The Complete Works of Aristotle One and Two

The same holds good where the relation is negative. For ‘that does not belong [30] to this’ does not always mean that this is not that, but sometimes that this is not of that or for that, e.g. there is not a motion of a motion or a becoming of a becoming, but there is a becoming of pleasure; so pleasure is not a becoming. Or again it may be said that there is a sign of laughter, but there is not a sign of a sign, consequently laughter is not a sign. This holds in the other cases too, in which a problem is refuted [35] because the genus is asserted in a particular way in relation to it. Again take the inference: opportunity is not the right time; for opportunity belongs to God, but the right time does not, since nothing is useful to God. We must take as terms opportunity, right time, God; but the proposition must be understood according to the case of the noun. For we state this universally without qualification, that the [49a1] terms ought always to be stated in the nominative, e.g. man, good, contraries, not in oblique cases, e.g. of man, of good, of contraries, but the propositions ought to be understood with reference to the cases of each term—either the dative, e.g. ‘equal to this’, or the genitive, e.g. ‘double of this’, or the accusative, e.g. ‘that which strikes or sees this’, or the nominative, e.g. ‘man is an animal’, or in whatever other way the [5] word falls in the proposition.

37 · The expressions ‘this belongs to that’ and ‘this holds true of that’ must be understood in as many ways as there are different categories, and these categories must be taken either with or without qualification, and further as simple or compound; the same holds good of negative expressions. We must consider these [10] points and define them better.

38 · A term which is repeated in the propositions ought to be joined to the first extreme, not to the middle. I mean for example that if a deduction should be made proving that there is knowledge of justice, that it is good, the expression ‘that it is good’ (or ‘qua good’) should be joined to the first term. Let A stand for [15] knowledge that it is good, B for good, C for justice. It is true to predicate A of B. For of the good there is knowledge that it is good. Also it is true to predicate B of C. For justice is identical with a good. In this way an analysis of the argument can be made. But if the expression ‘that it is good’ were added to B, there will be no [20] analysis; for A will be true of B, but B will not be true of C. For to predicate of justice the term ‘good that it is good’ is false and not intelligible. Similarly if it should be proved that the healthy is an object of knowledge quâ good, or goat-stag an object of knowledge quâ not existing, or man perishable quâ an object of sense: in [25] every case in which an addition is made to the predicate, the repetition must be joined to the extreme.

The position of the terms is not the same when something is deduced without qualification and when the deduction relates to some particular thing or way or condition, e.g. when the good is proved to be an object of knowledge and when it is proved to be an object of knowledge that it is good. If it has been proved to be an [30] object of knowledge without qualification, we must put as middle term that which is, but if we add the qualification ‘that it is good’, the middle term must be that which is something. Let A stand for knowledge that it is something, B stand for something, and C stand for good. It is true to predicate A of B; for ex hypothesi there is knowledge of that which is something, that it is something. B too is true of C; for that which C represents is something. Consequently A is true of C: there will [35] then be knowledge of the good, that it is good: for ex hypothesi the term something indicates the thing’s proper substance. But if being were taken as middle and being (without qualification) were joined to the extreme, not being something, we should not have had a deduction that there is knowledge of the good, that it is good, but that it is; e.g. let A stand for knowledge that it is, B for being, C for good. Clearly [49b1] then in particular deductions we must take the terms in the way stated.

39 · We ought also to substitute terms which have the same value, word for word, and phrase for phrase, and word and phrase, and always take a word in [5] preference to a phrase; for thus the setting out of the terms will be easier. For example if it makes no difference whether we say that the supposable is not the genus of the opinable or that the opinable is not identical with a particular kind of supposable (for what is meant is the same), it is better to take as terms the supposable and the opinable in preference to the phrase suggested.

40 · Since for pleasure to be good and for pleasure to be the good are not [10] identical, we must not set out the terms in the same way; but if the deduction is to prove that pleasure is the good, the term must be the good, but if the object is to prove that pleasure is good, the term will be good. Similarly in all other cases.

41 · It is not the same, either in fact or in speech, for A to belong to all of that to which B belongs, and for A to belong to all of that to all of which B belongs; [15] for nothing prevents B from belonging to C, though not to every C: e.g. let B stand for beautiful, and C for white. If beauty belongs to something white, it is true to say that beauty belongs to that which is white; but not perhaps to everything that is white. If then A belongs to B, but not to everything of which B is predicated, then [20] whether B belongs to every C or merely belongs to C, it is not necessary that A should belong, I do not say to every C, but even to C at all. But if A belongs to everything of which B is truly said, it will follow that A can be said of all of that of all of which B is said. If however A is said of that of all of which B may be said, [25] nothing prevents B belonging to C, and yet A not belonging to every C or to any C at all. If then we take three terms it is clear that the expression ‘A is said of all of which B is said’ means this, ‘A is said of all the things of which B is said’. And if B is said of all of a third term, so also is A; but if B is not said of all of the third term, there is no [30] necessity that A should be said of all of it.

We must not suppose that something absurd results through setting out the terms; for we do not use the existence of this particular thing, but imitate the geometrician who says that this line is a foot long, and straight, and without [35] breadth, when it is not,16 but does not use those propositions in the sense of deducing anything from them. For in general, unless there is something related as whole to part and something else related to this as part to whole, the prover does not prove [50a1] from them, and so no deduction is formed. We use the process of setting out terms like perception by sense, in the interests of the student—not as though it were impossible to demonstrate without them, as it is to demonstrate without the premisses of the deduction.

[5] 42 · We should not forget that in the same deduction not all conclusions are reached through one figure, but one through one figure, another through another. Clearly then we must analyse arguments in accordance with this. Since not every problem is proved in every figure, but certain problems in each figure, it is clear [10] from the conclusion in what figure the premisses should be sought.

43 · In reference to those arguments aiming at a definition which have been directed toward some part of the definition, we must take as a term the point to which the argument has been directed, not the whole definition; for so we shall be less likely to be disturbed by the length of the term: e.g. if a man proves that water is [15] a drinkable liquid, we must take as terms drinkable and water.

44 · Further we must not try to reduce hypothetical deductions; for with the given premisses it is not possible to reduce them. For they have not been proved by deduction, but assented to by agreement. For instance if a man should suppose that [20] unless there is one faculty of contraries, there cannot be one science, and should then argue that not every faculty is of contraries, e.g. of what is healthy and what is sickly; for the same thing will then be at the same time healthy and sickly. He has shown that there is not one faculty of all contraries, but he has not proved that there [25] is not a science. And yet one must agree. But the agreement does not come from a deduction, but from an hypothesis. This argument cannot be reduced; but the proof that there is not a single faculty can. The latter argument no doubt was a deduction; but the former was an hypothesis.

The same holds good of arguments which are brought to a conclusion per [30] impossibile. These cannot be analysed either; but the reduction to what is impossible can be analysed since it is proved by deduction, though the rest of the argument cannot, because the conclusion is reached from an hypothesis. But these differ from the previous arguments; for in the former a preliminary agreement must be reached if one is to accept the conclusion (e.g. an agreement that if there is [35] proved to be one faculty of contraries, then contraries fall under the same science); whereas in the latter, even if no preliminary agreement has been made, men still accept the reasoning, because the falsity is patent, e.g. the falsity of what follows from the assumption that the diagonal is commensurate, viz. that then odd numbers are equal to evens.

Many other arguments are brought to a conclusion by the help of an hypothesis; these we ought to consider and mark out clearly. We shall describe in the sequel 17 their differences, and the various ways in which hypothetical arguments [50b1] are formed; but at present this much must be clear, that it is not possible to resolve such deductions into the figures. And we have explained the reason.

45 · Whatever problems are proved in more than one figure, if they have [5] been deduced in one figure, can be reduced to another figure, e.g. a negative deduction in the first figure can be reduced to the second, and one in the middle figure to the first, not all however but some only. The point will be clear in the sequel. If A belongs to no B, and B to every C, then A belongs to no C. Thus the first [10] figure; but if the negative is converted, we shall have the middle figure. For B belongs to no A, and to every C. Similarly if the deduction is not universal but particular, i.e. if A belongs to no B, and B to some C. Convert the negative and you [15] will have the middle figure.

The universal deductions in the second figure can be reduced to the first, but only one of the two particular deductions. Let A belong to no B and to every C. Convert the negative, and you will have the first figure. For B will belong to no A, [20] and A to every C. But if the affirmative concerns B, and the negative C, C must be made first term. For C belongs to no A, and A to every B; therefore C belongs to no B. B then belongs to no C; for the negative is convertible. [25]

But if the deduction is particular, whenever the negative concerns the major extreme, reduction to the first figure will be possible, i.e. if A belongs to no B and to some C: convert the negative and you will have the first figure. For B will belong to no A, and A to some C. But when the affirmative concerns the major extreme, no [30] analysis will be possible, i.e. if A belongs to every B, but not to every C; for AB does not admit of conversion, nor would there be a deduction if it did.

Again deductions in the third figure cannot all be analysed into the first, though all in the first figure can be analysed into the third. Let A belong to B and B [35] to some C. Since the particular affirmative is convertible, C will belong to some B; but A belonged to every B; so that the third figure is formed. Similarly if the deduction is negative; for the particular affirmative is convertible; therefore A will belong to no B, and to some C.

Of the deductions in the last figure one only cannot be analysed into the first, [51a1] viz. when the negative is not universal: all the rest can be analysed. Let A and B be predicated of every C; then C can be converted partially with either A or B; C then belongs to some B. Consequently we shall get the first figure, if A belongs to every [5] C, and C to some B. If A belongs to every C and B to some C, the argument is the same; for C is convertible in reference to B. But if B belongs to every C and A to some C, the first term must be B; for B belongs to every C, and C to some A, [10] therefore B belongs to some A. But since the particular is convertible, A will belong to some B. If the deduction is negative, when the terms are universal we must take them in a similar way. Let B belong to every C, and A to no C; then C will belong to some B, and A to no C; and so C will be middle term. Similarly if the negative is [15] universal, the affirmative particular; for A will belong to no C, and C to some of the Bs. But if the negative is particular, no analysis will be possible, i.e. if B belongs to [20] every C, and A does not belong to some C: convert BC and both propositions will be particular.

It is clear that in order to analyse the figures into one another the proposition which concerns the minor extreme must be converted in both the figures; for when [25] this is altered, the transition to the other figure is made.

One of the deductions in the middle figure can, the other cannot, be analysed into the third figure. Whenever the universal is negative, analysis is possible. For if A belongs to no B and to some C, both B and C alike are convertible in relation to A, [30] so that B belongs to no A, and C to some A. A therefore is middle term. But when A belongs to every B, and not to some C, analysis will not be possible; for neither of the propositions is universal after conversion.

Deductions in the third figure can be analysed into the middle figure, [35] whenever the negative is universal, i.e. if A belongs to no C, and B to some of every C. For C then will belong to no A and to some B. But if the negative is particular, no analysis will be possible; for the particular negative does not admit of conversion.

[40] It is clear then that the same deductions cannot be analysed in these figures which could not be analysed into the first figure, and that when deductions are [51b1] reduced to the first figure these alone are confirmed by reduction to what is impossible.

