (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

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endekhomenon ek tou diorismou at 223,11 katallêlos, congruous, 129,9; 250,25 kataphasis, affirmation (157,19-160,30(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 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 translated it at 165,11; 188,15.16;

Greek-English Index 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, 136,10.13; 150,1; 181,28; 189,19.29.34; 192,9.12.29; 200,15;

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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

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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. 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; 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

Greek-English Index 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 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

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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;

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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 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; 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;

Greek-English Index 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

265

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; 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 References are to the pages of this book. a(-proposition), 4 AE-transformationc, 21-2, 60, 134-5, 137-40, App. 5, see also waste cases affirmation, affirmative, 138-9, App. 2, 4 AI-conversionc, 22, 28-31, 60 AI-conversionn, 9, 27, 60 AI-conversionu, 7, 59 Alexander’s proofs of unproveable syllogisms, 50, 160, 162-4 AO-transformationc, 139 Barbara1(NC_), 44-7, 53-4, 120-1, 177 Barbara1(NU_), see NU first-figure combinations Barbara1(UC_), 37-41, 85, 96-104, 109-10, 115, 118-19, App. 1, 2 Barbara1(UN_), 13-18 Baroco2(NC_), 163 Bocardo3(CC_), 57 n.55, 221 n.466 Bocardo3(CN_), 180-3 Bocardo3(CU_), 43-4, 53-4, 173-5, 180-1 Bocardo3(NC_), 45-6, 121, 225 n.520 Bocardo3(NNN), 12-13, 52-3 Bocardo3(NU_), 13, 96, 102 Bocardo3(UC_), 174 Camestres2(CN_), 49-50, 156-7 Camestres2(CU_), 41, 150-3 Camestres2(NC_), 50, 160, 163 Camestres2(UC_), 42-3, 149-50 Celarent1(CN_), 44, 124-5 Celarent1(NC_), 44-9, 58 n.74, 122-3, 128-9, 133, 152-3, 155, 166-7, 175-6 Celarent1(NU_), see NU first-figure combinations Celarent1(UC_), 37-8, 47-9, 104-12, 120, 132-3, 152-3, 155, 166-7, App. 1 (p. 229) Cesare2(CN_), 50, 157-9

Cesare2(CU_), 42-3, 149-50 Cesare2(NC_), 49-50, 155-6, 176 Cesare2(UC_), 41, 150-3, 171 Chrysippus, 87-92 circle argument, 45-7, 49, 121, 129 combination, 4 complete, 5-6, 83-4, 115, 133, 177; see dictum de omni et nullo completion, 7-8 conditionals, 85-96 contingency in the way specified, 20-1, 37-8, 41-2, 46-9, 78-9, 119-20, 124, 150-3, 160, 166-7, 170-2, 177-8, 179; see also Barbara1(UC_) and Barbara1(NC_), Celarent1(UC_), and Celarent1(NC_) contingency, 9, 12-13, 19-54, 78-81, 92-4, 104-12, 121, 132-3, 136-7, 138-40, 150-2, 156-7 contingently (holding), 19-20, 29-31 contradictories, 7, 98, 107-8, 159-60 contraries, 7, 138-40, 159-60 conversion, 7-8, 76-7; see also AI-conversion, EE-conversion, and II-conversion Darii1(NC_), 44-7, 58 n.74, 129, 177 Darii1(NU_), see NU first-figure combinations Darii1(UC_), 37-41, 115, 118 Darii1(UN_), 16-18 de re/de dicto distinction, 14-15, 23, 36, 55 n.22, 130-1, 164-5, 211 n.356 dictum de omni et nullo, 6, 11, 15-16, 75, 78, 83-4, 122, 124 Diodorus Cronus, 94 diorismos of contingency, 20-1, 78, 87; see also ‘contingency in the way specified’

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Subject Index

e(-proposition), 4 EA-transformationc, 21-2, 60, 110-11, 134-5, 136-8, 158, App. 5; see also waste cases EE-conversionc, 25, 27-35, 36-7, 60, 136-45, App. 5 EE-conversionn, 9-10, 25-7, 60 EE-conversionu, 7, 59 EI-tranformationc, 139 ekthesis, 13, 52 eternal recurrence, 87-8, 90-1 Eudemus, 15-16, 29, 83-4, 137, 141, 195 extreme, 4 Feliciano, Giovanni Bernardino, 190 n.92 Ferio1(NC_), 44-8, 51-2, 58 n.74, 126-9, 132, 136, 163, 167, 171, 179, 181-2 Ferio1(NU_) see NU first-figure combinations Ferio1(UC_), 37-41, 83-4, 115-16, 118, 131-2, 136, 153-4, 167, 172, App. 1, 2 Ferio1(UN_), 24-6 n.93, 22-5, App. 1 figure, 4 i(-proposition), 4 idiôs poios, 89-91 II-conversionc, 10-11, 22, 28-31, 60 II-conversionn, 9, 27, 60 II-conversionu, 8, 59 incompatibility acceptance argument, 16-18, 78-9; see also incompatibility rejection argument incompatibility rejection argument, 16-18, 32-3, 136-8 interpretations, 6-7, 32-3, 36, 39-41, 42-3, 50-1, 79-81, 99-102, 107, 110-12, 113-14, 126, 129-31, 137, 139, 142, 148-9, 151-2, 157-9, 160, 179, 182 IO-transformationc, 21-2, 60, 138-40, App. 5; see also waste cases major, 4 major term must be predicate of conclusion, 154, 156-7, 162, 168-9, 172-4, 179 middle, 4 minor, 4