It is clear from what we have said how we ought to reduce deductions, and that the figures may be analysed into one another.

[5] 46 · In establishing or refuting, it makes some difference whether we suppose the expressions ‘not to be this’ and ‘to be not-this’ are identical or different in meaning, e.g. ‘not to be white’ and ‘to be not-white’. For they do not mean the same thing, nor is ‘to be not-white’ the negation of ‘to be white’, but rather ‘not to be [10] white’. The reason for this is as follows. The relation of ‘he can walk’ to ‘he can not-walk’ is similar to the relation of ‘it is white’ to ‘it is not-white’; so is that of ‘he knows what is good’ to ‘he knows what is not-good’. For there is no difference between the expressions ‘he knows what is good’ and ‘he is knowing what is good’, or [15] ‘he can walk’ and ‘he is able to walk’: therefore there is no difference between their opposites ‘he cannot walk’—‘he is not able to walk’. If then ‘he is not able to walk’ means the same as ‘he is able not to walk’, these will belong at the same time to the same person (for the same man can both walk and not-walk, and is possessed of [20] knowledge of what is good and of what is not-good), but an affirmation and a denial which are opposed to one another do not belong at the same time to the same thing. As then not to know what is good is not the same as to know what is not good, so to be not-good is not the same as not to be good. For when two pairs correspond, if the one pair are different from one another, the other pair also must be different. Nor is [25] to be not-equal the same as not to be equal; for there is something underlying the one, viz. that which is not-equal, and this is the unequal, but there is nothing underlying the other. That is why not everything is either equal or unequal, but everything is equal or is not equal. Further the expressions ‘it is a not-white log’ and ‘it is not a white log’ do not belong at the same time. For if it is a not-white log, it [30] must be a log: but that which is not a white log need not be a log at all. Therefore it is clear that ‘it is not-good’ is not the denial of ‘it is good’. If then of every single thing either the affirmation or the negation is true if it is not a negation clearly it must in a sense be an affirmation. But every affirmation has a corresponding [35] negation. The negation then of this is ‘it is not not-good’.

The relation of these to one another is as follows. Let A stand for to be good, B for not to be good, let C stand for to be not-good and be placed under B, and let D stand for not to be not-good and be placed under A. Then either A or B will belong to everything, but they will never belong to the same thing; and either C or D will [40] belong to everything, but they will never belong to the same thing. And B must belong to everything to which C belongs. For if it is true to say it is not-white, it is [52a1] true also to say it is not white; for it is impossible that a thing should simultaneously be white and be not-white, or be a not-white log and be a white log; consequently if the affirmation does not belong, the denial must belong. But C does not always belong to B; for what is not a log at all, cannot be a not-white log either. On the [5] other hand D belongs to everything to which A belongs. For either C or D belongs to everything to which A belongs. But since a thing cannot be simultaneously not-white and white, D must belong to everything to which A belongs. For of that which is white it is true to say that it is not not-white. But A is not true of every D. For of that which is not a log at all it is not true to say A, viz. that it is a white log. [10] Consequently D is true, but A is not true, i.e. that it is a white log. It is clear also that A and C cannot together belong to the same thing, and that B and D may belong to the same thing.

Privative terms are similarly related to positive terms in respect of this [15] arrangement. Let A stand for equal, B for not equal, C for unequal, D for not unequal.

In many things also, to some of which something belongs which does not belong to others, the negation may be true in a similar way, viz. that all are not white or that each is not white, while that each is not-white or all are not-white is [20] false. Similarly also ‘every animal is not-white’ is not the negation of ‘every animal is white’ (for both are false) but rather ‘not every animal is white’.

Since it is clear that ‘it is not-white’ and ‘it is not white’ mean different things, and one is an affirmation, the other a denial, it is evident that the method of proving [25] each cannot be the same, e.g. that whatever is an animal is not white or may not be white, and that it is true to call it not-white; for this means that it is not-white. But we may prove that it is true to call it white or not-white in the same way—for both [30] are proved constructively by means of the first figure. For the expression ‘it is true’ stands on a similar footing to ‘it is’. For the negation of ‘it is true to call it white’ is not ‘it is true to call it not-white’ but ‘it is not true to call it white’. If then it is to be true to say that whatever is a man is musical or is not-musical, we must assume that [35] whatever is an animal either is musical or is not-musical; and the proof has been made. That whatever is a man is not musical is proved destructively in the three ways mentioned.

In general whenever A and B are such that they cannot belong at the same time to the same thing, and one of the two necessarily belongs to everything, and again C [52b1] and D are related in the same way, and A follows C but the relation cannot be converted, then D must follow B and the relation cannot be converted. And A and D may belong to the same thing, but B and C cannot. First it is clear from the [5] following consideration that D follows B. For since either C or D necessarily belongs to everything; and since C cannot belong to that to which B belongs, because it carries A along with it and A and B cannot belong to the same thing; it is clear that D must follow B. Again since C does not convert with A, but C or D belongs to [10] everything, it is possible that A and D should belong to the same thing. But B and C cannot belong to the same thing, because A follows C; and so something impossible results. It is clear then that B does not convert with D either, since it is possible that D and A should belong at the same time to the same thing.

It results sometimes even in such an arrangement of terms that one is deceived [15] through not apprehending the opposites rightly, one of which must belong to everything: e.g. we may reason that if A and B cannot belong at the same time to the same thing, but it is necessary that one of them should belong to whatever the other does not belong to; and again C and D are related in the same way; and A follows everything which C follows: it will result that B belongs necessarily to everything to [20] which D belongs—but this is false. Assume that F stands for the negation of A and B, and again that H stands for the negation of C and D. It is necessary then that either A or F should belong to everything; for either the affirmation or the denial [25] must belong. And again either C or H must belong to everything; for they are related as affirmation and denial. And ex hypothesi A belongs to everything to which C belongs. Therefore H belongs to everything to which F belongs. Again since either F or B belongs to everything, and similarly either H or D, and since H follows F, B must follow D; for we know this. If then A follows C, B must follow D. But this is false; for as we proved the relation of consequence is reversed in terms so [30] constituted. No doubt it is not necessary that A or F should belong to everything, or that F or B should belong to everything; for F is not the denial of A. For not-good is the negation of good; and not-good is not identical with neither good nor not-good. Similarly also with C and D. For two negations have been assumed in respect to one term.

BOOK II

1 · We have already explained the number of the figures, the character and number of the propositions, when and how a deduction is formed; further what we [53a1] must look for when refuting and establishing propositions, and how we should investigate a given problem in any branch of inquiry, also by what means we shall obtain principles appropriate to each subject. Since some deductions are universal, [5] others particular, all the universal deductions give more than one result, and of particular deductions the affirmative yield more than one, the negative yield only the stated conclusion. For all propositions are convertible save only the particular negative; and the conclusion states one thing about another. Consequently all other deductions yield more than one conclusion, e.g. if A has been proved to belong to [10] every or to some B, then B must belong to some A; and if A has been proved to belong to no B, then B belongs to no A. This is a different conclusion from the former. But if A does not belong to some B, it is not necessary that B should not belong to some A; for it may belong to every A.

This then is the reason common to all deductions whether universal or [15] particular. But it is possible to give another reason concerning those which are universal. For all the things that are subordinate to the middle term or to the conclusion may be proved by the same deduction, if the former are placed in the middle, the latter in the conclusion; e.g. if the conclusion AB is proved through C, then A must be said of all of whatever is subordinate to B or C; for if D is in B as in a [20] whole, and B is in A, then D will be in A. Again if E is in C as in a whole, and C is in A, then E will be in A. Similarly if the deduction is negative. In the second figure it [25] will be possible to deduce only that which is subordinate to the conclusion, e.g. if A belong to no B and to every C; we conclude that B belongs to no C. If then D is subordinate to C, clearly B does not belong to it. But that B does not belong to what is subordinate to A, is not clear by means of the deduction. And yet B does not belong to E, if E is subordinate to A. But while it has been proved through the [30] deduction that B belongs to no C, it has been assumed without proof that B does not belong to A, consequently it does not result through the deduction that B does not belong to E.

But in particular deductions there will be no necessity of inferring what is subordinate to the conclusion (for a deduction does not result when this is [35] particular), but whatever is subordinate to the middle term may be inferred, not however through the deduction, e.g. if A belongs to every B and B to some C. Nothing can be inferred about that which is subordinate to C; something can be inferred about that which is subordinate to B, but not through the preceding deduction. Similarly in the other figures: that which is subordinate to the conclusion [40] cannot be proved; the other subordinate can be proved, only not through the deduction, just as in the universal deductions what is subordinate to the middle term [53b1] is proved (as we saw) from a proposition which is not demonstrated; consequently either a conclusion is not possible there or else it is possible here too.

2 · It is possible for the premisses of the deduction to be true, or to be false, or [5] to be the one true, the other false. The conclusion is either true or false necessarily. From true premisses it is not possible to draw a false conclusion; but a true conclusion may be drawn from false premisses—true however only in respect to the fact, not to the reason. The reason cannot be established from false premisses: why this is so will be explained in the sequel.18 [10]

First then that it is not possible to draw a false conclusion from true premisses, is made clear by this consideration. If it is necessary that B should be when A is, it is necessary that A should not be when B is not. If then A is true, B must be true: [15] otherwise it will turn out that the same thing both is and is not at the same time. But this is impossible. (Let it not, because A is laid down as a single term, be supposed that it is possible, when a single fact is given, that something should necessarily result. For that is not possible. For what results necessarily is the conclusion, and [20] the means by which this comes about are at the least three terms, and two relations or propositions. If then it is true that A belongs to all that to which B belongs, and that B belongs to all that to which C belongs, it is necessary that A should belong to all that to which C belongs, and this cannot be false; for then the same thing will belong and not belong at the same time. So A is posited as one thing, being two premisses taken together.) The same holds good of negative deductions: it is not [25] possible to prove a false conclusion from truths.

But from what is false a true conclusion may be drawn, whether both the premisses are false or only one (provided that this is not either of the premisses indifferently, but the second, if it is taken as wholly false; but if it is not taken as [30] wholly false, it does not matter which of the two is false). Let A belong to the whole of C, but to no B, neither let B belong to C. This is possible, e.g. animal belongs to no stone, nor stone to any man. If then A is taken to belong to every B and B to every C, A will belong to every C; consequently though both the premisses are false the [35] conclusion is true; for every man is an animal. Similarly with the negative. For it is possible that neither A nor B should belong to any C, although A belongs to every B, e.g. if the same terms are taken and man is put as middle; for neither animal nor man belongs to any stone, but animal belongs to every man. Consequently if one [40] term is taken to belong to none of that to which it does belong, and the other term is taken to belong to all of that to which it does not belong, though both the premisses [54a1] are false the conclusion will be true. A similar proof may be given if each premiss is partially false.

But if one only of the premisses is false, when the first premiss is wholly false, i.e. AB, the conclusion will not be true, but if BC is wholly false, a true conclusion [5] will be possible. I mean by wholly false the contrary of the truth, e.g. if what belongs to none is assumed to belong to all, or if what belongs to all is assumed to belong to none. Let A belong to no B, and B to every C. If then the proposition BC which I take is true, and AB is wholly false, viz. that A belongs to every B, it is impossible [10] that the conclusion should be true; for A belonged to none of the Cs, since A belonged to nothing to which B belonged, and B belonged to every C. Similarly there cannot be a true conclusion if A belongs to every B, and B to every C, but while the true proposition BC is assumed, the wholly false AB is also assumed, viz. that A belongs to nothing to which B belongs—here the conclusion must be false. For A [15] will belong to every C, since A belongs to everything to which B belongs, and B to every C. It is clear then that when the first premiss is wholly false, whether affirmative or negative, and the other premiss is true, the conclusion cannot be true.

But if the premiss is not wholly false, a true conclusion is possible. For if A belongs to every C and to some B, and if B belongs to every C, e.g. animal to every [20] swan and to some white thing, and white to every swan, then if we assume that A belongs to every B, and B to every C, A will belong to every C truly; for every swan is an animal. Similarly if AB is negative. For it is possible that A should belong to some B and to no C, and that B should belong to every C, e.g. animal to some white [25] thing, but to no snow, and white to all snow. If then one should assume that A belongs to no B, and B to every C, then A will belong to no C.

But if the proposition AB, which is assumed, is wholly true, and BC is wholly false, a true deduction will be possible; for nothing prevents A belonging to every B [30] and to every C, though B belongs to no C, e.g. these being species of the same genus which are not subordinate one to the other—for animal belongs both to horse and to man, but horse to no man. If then it is assumed that A belongs to every B and B to every C, the conclusion will be true, although the proposition BC is wholly false. [35] Similarly if the proposition AB is negative. For it is possible that A should belong neither to any B nor to any C, and that B should not belong to any C, e.g. a genus to species of another genus—for animal belongs neither to music nor to medicine, nor does music belong to the medicine. If then it is assumed that A belongs to no B, and [54b1] B to every C, the conclusion will be true.

And if BC is not wholly false but in part only, even so that conclusion may be true. For nothing prevents A belonging to the whole of B and of C, while B belongs [5] to some C, e.g. a genus to its species and difference—for animal belongs to every man and to every footed thing, and man to some footed things though not to all. If then it is assumed that A belongs to every B, and B to every C, A will belong to every C; and this ex hypothesi is true. Similarly if the proposition AB is negative. For it is [10] possible that A should neither belong to any B nor to any C, though B belongs to some C, e.g. a genus to the species of another genus and its difference for animal neither belongs to any wisdom nor to any speculative science, but wisdom belongs to some speculative sciences. If then it should be assumed that A belongs to no B, and B to every C, A will belong to no C; and this ex hypothesi is true. [15]

In particular deductions it is possible when the first proposition is wholly false, and the other true, that the conclusion should be true; also when the first is false in part, and the other true; and when the first is true, and the particular is false; and [20] when both are false. For nothing prevents A belonging to no B, but to some C, and B to some C, e.g. animal belongs to no snow, but to some white thing, and snow to some white thing. If then snow is taken as middle, and animal as first term, and it is assumed that A belongs to the whole of B, and B to some C, then AB is wholly false, [25] BC true, and the conclusion true. Similarly if the proposition AB is negative; for it is possible that A should belong to the whole of B, but not to some C, although B belongs to some C, e.g. animal belongs to every man, but does not follow some white, [30] but man belongs to some white; consequently if man be taken as middle term and it is assumed that A belongs to no B but B belongs to some C, the conclusion will be true although the proposition AB is wholly false.

If the proposition AB is false in part, the conclusion may be true. For nothing [35] prevents A belonging both to some B and to some C, and B belonging to some C, e.g. animal to something beautiful and to something great, and beautiful belonging to something great. If then A is assumed to belong to every B, and B to some C, the [55a1] proposition AB will be partially false, BC will be true, and the conclusion true. Similarly if the proposition AB is negative. For the same terms will serve, and in the same positions, to prove the point.

[5] Again if AB is true, and BC is false, the conclusion may be true. For nothing prevents A belonging to the whole of B and to some C, while B belongs to no C, e.g. animal to every swan and to some black things, though swan belongs to no black thing. Consequently if it should be assumed that A belongs to every B, and B to [10] some C, the conclusion will be true, although BC is false. Similarly if the proposition AB is negative. For it is possible that A should belong to no B, and not to some C, while B belongs to no C, e.g. a genus to the species of another genus and to the [15] accident of its own species—for animal belongs to no number and to some white things, and number belongs to nothing white. If then number is taken as middle, and it is assumed that A belongs to no B, and B to some C, then A will not belong to some C, which ex hypothesi is true. And the proposition AB is true, BC false.

[20] Also if AB is partially false, and BC is false too, the conclusion may be true. For nothing prevents A belonging to some B and to some C, though B belongs to no C, e.g. if B is the contrary of C, and both are accidents of the same genus—for animal belongs to some white things and to some black things, but white belongs to [25] no black thing. If then it is assumed that A belongs to every B, and B to some C, the conclusion will be true. Similarly if AB is negative; for the same terms arranged in the same way will serve for the proof.

Also though both premisses are false the conclusion may be true. For it is [30] possible that A may belong to no B and to some C, while B belongs to no C, e.g. a genus in relation to the species of another genus, and to the accident of its own species for animal belongs to no number, but to some white things, and number to nothing white. If then it is assumed that A belongs to every B and B to some C, the [35] conclusion will be true, though both propositions are false. Similarly also if AB is negative. For nothing prevents A belonging to the whole of B, and not to some C, while B belongs to no C, e.g. animal belongs to every swan, and not to some black things, and swan belongs to nothing black. Consequently if it is assumed that A [55b1] belongs to no B, and B to some C, then A does not belong to some C. The conclusion then is true, but the propositions are false.

3 · In the middle figure it is possible in every way to reach a true conclusion through false premisses, whether the deductions are universal or particular, viz. [5] when both propositions are wholly false; when each is partially false; when one is true, the other [wholly] false (it does not matter which of the two premisses is false). [if both premisses are partially false; if one is quite true, the other partially false; if [10] one is wholly false, the other partially true.]19 For if A belongs to no B and to every C, e.g. animal to no stone and to every horse, then if the propositions are stated contrariwise and it is assumed that A belongs to every B and to no C, though the propositions are wholly false they will yield a true conclusion. Similarly if A belongs to every B and to no C; for we shall have the same deduction. [15]

Again if one premiss is wholly false, the other wholly true; for nothing prevents A belonging to every B and to every C, though B belongs to no C, e.g. a genus to its co-ordinate species. For animal belongs to every horse and man, and no man is a [20] horse. If then it is assumed that animal belongs to all of the one, and none of the other, the one premiss will be wholly false, the other wholly true, and the conclusion will be true whichever term the negative statement concerns.

Also if one premiss is partially false, the other wholly true. For it is possible that A should belong to some B and to every C, though B belongs to no C, e.g. [25] animal to some white things and to every raven, though white belongs to no raven. If then it is assumed that A belongs to no B, but to the whole of C, the proposition AB is partially false, AC wholly true, and the conclusion true. Similarly if the negative is transposed: the proof can be made by means of the same terms. Also if the [30] affirmative proposition is partially false, the negative wholly true, a true conclusion is possible. For nothing prevents A belonging to some B, but not to C as a whole, while B belongs to no C, e.g. animal belongs to some white things, but to no pitch, and white belongs to no pitch. Consequently if it is assumed that A belongs to the [35] whole of B, but to no C, AB is partially false, AC is wholly true, and the conclusion is true.

And if both the propositions are partially false, the conclusion may be true. For it is possible that A should belong to some B and to some C, and B to no C, e.g. animal to some white things and to some black things, though white belongs to [56a1] nothing black. If then it is assumed that A belongs to every B and to no C, both propositions are partially false, but the conclusion is true. Similarly, if the negative is transposed, the proof can be made by means of the same terms.

It is clear also that the same holds for particular deductions. For nothing [5] prevents A belonging to every B and to some C, though B does not belong to some C, e.g. animal to every man and to some white things, though man will not belong to some white things. If then it is stated that A belongs to no B and to some C, the universal proposition is wholly false, the particular is true, and the conclusion is [10] true. Similarly if AB is affirmative; for it is possible that A should belong to no B, and not to some C, though B does not belong to some C, e.g. animal belongs to nothing inanimate, and to some white things, and inanimate will not belong to some [15] white things. If then it is stated that A belongs to B and not to some C, the AB which is universal is wholly false, AC is true, and the conclusion is true. Also a true conclusion is possible when the universal is true, and the particular is false. For nothing prevents A following neither B nor C at all, while B does not belong to some [20] C, e.g. animal belongs to no number nor to anything inanimate, and number does not follow some inanimate things. If then it is stated that A belongs to no B and to some C, the conclusion will be true, and the universal proposition true, but the particular false. Similarly if the premiss which is stated universally affirmative. For [25] it is possible that A should belong both to B and to C as wholes, though B does not follow some C, e.g. a genus in relation to its species and difference—for animal follows every man and footed things as a whole, but man does not follow every footed thing. Consequently if it is assumed that A belongs to the whole of B, but [30] does not belong to some C, the universal proposition is true, the particular false, and the conclusion true.

It is clear too that though both propositions are false they may yield a true conclusion, since it is possible that A should belong both to B and to C as wholes, [35] though B does not follow some C. For if it is assumed that A belongs to no B and to some C, the propositions are both false, but the conclusion is true. Similarly if the universal proposition is affirmative and the particular negative. For it is possible that A should follow no B and every C, though B does not belong to some C, e.g. animal follows no science but every man, though science does not follow every man. [56b1] If then A is assumed to belong to the whole of B, and not to follow some C, the propositions are false but the conclusion is true.

[5] 4 · In the last figure a true conclusion may come through what is false, alike when both are wholly false, when each is partly false, when one is wholly true, the other false, when one is partly false, the other wholly true, and vice versa, and in every other way in which it is possible to alter the propositions. For nothing prevents [10] neither A nor B from belonging to any C, while A belongs to some B, e.g. neither man nor footed follows anything inanimate, though man belongs to some footed things. If then it is assumed that A and B belong to every C, the propositions will be wholly false, but the conclusion true. Similarly if one is negative, the other [15] affirmative. For it is possible that B should belong to no C, but A to every C, and that A should not belong to B, e.g. black belongs to no swan, animal to every swan, and animal not to everything black. Consequently if it is assumed that B belongs to every C, and A to no C, A will not belong to some B; and the conclusion is true, [20] though the propositions are false.

Also if each is partly false, the conclusion may be true. For nothing prevents both A and B from belonging to some C while A belongs to some B, e.g. white and beautiful belong to some animals, and white to some beautiful things. If then it is [25] stated that A and B belong to every C, the propositions are partially false, but the conclusion is true. Similarly if AC is stated as negative. For nothing prevents A from not belonging, and B from belonging, to some C, while A does not belong to every B, e.g. white does not belong to some animals, beautiful belongs to some [30] animals, and white does not belong to everything beautiful. Consequently if it is assumed that A belongs to no C, and B to every C, both propositions are partly false, but the conclusion is true. Similarly if one is wholly false, the other wholly true. For it is possible that [35] both A and B should follow every C, though A does not belong to some B, e.g. animal and white follow every swan, though animal does not belong to everything white. Taking such terms, if one assumes that B belongs to the whole of C, but A does not belong to C at all, BC will be wholly true, AC wholly false, and the conclusion true.