necessarily (holding), 30-1 necessity, 51, 90, 105-6, 107-9, 113-14, 120, 179, App. 1, 3 negation, negative, 107-9, 138-9, App. 2, 4 (p. 237) Nemesius, 190 n.90 non-syllogistic combinations, 6-7 NU first-figure combinations (i.e. Barbara1(NU_), Celarent1(NU_), Darii1(NU_), Ferio1(NU_)), 13-18, 43-4, 48-9, 53, 115-16, 123, 126-7, 128-9, 132, 155-6, 173-4 o(-proposition), 4 OI-transformationc, 21-2, 60, 79, 138-40, 164, 178, App. 5; see also waste cases OO-conversionc, 32, 41-2, 151, 154, 169, 172, 179, 181 opposites, 78-9, 96-7, 106-9, 121, 122-4, 128-9, 132, 138-40, 141-5, 159-60 peculiarly qualified (idiôs poios), 89-91 peiorem rule, 15-16, 35, 44, 49, 52-4, 83-4 premiss, 4 Philo the Dialectician (‘of Megara’), 94 propositions, 4 quality, 4 quantity, 4 reductio ad impossibile, 8, 43-4, 50-1, 78-9, 85-6, 97-9, 102-9, 114-16, 119-25, 127-9, 131-3, 137-8, 140-47, 150, 160, 162, 176 reduction, 77-8 Sosigenes, App. 3 Stoics, 87-92 temporal interpretation of modality, 23-5, 28-31, 139-40, 189 n.67 terms, 4; see interpretations Theophrastus, 3, 15-16, 29, 43-4, 52-4, 83-4, 111, 202, 137, 141, 157, 174, 195 n.158, App. 3 transformation, 56 n.38 two affirmative premisses in the second figure, 134-5, 145, 203 n.256 U-for-C substitution, 39-40, 43-4, 48-9, 53-4, 85-6, 98-105, 115, 120-1,

Subject Index 123-4, 128-9, 132-3; see Barbara1(UC_), Celarent1(UC_), Darii1(UC_), Ferio1(UC_) universal (propositions), 14, 30-1, 39-40, 99-102, 151-2 unqualified, 9 unqualifiedly (holding), 29-31

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waste cases, 21-2, 76-7, 79, 112-13, 126, 153, 154, 161, 164, 169, 172, 178, 180-2 weak two-sided Theophrastean contingency, 32-5, 105-11, 119-21, 125, 141-5, 146-7, 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, 191 n.106 25,26-26,14, 23-5 26,29-27,1, 56 n.33 31,5-6, App. 2 36,7-25, 26 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. 238) 39,4-11, 32 39,19-23, 20 41,21-4, 137 54,12-18, 6 60,27-61,1, 6 On the difference between Aristotle and his associates concerning mixtures 99, 102, 121, 128, 160, 176 Quaest. 31,9-10, 191 n.107

25a7-10, 135 25a27-36, 10-11, 25-7 25a29-32, 10, 25-6 25a32-4, 10 25a37-9, 19, 191 n.97 25a37-b3, 22, 28 25b3-14, 31 25b3-25, 136 25b14-15, 19 25b14-19, 32 26a17, 180 26b21, 198 n.197 28a7-9, 134, 145, 207 n.310 2.2-4, 95 Int. 17b22-3, 172-3 19a23-6, App. 3 23a6-11, App. 4 (pp. 237-8) 23a7-20, 19 chs. 12-13, App. 4 (fragments, ed. Wehrli) 16, 209 n.335 19, 188 n.51

EUDEMUS

AMMONIUS

in Int. 153,13-154,2, App. 3 221,11-229,11, 209 n.342 245,1-32, App. 4

NEMESIUS

De natura hominis (Morani) 2.81 190 n.90 PHILOPONUS

[AMMONIUS]

in An.Pr. 39,10-25, App. 3 n.7

ARISTOTLE

An. Pr. (outside 29b29-40b15) 24b22-4, 6 24b26-30, 6 24b29-30, 185 n.6 25a1-2, 23 25a2-3, 56 n.33

in An. Pr. 53,15-56,5, 209 n.342 126,8-29, App. 3 205,13-27, 58 n.74 205,13-27, 202 n.246 226,5, 217 n.429 SIMPLICIUS

in Cael. 316,25-9, 191 n.107

Index Locorum STEPHANUS

in Int. 38,14-31, App. 3 n.2

STOBAEUS

Eclogae I.79.1, 190 n.91 (dialectic fragments, ed. Hülser) 992, 191 n.106 994, 189 n.71

STOICS

(von Arnim) II.202a, 189 n.71

SVF

II.624, 190 n.89 (fragments, FHSG) 100C, App. 3 n.8 102A, 209 n.335 102B, 210 n.351 102C, 209 n.335 103B, 196 n.175 107A, 188 n.51 108B, 216 n.422 109A, 58 n.74, 202 n.246

THEOPHRASTUS

[THEMISTIUS]

in An. Pr. 25,20-1, 188 n.65

270