Again if one is wholly true, the other partly false, the conclusion may be true. [10] For it is possible that B should belong to every C, and A to some C, while A belongs to some B, e.g. biped belongs to every man, beautiful not to every man, and beautiful to some bipeds. If then it is assumed that both A and B belong to the whole of C, BC is wholly true, AC partly false, the conclusion true. Similarly if AC is true [15] and BC partly false, a true conclusion is possible: this can be proved, if the same terms as before are transposed. Also the conclusion may be true if one is negative, the other affirmative. For since it is possible that B should belong to the whole of C, and A to some C, and, when they are so, that A should not belong to every B, [20] therefore if it is assumed that B belongs to the whole of C, and A to no C, the negative is partly false, the other wholly true, and the conclusion is true. Again since it has been proved that if A belongs to no C and B to some C, it is possible that A should not belong to some B, it is clear that if AC is wholly true, and BC partly false, [25] it is possible that the conclusion should be true. For if it is assumed that A belongs to no C, and B to every C, AC is wholly true, and BC is partly false.

It is clear also in the case of particular deductions that a true conclusion may come through what is false, in every possible way. For the same terms must be taken [30] as have been taken when the propositions are universal, positive terms in positive deductions, negative terms in negative. For it makes no difference to the setting out of the terms, whether one assumes that what belongs to none belongs to all or that what belongs to some belongs to all. The same applies to negative deductions. [35]

It is clear then that if the conclusion is false, the premisses of the argument must be false, either all or some of them; but when the conclusion is true, it is not necessary that the premisses should be true, either one or all, yet it is possible, though no part of the deduction is true, that the conclusion may none the less be true; but not necessarily. The reason is that when two things are so related to one [57b1] another, that if the one is, the other necessarily is, then if the latter is not, the former will not be either, but if the latter is, it is not necessary that the former should be. But it is impossible that the same thing should be necessitated by the being and by the not-being of the same thing. I mean, for example, that it is impossible that B should necessarily be great if A is white and that B should necessarily be great if A [5] is not white. For whenever if this, A, is white it is necessary that that, B, should be great, and if B is great that C should not be white, then it is necessary if A is white that C should not be white. And whenever it is necessary, if one of two things is, that the other should be, it is necessary, if the latter is not, that the former should not be. [10] If then B is not great A cannot be white. But if, if A is not white, it is necessary that B should be great, it necessarily results that if B is not great, B itself is great. But [15] this is impossible. For if B is not great, A will necessarily not be white. If then if this is not white B must be great, it results that if B is not great, it is great, just as if it were proved through three terms.

5 · Circular and reciprocal proof means proof by means of the conclusion [20] and by taking one of the propositions with its predication reversed and inferring the other which was assumed in the original deduction: e.g. suppose we had to prove that A belongs to every C, and it has been proved through B; suppose that A should now be proved to belong to B by assuming that A belongs to C, and C to B before; [25] but the reverse was assumed, viz. that B belongs to C. Or suppose it is necessary to prove that B belongs to C, and A is assumed to belong to C, which was the conclusion and B to belong to A: the reverse was assumed before viz. that A belongs to B. In no other way is reciprocal proof possible. For if another term is taken as [30] middle, the proof is not circular; for neither of the propositions assumed is the same as before; and if one of them is assumed, only one can be for if both of them are taken the same conclusion as before will result; but it must be different.

If the terms are not convertible, one of the propositions from which the deduction results must be undemonstrated; for it is not possible to demonstrate [35] through these terms that the third belongs to the middle or the middle to the first. If the terms are convertible, it is possible to demonstrate everything reciprocally, e.g. if A and B and C are convertible with one another. Suppose AC has been proved through B as middle term, and again AB through the conclusion and the proposition BC converted, and similarly BC through the conclusion and the proposition AB [58a1] converted. But it is necessary to prove both proposition CB and BA; for we have used these alone without demonstrating them. If then it is assumed that B belongs to every C, and C to every A, we shall have a deduction relating B to A. Again if it is [5] assumed that C belongs to every A, and A to every B, C must belong to every B. In both these deductions the proposition CA has been assumed without being demonstrated: the others had been proved. Consequently if we succeed in demon-strating [10] this, all will have been proved reciprocally. If then it is assumed that C belongs to every B, and B to every A, both the propositions assumed have been demonstrated, and C must belong to A.

It is clear then that only if the terms are convertible is circular and reciprocal demonstration possible (if the terms are not convertible, the matter stands as we [15] said above). But it turns out that even in these we use for the demonstration the very thing that is being proved; for C is proved of B, and B of A, by assuming that C is said of A, and C is proved of A through these propositions, so that we use the [20] conclusion for the demonstration.

In negative deductions reciprocal proof is as follows. Let B belong to every C, and A to no B: we conclude that A belongs to no C. If again it is necessary to [25] conclude that A belongs to no B (which was previously assumed) A must belong to no C, and C to every B: thus the proposition is reversed. If it is necessary to conclude that B belongs to C, AB must no longer be converted as before; for the proposition that B belongs to no A is identical with the proposition that A belongs to no B. But we must assume that B belongs to all of that to none of which A belongs. Let A belong to no C (which was the conclusion) and assume that B belongs to all of that [30] to none of which A belongs. It is necessary then that B should belong to every C. Consequently each of the three propositions has been made a conclusion, and this is circular demonstration, to assume the conclusion and the reverse of one of the propositions, and deduce the remaining one. [35]

In particular deductions it is not possible to demonstrate the universal proposition through the others, but the particular can be demonstrated. Clearly it is impossible to demonstrate the universal; for what is universal is proved through propositions which are universal, but the conclusion is not universal, and the proof must start from the conclusion and the other proposition. Further a deduction cannot be made at all if the other proposition is converted; for the result is that both [58b1] propositions are particular. But the particular may be proved. Suppose that A has been proved of some C through B. If then it is assumed that B belongs to every A, and the conclusion is retained, B will belong to some C; for we obtain the first figure [5] and A is middle. But if the deduction is negative, it is not possible to prove the universal proposition, for the reason given above. But it is possible to prove the particular, if AB is converted as in the universal cases, i.e. B belongs to some of that to some of which A does not belong: otherwise no deduction results because the [10] particular proposition is negative.

6 · In the second figure it is not possible to prove an affirmative proposition in this way, but a negative may be proved. An affirmative is not proved because both propositions are not affirmative (for the conclusion is negative) but an [15] affirmative is (as we saw) proved from premisses which are both affirmative. The negative is proved as follows. Let A belong to every B, and to no C: we conclude that B belongs to no C. If then it is assumed that B belongs to every A, it is necessary that [20] A should belong to no C; for we get the second figure, with B as middle. But if AB is negative, and the other affirmative, we shall have the first figure. For C belongs to every A, and B to no C, consequently B belongs to no A; neither, then, does A belong [25] to B. Through the conclusion, therefore, and one proposition, we get no deduction, but if another is assumed in addition, a deduction will be possible. But if the deduction is not universal, the universal proposition cannot be proved, for the same reason as we gave above; but the particular can be proved whenever the universal is affirmative. Let A belong to every B, and not to every C: the conclusion [30] is BC. If then it is assumed that B belongs to every A, but not to every C. A will not belong to some C, B being middle. But if the universal is negative, the proposition AC will not be proved by the conversion of AB; for it turns out that either both or [35] one of the propositions is negative; consequently a deduction will not be possible. But the proof will proceed as in the universal cases, if it is assumed that A belongs to some of that to some of which B does not belong.

7 · In the third figure, when both propositions are taken universally, it is not [40] possible to prove them reciprocally; for that which is universal is proved through [59a1] propositions which are universal, but the conclusion in this figure is always particular, so that it is clear that it is not possible at all to prove through this figure the universal proposition. But if one is universal, the other particular, proof will [5] sometimes be possible, sometimes not. When both are affirmative, and the universal concerns the minor extreme, proof will be possible, but when it concerns the other extreme, impossible. Let A belong to every C and B to some C: the conclusion is AB. If then it is assumed that C belongs to every A, it has been proved that C belongs to [10] some B, but that B belongs to some C has not been proved. And yet it is necessary, if C belongs to some B, that B should belong to some C. But it is not the same that this should belong to that, and that to this; but we must assume besides that if this belongs to some of that, that belongs to some of this. But if this is assumed the deduction no longer results from the conclusion and the other proposition. But if B [15] belongs to every C, and A to some C, it will be possible to prove AC, when it is assumed that C belongs to every B, and A to some B. For if C belongs to every B and A to some B, it is necessary that A should belong to some C, B being middle.

And whenever one is affirmative, the other negative, and the affirmative is [20] universal, the other can be proved. Let B belong to every C, and A not to some C: the conclusion is that A does not belong to some B. If then it is assumed further that C belongs to every B, it is necessary that A should not belong to some C, B being [25] middle. But when the negative is universal, the other is not proved, except as before, viz. if it is assumed that that belongs to some of that, to some of which this does not belong, e.g. if A belongs to no C, and B to some C: the conclusion is that A does not belong to some C. If then it is assumed that C belongs to some of that to some of which A does not belong, it is necessary that C should belong to some B. In no other [30] way is it possible by converting the universal proposition to prove the other; for in no other way can a deduction be formed.

[It is clear then that in the first figure reciprocal proof is made both through the third and through the first figure—if the conclusion is affirmative through the [35] first; if the conclusion is negative through the last. For it is assumed that that belongs to all of that to none of which this belongs. In the middle figure, when the deduction is universal, proof is possible through the second figure and through the first, but when particular through the second and the last. In the third figure all proofs are made through itself. It is clear also that in the third figure and in the [40] middle figure those deductions which are not made through those figures themselves either are not of the nature of circular proof or are imperfect.]20

[59b1] 8 · To convert is to alter the conclusion and make a deduction to prove that either the extreme does not belong to the middle or the middle to the last term. For it is necessary, if the conclusion has been converted and one of the propositions [5] stands, that the other should be destroyed. For if it should stand, the conclusion also must stand. It makes a difference whether the conclusion is converted into its opposite or into its contrary. For the same deduction does not result whichever form the conversion takes. This will be made clear by the sequel. (By opposition I mean the relation of ‘to every’ to ‘not to every’, and of ‘to some’ to ‘to none’; by contrarily I [10] mean the relation of ‘to every’ to ‘to none’, and of ‘to some’ to ‘not to some’.) Suppose that A has been proved of C, through B as middle term. If then it should be assumed that A belongs to no C, but to every B, B will belong to no C. And if A belongs to no C, and B to every C, A will belong, not to no B at all, but not to every B. For (as we saw) the universal is not proved through the last figure. In a word it is not [15] possible to refute universally by conversion the proposition which concerns the major extreme; for the refutation always proceeds through the third figure; since it is necessary to take both propositions in reference to the minor extreme. Similarly if the deduction is negative. Suppose it has been proved that A belongs to no C [20] through B. Then if it is assumed that A belongs to every C, and to no B, B will belong to no C And if A and B belong to every C, A will belong to some B; but in the original premiss it belonged to no B.

If the conclusion is converted into its opposite, the deductions will be opposite [25] and not universal. For one proposition is particular, so that the conclusion also will be particular. Let the deduction be affirmative, and let it be converted as stated. Then if A belongs not to every C, but to every B, B will belong not to every C. And if [30] A belongs not to every C, but B belongs to every C, A will belong not to every B. Similarly if the deduction is negative. For if A belongs to some C, and to no B, B will belong, not to no C at all, but not to some C. And if A belongs to some C, and B to [35] every C, as was originally assumed, A will belong to some B.

In particular deductions when the conclusion is converted into its opposite, both propositions may be refuted; but when it is converted into its contrary, neither. For the result is no longer, as in the universal cases, a refutation in which the [40] conclusion reached by conversion lacks universality, but no refutation at all. Suppose that A has been proved of some C. If then it is assumed that A belongs to no [60a1] C, and B to some C, A will not belong to some B; and if A belongs to no C, but to every B, B will belong to no C. Thus both are refuted. But neither can be refuted if the conclusion is converted into its contrary. For if A does not belong to some C, but [5] to every B, then B will not belong to some C. But the original premiss is not yet refuted; for it is possible that B should belong to some C, and should not belong to some C. The universal AB cannot be affected by a deduction at all; for if A does not belong to some C, but B belongs to some C, neither of the propositions is universal. [10] Similarly if the deduction is negative; for if it should be assumed that A belongs to every C, both are refuted; but if the assumption is that A belongs to some C, neither is. The demonstration is the same as before.

9 · In the second figure it is not possible to refute the proposition which [15] concerns the major extreme by establishing something contrary to it, whichever form the conversion may take. For the conclusion will always be in the third figure, and in this figure (as we saw) there is no universal deduction. The other can be refuted in a manner similar to the conversion: I mean, if the conclusion is converted into its contrary, contrarily; if into its opposite, oppositely. Let A belong to every B [20] and to no C: conclusion BC. If then it is assumed that B belongs to every C, and AB stands, A will belong to every C, since the first figure is produced. If B belongs to [25] every C, and A to no C, then A belongs not to every B: the figure is the last. But if BC is converted into its opposite, AB will be proved as before, AC oppositely. For if B belongs to some C, and A to no C, then A will not belong to some B. Again if B [30] belongs to some C, and A to every B, A will belong to some C, so that the deduction is opposite. A similar proof can be given if the propositions are the other way about.

If the deduction is particular, when the conclusion is converted into its contrary neither proposition can be refuted, as also happened in the first figure, but [35] if the conclusion is converted into its opposite, both can be refuted. Suppose that A belongs to no B, and to some C: the conclusion is BC. If then it is assumed that B belongs to some C, and AB stands, the conclusion will be that A does not belong to some C. But the original statement has not been refuted; for it is possible that A [40] should belong to some C and also not to some C. Again if B belongs to some C and A to some C, no deduction will be possible; for neither of the assumptions is universal. [60b1] Consequently AB is not refuted. But if the conclusion is converted into its opposite, both can be refuted. For if B belongs to every C, and A to no B, A will belong to no C; but it was assumed to belong to some C. Again if B belongs to every C and A to some C, A will belong to some B. The same demonstration can be given if the [5] universal is affirmative.

10 · In the third figure when the conclusion is converted into its contrary, neither of the propositions can be refuted in any of the deductions, but when the conclusion is converted into its opposite, both may be refuted and in all the moods. [10] Suppose it has been proved that A belongs to some B, C being taken as middle, and the propositions being universal. If then it is assumed that A does not belong to some B, but B belongs to every C, no deduction is formed about A and C. Nor if A does not belong to some B, but belongs to every C, will a deduction be possible about B [15] and C. A similar proof can be given if the propositions are not universal. For either both propositions arrived at by the conversion must be particular, or the universal must refer to the minor extreme. But we found that no deduction is possible thus either in the first or in the middle figure. But if the conclusion is converted into its [20] opposite, both the propositions can be refuted. For if A belongs to no B, and B to every C, then A belongs to no C; again if A belongs to no B, and to every C, B belongs to no C. And similarly if one is not universal. For if A belongs to no B, and B to some C, A will not belong to some C; if A belongs to no B, and to every C, B will [25] belong to no C.

From what has been said it is clear how a deduction results in each figure when [5] the conclusion is converted; and when it is contrary to the proposition, and when opposite. It is clear that in the first figure the deductions are formed through the middle and the last figures, and the proposition which concerns the minor extreme is always refuted through the middle figure, that which concerns the major through [10] the last figure. In the second figure deductions proceed through the first and the last figures, and the proposition which concerns the minor extreme is always refuted through the first figure, that which concerns the major extreme through the last. In the third figure the deductions proceed through the first and the middle figures; the proposition which concerns the major is always refuted through the first figure, that [15] which concerns the minor through the middle figure.

11 · It is clear then what conversion is, how it is effected in each figure, and what deduction results. Deduction per impossibile is proved when the contradictory of the conclusion is posited and another proposition is assumed; it can be made in all [20] the figures. For it resembles conversion, differing only in this: conversion takes place after a deduction has been formed and both the propositions have been assumed, but a reduction to the impossible takes place not because the opposite has been agreed to already, but because it is clear that it is true. The terms are alike in [25] both, and the premisses of both are assumed in the same way. For example if A belongs to every B, C being middle, then if it is supposed that A does not belong to every B or belongs to no B, but to every C (which was true), it follows that C belongs to no B or not to every B. But this is impossible; consequently the supposition is [30] false; its opposite then is true. Similarly in the other figures; for whatever moods admit of conversion admit also of deduction per impossibile.

All the problems can be proved per impossibile in all the figures, excepting the universal affirmative, which is proved in the middle and third figures, but not in the [35] first. Suppose that A belongs not to every B, or to no B, and take besides another proposition concerning either of the terms, viz. that C belongs to every A, or that B belongs to every D; thus we get the first figure. If then it is supposed that A does not [40] belong to every B, no deduction results whichever term the assumed proposition concerns; but if it is supposed that A belongs to no B, when BD is assumed as well we [61b1] shall deduce what is false, but not the problem proposed. For if A belongs to no B, and B belongs to every D, A belongs to no D. Let this be impossible: it is false then [5] that A belongs to no B. But the universal affirmative is not necessarily true if the universal negative is false. But if CA is assumed as well, no deduction results, nor does it do so when it is supposed that A does not belong to every B. Consequently it is clear that the universal affirmative cannot be proved in the first figure per [10] impossibile.

But the particular affirmative and the universal and particular negatives can all be proved. Suppose that A belongs to no B, and let it have been assumed that B belongs to every or to some C. Then it is necessary that A should belong to no C or not to every C. But this is impossible (for let it be true and clear that A belongs to [15] every C); consequently if this is false, it is necessary that A should belong to some B. But if the other proposition assumed relates to A, no deduction will be possible. Nor can a conclusion be drawn when the contrary of the conclusion is supposed, i.e. that A does not belong to some B. Clearly then we must suppose the opposite.

Again suppose that A belongs to some B, and let it have been assumed that C [20] belongs to every A. It is necessary then that C should belong to some B. But let this be impossible, so that the supposition is false: in that case it is true that A belongs to no B. We may proceed in the same way if CA has been taken as negative. But if the proposition assumed concerns B, no deduction will be possible. If the contrary is [25] supposed, we shall have a deduction and an impossible conclusion, but the problem in hand is not proved. Suppose that A belongs to every B, and let it have been assumed that C belongs to every A. It is necessary then that C should belong to every B. But this is impossible, so that it is false that A belongs to every B. But we have not yet shown it to be necessary that A belongs to no B, if it does not belong to [30] every B. Similarly if the other proposition taken concerns B; we shall have a deduction and a conclusion which is impossible, but the supposition is not refuted. Therefore it is the opposite that we must suppose.

To prove that A does not belong to every B, we must suppose that it belongs to [35] every B; for if A belongs to every B, and C to every A, then C belongs to every B; so that if this is impossible, the supposition is false. Similarly if the other proposition assumed concerns B. The same results if CA is negative; for thus also we get a deduction. But if the negative concerns B, nothing is proved. If the supposition is [40] that A belongs not to every but to some B, it is not proved that A belongs not to every B, but that it belongs to no B. For if A belongs to some B, and C to every A, then C [62a1] will belong to some B. If then this is impossible, it is false that A belongs to some B; consequently it is true that A belongs to no B. But if this is proved, the truth is refuted as well; for the original conclusion was that A belongs to some B, and does not belong to some B. Further nothing impossible results from the supposition; for [5] then the supposition would be false, since it is impossible to deduce a false conclusion from true premisses; but in fact it is true; for A belongs to some B. Consequently we must not suppose that A belongs to some B, but that it belongs to every B. Similarly if we should be proving that A does not belong to some B; for if [10] not to belong to some and to belong not to every are the same, the demonstration of both will be identical.

It is clear then that not the contrary but the opposite ought to be supposed in all the deductions. For thus we shall have the necessity, and the claim we make will be reputable. For if of everything either the affirmation or the negation holds good, then if it is proved that the negation does not hold, the affirmation must be true. [15] Again if it is not admitted that the affirmation is true, the claim that the negation is true will be reputable. But in neither way does it suit to maintain the contrary; for it is not necessary that if the universal negative is false, the universal affirmative should be true, nor is it reputable that if the one is false the other is true.

12 · It is clear then that in the first figure all problems except the universal [20] affirmative are proved per impossibile. But in the middle and the last figures this also is proved. Suppose that A does not belong to every B, and let it have been assumed that A belongs to every C. If then A belongs not to every B, but to every C, [25] C will not belong to every B. But this is impossible (for suppose it to be clear that C belongs to every B); consequently the supposition is false. It is true then that A belongs to every B. But if the contrary is supposed, we shall have a deduction and a result which is impossible; but the problem in hand is not proved. For if A belongs to [30] no B, and to every C, C will belong to no B. This is impossible; so that it is false that A belongs to no B. But though this is false, it does not follow that it is true that A belongs to every B.

If we want to prove that A belongs to some B, suppose that A belongs to no B, and let A belong to every C. It is necessary then that C should belong to no B. Consequently, if this is impossible, A must belong to some B. But if it is supposed [35] that A does not belong to some B, we shall have the same results as in the first figure.

Again suppose that A belongs to some B, and let A belong to no C. It is necessary then that C should not belong to some B. But originally it belonged to every B; consequently the supposition is false; A then will belong to no B. [40]

If we want to prove that A does not belong to every B, suppose it does belong to every B, and to no C. It is necessary then that C should belong to no B. But this is [62b1] impossible; so that it is true that A does not belong to every B. It is clear then that all the deductions can be formed in the middle figure.

13 · Similarly they can all be formed in the last figure. Suppose that A does [5] not belong to some B, but C belongs to every B; then A does not belong to some C. If then this is impossible, it is false that A does not belong to some B; so that it is true that A belongs to every B. But if it is supposed that A belongs to no B, we shall have a deduction and a conclusion which is impossible; but the problem in hand is not proved; for if the contrary is supposed, we shall have the same results as before. [10]

But to prove that A belongs to some B, this supposition must be made. If A belongs to no B, and C to some B, A will belong not to every C. If then this is false, it is true that A belongs to some B.

To prove that A belongs to no B, suppose A belongs to some B, and let it have [15] been assumed that C belongs to every B. Then it is necessary that A should belong to some C. But ex hypothesi it belongs to no C, so that it is false that A belongs to some B. But if it is supposed that A belongs to every B, the problem is not proved.

But this supposition must be made if we are to prove that A belongs not to [20] every B. For if A belongs to every B and C to every B, then A belongs to some C. But this we assumed not to be so, so it is false that A belongs to every B. But in that case it is true that A belongs not to every B. If however it is supposed that A belongs to some B, we shall have the same result as before.

[25] It is clear then that in all the deductions which proceed per impossibile the opposite must be supposed. And it is plain that in the middle figure an affirmative conclusion, and in the last figure a universal conclusion, are proved in a way.

14 · Demonstration per impossibile differs from probative demonstration in [30] that it posits what it wishes to refute by reduction to a statement admitted to be false; whereas probative demonstration starts from admitted positions. Both, indeed, take two propositions that are admitted, but the latter takes the premisses from which the deduction starts, the former takes one of these, along with the [35] contradictory of the conclusion. Also in the probative case it is not necessary that the conclusion should be familiar, nor that one should suppose beforehand that it is true or not; in the other it is necessary to suppose beforehand that it is not true. It makes no difference whether the conclusion is affirmative or negative; the method is the same in both cases.

[40] Everything which is concluded probatively can be proved per impossibile, and that which is proved per impossibile can be proved probatively, through the same [63a1] terms. Whenever the deduction is formed in the first figure, the truth will be found in the middle or the last figure, if negative in the middle, if affirmative in the last. Whenever the deduction is formed in the middle figure, the truth will be found in [5] the first, whatever the problem may be. Whenever the deduction is formed in the last figure, the truth will be found in the first and middle figures, if affirmative in the first, if negative in the middle. Suppose that A has been proved to belong to no B, or not to every B, through the first figure. Then the supposition must have been that [10] A belongs to some B, and it was assumed that C belongs to every A and to no B. For thus the deduction was made and the impossible conclusion reached. But this is the middle figure, if C belongs to every A and to no B. And it is clear from this that A belongs to no B. Similarly if A has been proved not to belong to every B. For the [15] supposition is that A belongs to every B; and it was assumed that C belongs to every A but not to every B. Similarly too, if CA should be negative; for thus also we have the middle figure. Again suppose it has been proved that A belongs to some B. The [20] supposition here is that A belongs to no B; and it was assumed that B belongs to every C, and A either to every or to some C; for in this way we shall get what is impossible. But if A and B belong to every C, we have the last figure. And it is clear from this that A must belong to some B. Similarly if B or A should be assumed to belong to some C.

[25] Again suppose it has been proved in the middle figure that A belongs to every B. Then the supposition must have been that A belongs not to every B, and it was assumed that A belongs to every C, and C to every B; for thus we shall get what is impossible. But if A belongs to every C, and C to every B, we have the first figure. Similarly if it has been proved that A belongs to some B; for the supposition then [30] must have been that A belongs to no B, and it was assumed that A belongs to every C, and C to some B. If the deduction is negative, the supposition must have been that A belongs to some B, and it was assumed that A belongs to no C, and C to every B, so that the first figure results. If the deduction is not universal, but proof has been [35] given that A does not belong to some B, we may infer in the same way. The supposition is that A belongs to every B, and it was assumed that A belongs to no C, and C belongs to some B; for thus we get the first figure.

Again suppose it has been proved in the third figure that A belongs to every B. [40] Then the supposition must have been that A belongs not to every B, and it was assumed that C belongs to every B, and A belongs to every C; for thus we shall get [63b1] what is impossible. And this is the first figure. Similarly if the demonstration establishes a particular proposition: the supposition then must have been that A belongs to no B, and it was assumed that C belongs to some B, and A to every C. If the deduction is negative, the supposition must have been that A belongs to some B, [5] and it was assumed that C belongs to no A and to every B; and this is the middle figure. Similarly if the demonstration is not universal. The supposition will then be that A belongs to every B, and it was assumed that C belongs to no A and to some B;[10] and this is the middle figure.

It is clear then that it is possible through the same terms to prove each of the problems probatively as well. Similarly it will be possible if the deductions are probative to reduce them ad impossibile in the terms which have been taken, [15] whenever the opposite of the conclusion is taken as a premiss. For the deductions become identical with those which are obtained by means of conversion, so that we obtain immediately the figures through which each problem will be solved. It is clear then that every problem can be proved in both ways, i.e. per impossibile and [20] probatively, and it is not possible to separate one method from the other.

15 · In what figure it is possible to draw a conclusion from propositions which are opposed, and in what figure this is not possible, will be made clear in this way. Verbally four kinds of opposition are possible, viz. ‘to every’-‘to no’, ‘to [25] every’-‘not to every’, ‘to some’-‘to no’, ‘to some’-‘not to some’; but in reality there are only three, for ‘to some’ is only verbally opposed to ‘not to some’. Of these I call those which are universal contraries (‘to every’-‘to no’, e.g. ‘every science is good’, ‘no science is good’); the others I call opposites. [30]

In the first figure no deduction whether affirmative or negative can be made out of opposed propositions: no affirmative deduction is possible because both propositions must be affirmative, but opposites are the one affirmative, the other negative; no negative deduction is possible because opposites affirm and deny the [35] same predicate of the same subject, and the middle term in the first figure is not predicated of both extremes, but one thing is denied of it, and it is affirmed of something else and such propositions are not opposed.

In the middle figure a deduction can be made both of opposites and of contraries. Let A stand for good, let B and C stand for science. If then one assumes [64a1] that every science is good, and no science is good, A belongs to every B and to no C, so that B belongs to no C; no science, then, is a science. Similarly if after assuming [5] that every science is good one assumed that the science of medicine is not good; for A belongs to every B but to no C, so that a particular science will not be a science. Again, if A belongs to every C but to no B, and B is science, C medicine, and A supposition; for after assuming that no science is supposition, one has assumed that [10] a particular science is supposition. This differs from the preceding deduction because the relations between the terms are converted: before, the affirmative concerned B, now it concerns C. Similarly if one proposition is not universal; for the middle term is always that which is said negatively of one extreme, and affirmatively [15] of the other. Consequently it is possible that opposites may lead to a conclusion, though not always or in every mood, but only if the terms subordinate to the middle are such that they are either identical or related as whole to part. Otherwise it is impossible; for the propositions cannot anyhow be either contraries or opposites.

[20] In the third figure an affirmative deduction can never be made out of opposite propositions, for the reason given in reference to the first figure; but a negative deduction is possible whether the terms are universal or not. Let B and C stand for [25] science, A for medicine. If then one should assume that all medicine is science and that no medicine is science, he has assumed that B belongs to every A and C to no A, so that some science will not be a science. Similarly if the proposition BA is not assumed universally; for if some medicine is science and again no medicine is [30] science, it results that some science is not science. The propositions are contrary if the terms are taken universally; if one is particular, they are opposite.

We must recognize that it is possible to take opposites in the way we said, viz. [35] ‘all science is good’ and ‘no science is good’ or ‘some science is not good’. This does not usually escape notice. But it is possible to establish one of the opposites by way of other questions, or to assume it in the way suggested in the Topics.21 Since there are three oppositions to affirmations, it follows that opposites may be assumed in six ways—either to all and to no, or to all and not to all, or to some and to no; and the [64b1] relations between the terms may be converted; e.g. A may belong to every B and to no C, or to every C and to no B, or to every of the one, not to every of the other; here too the relation between the terms may be converted. Similarly in the third figure. [5] So it is clear in how many ways and in what figures a deduction can be made by means of propositions which are opposed.

It is clear too that from false premisses it is possible to draw a true conclusion, as has been said before, but it is not possible if the premisses are opposed. For the [10] deduction is always contrary to the fact, e.g. if a thing is good, it is deduced that it is not good, if an animal, that it is not an animal, because the deduction springs out of a contradiction and the terms presupposed are either identical or related as whole and part. It is evident also that in fallacious reasonings nothing prevents a contradiction to the supposition from resulting, e.g. if something is odd, that it is not [15] odd. For the deduction owed its contrariety to its opposite premisses: if we assume such premisses we shall get a result that contradicts our supposition. But we must recognize that contraries cannot be inferred from a single deduction in such a way that we conclude that what is not good is good, or anything of that sort, unless a proposition of that form is at once assumed (e.g. every animal is white and not [20] white, and man is an animal). Either we must introduce the contradiction by an additional assumption, assuming, e.g., that every science is supposition, and then assuming that medicine is a science, but none of it is supposition (which is the mode in which refutations are made); or we must argue from two deductions. In no other [25] way than this, as was said before, is it possible that the assumptions should be really contrary.

16 · To beg and assume the point at issue is a species of failure to demonstrate the problem proposed; but this happens in many ways. A man may not [30] deduce at all, or he may argue from premisses which are more unknown or equally unknown, or he may establish what is prior by means of what is posterior; for demonstration proceeds from what is more convincing and prior. Now begging the point at issue is none of these; but since some things are naturally known through [35] themselves, and other things by means of something else (the first principles through themselves, what is subordinate to them through something else), whenever a man tries to prove by means of itself what is not known by means of itself, then he begs the point at issue. This may be done by claiming what is at issue at once; it is also possible to make a transition to other things which would naturally be proved through the point at issue, and demonstrate it through them, e.g. if A should [65a1] be proved through B, and B through C, though it was natural that C should be proved through A; for it turns out that those who reason thus are proving A by means of itself. This is what those persons do who suppose that they are constructing parallel lines; for they fail to see that they are assuming facts which it [5] is impossible to demonstrate unless the parallels exist. So it turns out that those who reason thus merely say a particular thing is, if it is: in this way everything will be known by means of itself. But that is impossible.

If then it is uncertain whether A belongs to C, and also whether A belongs to B, [10] and if one should assume that A does belong to B, it is not yet clear whether he begs the point at issue, but it is evident that he is not demonstrating; for what is as uncertain as the question to be answered cannot be a principle of a demonstration. If however B is so related to C that they are identical, or if they are plainly convertible, or the one inheres in the other, the point at issue is begged. For one [15] might equally well prove that A belongs to B through those terms if they are convertible. (As it is, things prevent such a demonstration, but the method does not.) But if one were to make the conversion, then he would be doing what we have described and effecting a reciprocal proof with three propositions.

If then begging the point at issue is proving by means of itself what is not clear by means of itself, in other words failing to prove when the failure is due to the thesis to be proved and that through which it is proved being equally uncertain, either because predicates which are identical belong to the same subject, or because the same predicate belongs to subjects which are identical, the point at issue may be [30] begged in the middle and third figures in both ways, though, if the deduction is affirmative, only in the third and first figures. If the deduction is negative, it occurs when identical predicates are denied of the same subject; and both propositions do not beg the point at issue in the same way (similarly in the middle figure), because [35] the terms in negative deductions are not convertible. In demonstrations the point at issue is begged when the terms are really related in the manner described, in dialectical arguments when they are believed to be so related.

17 · The objection that this is not the reason why the result is false, which we [65b1] frequently make in argument, arises first in the case of a reductio ad impossibile, when it is used to contradict that which was being proved by the reduction. For unless a man has contradicted this proposition he will not say, ‘That is not the reason’, but urge that something false has been assumed in the earlier parts of the argument; nor will he use the formula in the case of a probative demonstration; for here what one contradicts is not posited. Further when anything is refuted [5] probatively by ABC, it cannot be objected that the deduction does not depend on the assumption laid down. For we say that something comes about not for that reason, when the deduction is concluded in spite of the refutation of this; but that is not possible in probative cases; since if an assumption is refuted, a deduction can no longer be drawn in reference to it. It is clear then that ‘Not for that reason’ can only [10] be used in the case of a reductio ad impossibile, and when the original supposition is so related to the impossible conclusion, that the conclusion results indifferently whether the supposition is made or not.

The most obvious case in which the falsity does not come about by reason of [15] the supposition is when a deduction drawn from middle terms to an impossible conclusion is independent of the supposition, as we have explained in the Topics.22 For to put that which is not the cause as the cause, is just this: e.g. if a man, wishing to prove that the diagonal of the square is incommensurate with the side, should try to prove Zeno’s theorem that motion is impossible, and so establish a reductio ad [20] impossibile; for the falsity has no connexion at all with the original assumption. Another case is where the impossible conclusion is connected with the supposition, but does not result from it. This may happen whether one traces the connexion [25] upwards or downwards, e.g. if it is laid down that A belongs to B, B to C, and C to D, and it is false that B belongs to D; for if we eliminated A and assumed all the same that B belongs to C and C to D, the false conclusion would not depend on the original supposition. Or again trace the connexion upwards; e.g. suppose that A belongs to B, E to A, and F to E, it being false that F belongs to A. In this way too [30] the impossible conclusion would result, though the original supposition were eliminated. But the impossible conclusion ought to be connected with the original terms: in this way it will depend on the supposition, e.g. when one traces the connexion downwards, the impossible conclusion must be connected with the term which is predicate; for if it is impossible that A should belong to D, the false [35] conclusion will no longer result after A has been eliminated. If one traces the connexion upwards, the impossible conclusion must be connected with the term which is subject; for if it is impossible that F should belong to B, the impossible conclusion will disappear if B is eliminated. Similarly when the deductions are negative.

It is clear then that when the impossibility is not related to the original terms, [66a1] the falsity does not result by reason of the supposition. Or perhaps even so it may sometimes be independent. For if it were laid down that A belongs not to B but to K, and that K belongs to C and C to D), the impossible conclusion would still stand (similarly if one takes the terms in an ascending series); consequently since the [5] impossibility results whether the first assumption is suppressed or not, it does not hold by reason of the supposition. Or perhaps we ought not to understand the statement that the false conclusion results even if the assumption does not hold, in the sense that if something else were supposed the impossibility would result; but rather in the sense that when it is eliminated, the same impossibility results through [10] the remaining propositions; since it is not perhaps absurd that the same false result should follow from several suppositions, e.g. that parallels meet, both on the assumption that the interior angle is greater than the exterior and on the assumption that a triangle contains more than two right angles. [15]

18 · A false argument comes about by reason of the first falsity in it. Every deduction is made out of two or more propositions. If then it is drawn from two, one or both of them must be false; for (as was proved) a false deduction cannot be drawn from true premisses. But if from more than two, e.g. if C is established through A [20] and B, and these through D, E, F, and G, one of these higher propositions must be false, and the argument fails by reason of this; for A and B are inferred by means of them. Therefore the conclusion and the falsity come about by reason of one of them.

19 · In order to avoid being argued down, we must take care, whenever an [25] opponent sets up an argument without disclosing the conclusions, not to grant him the same term twice over in his propositions, since we know that a deduction cannot be drawn without a middle term, and that term which is stated more than once is the middle. How we ought to watch out for the middle in reference to each conclusion, is evident from our knowing what kind of thesis is proved in each figure. This will [30] not escape us since we know how we are maintaining the argument.

That which we urge men to beware of in their admissions, they ought in attack to try to conceal. This will be possible first, if, instead of drawing the conclusions of [35] preliminary deductions, they make the necessary assumptions and leave the conclusions in the dark; secondly if instead of inviting assent to propositions which are closely connected they take as far as possible those that are not connected by middle terms. For example suppose that A is to be inferred to be true of F; B, C, D, and E being middle terms. One ought then to ask whether A belongs to B, and next whether D belongs to E, instead of asking whether B belongs to C; after that he may [66b1] ask whether B belongs to C, and so on. If the deduction is drawn through one middle term, he ought to begin with that: in this way he will most likely deceive his opponent.

20 · Since we know when a deduction can be formed and how its terms must [5] be related, it is clear when refutation will be possible and when impossible. A refutation is possible whether everything is conceded, or the answers alternate (one, I mean, being affirmative, the other negative). For, as has been shown, a deduction is possible both in the former and in the latter case: consequently, if what is laid [10] down is contrary to the conclusion, a refutation must take place; for a refutation is a deduction which establishes the contradictory. But if nothing is conceded, a refutation is impossible; for no deduction is possible (as we saw) when all the terms are negative; therefore no refutation is possible. For if a refutation were possible, a [15] deduction must be possible; although if a deduction is possible it does not follow that a refutation is possible. Similarly refutation is not possible if nothing is conceded universally; since refutation and deduction are defined in the same way.

21 · It sometimes happens that just as we are deceived in the arrangement of the terms, so error may arise in our thought about them, e.g. if it is possible that the [20] same predicate should belong to more than one subject primarily, but although knowing the one, a man may forget the other and think the predicate belongs to none of it. Suppose that A belongs to B and to C in virtue of themselves, and that B and C belong to every D in the same way. If then a man thinks that A belongs to every B, and B to D, but A to no C, and C to every D, he will have knowledge and [25] ignorance of the same thing in respect of the same thing. Again if a man were to make a mistake about the members of a single series; e.g. suppose A belongs to B, B to C, and C to D, but someone thinks that A belongs to every B, but to no C: he will [30] both know that A belongs to C, and believe that it does not. Does he then actually maintain after this that what he knows, he does not believe? For he knows in a way that A belongs to C through B, knowing the particular by virtue of his universal knowledge; so that what he knows in a way, this he maintains he does not believe at all; but that is impossible.

[35] In the former case, where the middle term does not belong to the same series, it is not possible to believe both the propositions with reference to each of the two middle terms: e.g. that A belongs to every B, but to no C, and both B and C belong to every D. For it turns out that the first proposition is either wholly or partially contrary. For if he believes that A belongs to everything to which B belongs, and he [67a1] knows that B belongs to D, then he knows that A belongs to D. Consequently if again he thinks that A belongs to nothing to which C belongs, he does not think that A belongs to some of that to which B belongs; but if he thinks that A belongs to everything to which B belongs, and again does not think that A belongs to some of that to which B belongs, these beliefs are wholly or partially contrary. [5]

In this way then it is not possible to believe; but nothing prevents a man believing one proposition of each deduction or both of one: e.g. A belongs to every B, and B to D, and again A belongs to no C. An error of this kind is similar to the error into which we fall concerning particulars: e.g. if A belongs to everything to which B belongs, and B to every C, A will belong to every C. If then a man knows that A [10] belongs to everything to which B belongs, he knows also that A belongs to C. But nothing prevents his being ignorant that C exists; e.g. let A stand for two right angles, B for triangle, C for a sensible triangle. A man might believe that C did not exist, though he knew that every triangle contains two right angles; consequently he [15] will know and not know the same thing at the same time. For knowing that every triangle has its angles equal to two right angles is not simple—it may obtain either by having universal knowledge or by particular. Thus by universal knowledge he knows that C contains two right angles, but not by particular; consequently his [20] knowledge will not be contrary to his ignorance. The argument in the Meno23 that learning is recollection may be criticized in a similar way. For it never happens that a man has foreknowledge of the particular, but in the process of induction he receives a knowledge of the particulars, as though by an act of recognition. For we know some things directly; e.g. that the angles are equal to two right angles, if we [25] see that the figure is a triangle. Similarly in all other cases.

By universal knowledge then we see the particulars, but we do not know them by the kind of knowledge which is proper to them; consequently it is possible that we may make mistakes about them, but not that we should have the knowledge and error that are contrary to one another: rather we have universal knowledge but make a mistake in regard to the particular. Similarly in the cases stated above. The [30] error in respect of the middle term is not contrary to the knowledge obtained through the deduction, nor is the belief in respect of the middle terms. Nothing prevents a man who knows both that A belongs to the whole of B, and that B again belongs to C, thinking that A does not belong to C, e.g. knowing that every mule is [35] sterile and that this is a mule, and thinking that this animal is with foal; for he does not know that A belongs to C, unless he considers the two things together. So it is evident that if he knows the one and does not know the other, he will fall into error. And this is the relation of universal knowledge to particular. For we know no sensible thing, once it has passed beyond the range of our senses, even if we happen [67b1] to have perceived it, except by means of the universal and by possessing (but not actualising) particular. For knowing is spoken of in three ways: it may be either universal knowledge or knowledge proper to the matter in hand or actualising such [5] knowledge; consequently three kinds of error also are possible. Nothing then prevents a man both knowing and being mistaken about the same thing, provided that his knowledge and his error are not contrary. And this happens also to the man who knows each proposition separately and who has not previously considered the particular question. For when he believes that the mule is with foal he does not have [10] knowledge actualised, nor on the other hand has his belief caused an error contrary to his knowledge; for the error contrary to the universal knowledge would be a deduction.

But he who believes the essence of good is the essence of bad will believe the same thing to be the essence of good and the essence of bad. Let A stand for the essence of good and B for the essence of bad, and again C for the essence of good. [15] Since then he believes B and C identical, he will believe that C is B, and similarly that B is A; consequently that C is A. For just as we saw that if B is true of all of which C is true, and A is true of all of which B is true, and A is true of all of which B [20] is true, A is true of C, similarly with believing. Similarly also with being; for we saw that if C is the same as B, and B as A, C is the same as A. Similarly therefore with opining. Perhaps then this is necessary if a man will grant the first point. But presumably that is false, that any one could think the essence of good to be the [25] essence of bad (save accidentally—for it is possible to believe this in many different ways). But we must consider this matter better.

22 · Whenever the extremes are convertible it is necessary that the middle should be convertible with both. For if A belongs to C through B, then if A and C are [30] convertible and C belongs to everything to which A belongs, B is convertible with A, and B belongs to everything to which A belongs, through C as middle; and C is convertible with B through A as middle. Similarly in the negative case, e.g. if B belongs to C, but A does not belong to B, neither will A belong to C. If then B is [35] convertible with A, C will be convertible with A. Suppose B does not belong to A; neither then will C; for ex hypothesi B belonged to every C. And if C is convertible with B, A is convertible with it too; for C is said of that of all of which B is said. And if C is convertible in relation to A as well, B also will be convertible. For C belongs to [68a1] that to which B belongs; but C does not belong to that to which A belongs. And this alone starts from the conclusion: the others differ here from the affirmative deduction.

[5] Again if A and B are convertible, and similarly C and D, and if A or C must belong to anything whatever, then B and D will be such that one or other belongs to anything whatever. For since B belongs to that to which A belongs, and D belongs to that to which C belongs, and since A or C belongs to everything, but not together, it is clear that B or D belongs to everything, but not together. For two deductions have been put together. Again if A or B belongs to everything and if C or D belongs to everything, but they do not belong together, then when A and C are convertible B and D are convertible. For if B does not belong to something to which D belongs, it is [15] clear that A belongs to it. But if A then C; for they are convertible. Therefore C and D belong together. But this is impossible. For example if that which is uncreated is incorruptible and that which is incorruptible is uncreated, it is necessary that what [10] is created should be corruptible and what is corruptible should have been created.24

When A belongs to the whole of B and to C and is affirmed of nothing else, and B also belongs to every C, it is necessary that A and B should be convertible; for since A is said of B and C only, and B is affirmed both of itself and of C, it is clear [20] that B will be said of everything of which A is said, except A itself. Again when A and B belong to the whole of C, and C is convertible with B, it is necessary that A should belong to every B; for since A belongs to every C, and C to B by conversion, A will belong to every B.

When, of two opposites A and B, A is preferable to B, and similarly D is [25] preferable to C, then if A and C together are preferable to B and D together, A is preferable to D. For A is as much to be pursued as B is to be avoided, since they are opposites; and C is similarly related to D, since they also are opposites. If then A is as desirable as D, B is as much to be avoided as C (since each is to the same extent as [30] each—the one an object of aversion, the other an object of desire). Therefore A and C together will be as much to be desired or avoided as B and D together. But since A and C are preferable to B and D, A cannot be as desirable as D; for then B along with D would be as desirable as A along with C. But if D is preferable to A, then B must be less to be avoided than C; for the less is opposed to the less. But the greater [35] good and lesser evil are preferable to the lesser good and greater evil: the whole BD, then, is preferable to the whole AC. But ex hypothesi this is not so. A then is preferable to D, and C consequently is less to be avoided than B. If then every lover in virtue of his love would prefer A, viz. that the beloved should be such as to grant a [68b1] favour, and yet should not grant it (for which C stands), to the beloved’s granting the favour (represented by D) without being such as to grant it (represented by B), it is clear that A (being of such a nature) is preferable to granting the favour. To receive affection then is preferable in love to sexual intercourse. Love then aims at affection rather than at intercourse. And if it aims most at affection, then this is its end. Intercourse then either is not an end at all or is an end relative to the receiving [5] of affection. And indeed the same is true of the other desires and arts.

23 · It is clear then how the terms are related in conversion, and in respect of being preferable or more to be avoided. We must now state that not only dialectical and demonstrative deductions are formed by means of the aforesaid [10] figures, but also rhetorical deductions and in general any form of persuasion, however it may be presented. For every belief comes either through deduction or from induction.

Now induction, or rather the deduction which springs out of induction, consists [15] in deducing a relation between one extreme and the middle by means of the other extreme, e.g. if B is the middle term between A and C, it consists in proving through C that A belongs to B. For this is the manner in which we make inductions. For example, let A stand for long-liver, B for bileless, and C for the particular long-lived [20] animals, e.g. man, horse, mule. A then belongs to the whole of C; [for whatever is bileless is long-lived].25 But B also (not possessing bile) belongs to every C. If then C is convertible with B, and the middle term is not wider in extension, it is necessary that A should belong to B. For it has already been proved that if two things belong to [25] the same thing, and the extreme is convertible with one of them, then the other predicate will belong to one that is converted. But we must apprehend C as made up of all the particulars. For induction proceeds through an enumeration of all the cases.

[30] Such is the deduction which establishes primary and immediate propositions; for where there is a middle term the deduction proceeds through the middle term; when there is no middle term, through induction. And in a way induction is opposed to deduction; for the latter proves the extreme to belong to the third term by means of the middle, the former proves the extreme to belong to the middle by means of the [35] third. In the order of nature, deduction through the middle term is prior and more familiar, but deduction through induction is clearer to us.

24 · We have an example when the extreme is proved to belong to the middle by means of a term which resembles the third. It must be familiar both that the middle belongs to the third term, and that the first belongs to that which resembles the third. For example let A be evil, B making war against neighbours, C [69a1] Athenians against Thebans, D Thebans against Phocians. If then we wish to prove that to fight with the Thebans is an evil, we must assume that to fight against neighbours is an evil. Conviction of this is obtained from similar cases, e.g. that the [5] war against the Phocians was an evil to the Thebans. Since then to fight against neighbours is an evil, and to fight against the Thebans is to fight against neighbours, it is clear that to fight against the Thebans is an evil. Now it is clear that B belongs to C and to D (for both are cases of making war upon one’s neighbours) and that A [10] belongs to D (for the war against the Phocians did not turn out well for the Thebans); but that A belongs to B will be proved through D. Similarly if the conviction in the relation of the middle term to the extreme should be produced by several similar cases. Clearly then an example stands neither as part to whole, nor [15] as whole to part, but rather as part to part, when both are subordinate to the same term, and one of them is familiar. It differs from induction, because induction starting from all the particular cases proves (as we saw) that the extreme belongs to the middle, and does not connect the deduction to the extreme, whereas argument by example does make this connexion and does not draw its proof from all the particular cases.

[20] 25 · By reduction we mean an argument in which the first term clearly belongs to the middle, but the relation of the middle to the last term is uncertain though equally or more convincing than the conclusion; or again an argument in which the terms intermediate between the last term and the middle are few. For in any of these cases it turns out that we approach more nearly to knowledge. For [25] example let A stand for what can be taught, B for knowledge, C for justice. Now it is clear that knowledge can be taught; but it is uncertain whether virtue is knowledge. If now BC is equally or more convincing than AC, we have a reduction; for we are nearer to knowledge, since we have made an extra assumption, being before without knowledge that A belongs to C.26 Or again suppose that the terms intermediate between B and C are few; for thus too we are nearer knowledge. For example let D [30] stand for squaring, E for rectilinear figure, F for circle. If there were only one term intermediate between E and F (viz. that the circle is made equal to a rectilinear figure by the help of lunules), we should be near to knowledge. But when BC is not more convincing than AC, and the intermediate terms are not few, I do not call this [35] reduction; nor again when BC is immediate—for such a statement is knowledge.

26 · An objection is a proposition contrary to a proposition. It differs from a proposition, because it may be particular, but a proposition either cannot be particular at all or not in universal deductions. An objection is brought in two ways [69b1] and through two figures; in two ways because every objection is either universal or particular, by two figures because objections are brought in opposition to the proposition, and opposites can be proved only in the first and third figures. When a [5] man claims that something belongs to all of a given subject, we object either that it belongs to none or that it does not belong to some; and of these, the former is proved from the first figure, the latter from the third. For example let A stand for there being a single science, B for contraries. If a man proposes that contraries are subjects of a single science, the objection may be either that opposites are never [10] subjects of a single science, and contraries are opposites, so that we get the first figure; or that the knowable and the unknowable are not subjects of a single science—this is the third figure; for it is true of C (the knowable and the unknowable) that they are contraries, and it is false that they are the subjects of a single science.

Similarly if the proposition is negative. For if a man claims that contraries are [15] not subjects of a single science, we reply either that all opposites or that certain contraries, e.g. what is healthy and what is sickly, are subjects of the same science: the former argument issues from the first, the latter from the third figure.

In general, in all cases if a man urges a universal objection he must frame his contradiction with reference to the universal of the terms proposed, e.g. if a man [20] claims that contraries are not subjects of the same science, his opponent must reply that there is a single science of all opposites. Thus we must have the first figure; for the term which is universal relative to the original subject becomes the middle term.

If the objection is particular, the objector must frame his contradiction with reference to a term relatively to which the subject of the proposition is universal, e.g. he will point out that the knowable and the unknowable are not subjects of the same [25] science; for contraries are universal relatively to these. And we have the third figure; for the particular term assumed is middle, e.g. the knowable and the unknowable. Premisses from which it is possible to draw the contrary conclusion are what we start from when we try to make objections. Consequently we bring objections in these figures only; for in them only are opposite deductions possible, [30] since the second figure cannot produce an affirmative conclusion.

Besides, an objection in the middle figure would require a fuller argument, e.g. if it should not be granted that A belongs to B, because C does not follow B. This can [35] be made clear only by other propositions. But an objection ought not to turn off into other things, but have its other proposition quite clear immediately. [For this reason also this is the only figure from which proof by signs cannot be obtained.]27

We must consider too the other kinds of objection, namely the objection from contraries, from similars, and from common opinion, and inquire whether a [70a1] particular objection cannot be elicited from the first figure or a negative objection from the second.28

27 · A probability and a sign are not identical, but a probability is a reputable proposition: what men know to happen or not to happen, to be or not to be, [5] for the most part thus and thus, is a probability, e.g. envious men hate, those who are loved show affection. A sign is meant to be a demonstrative proposition either necessary or reputable; for anything such that when it is another thing is, or when it has come into being the other has come into being before or after, is a sign of the [10] other’s being or having come into being. An enthymeme is a deduction starting from probabilities or signs,29 and a sign may be taken in three ways, corresponding to the position of the middle term in the figures. For it may be taken as in the first figure or the second or the third. For example the proof that a woman is with child because she has milk is in the first figure: for to have milk is the middle term. Let A [15] represent to be with child, B to have milk, C woman. The proof that wise men are good, since Pittacus is good, comes through the last figure. Let A stand for good, B for wise men, C for Pittacus. It is true then to predicate both A and B of C—only men do not say the latter, because they know it, though they state the former. The [20] proof that a woman is with child because she is pale is meant to come through the middle figure; for since paleness follows women with child and is a concomitant of this woman, people suppose it has been proved that she is with child. Let A stand for paleness, B for being with child, C for woman.

Now if the one proposition is stated, we have only a sign, but if the other is [25] stated as well, a deduction, e.g. Pittacus is generous; for ambitious men are generous and Pittacus is ambitious. Or again: Wise men are good; for Pittacus is not only good but wise. In this way then deductions are formed, only that which [30] proceeds through the first figure is irrefutable if it is true (for it is universal), that which proceeds through the last figure is refutable even if the conclusion is true, since the deduction is not universal nor relevant to the matter in question; for though Pittacus is good, it is not therefore necessary that all other wise men should be good. But the deduction which proceeds through the middle figure is always [35] refutable in any case; for a deduction can never be formed when the terms are related in this way; for though a woman with child is pale, and this woman also is pale, it is not necessary that she should be with child. Truth then may be found in signs whatever their kind, but they have the differences we have stated.

We must either divide signs in the way stated, and among them designate the [70b1] middle term as the evidence (for people call that the evidence which makes us know, and the middle term above all has this character), or else we must call the arguments derived from the extremes signs, that derived from the middle term the evidence; for that which is proved through the first figure is most reputable and [5] most true.

It is possible to infer character from physical features, if it is granted that the body and the soul are changed together by the natural affections (No doubt by learning music a man has made some change in his soul, but this is not one of those affections which are natural to us; but rather such natural motions as anger and [10] desire.) If then this were granted and also that there is one sign for one affection, and if we could state the affection and sign proper to each kind of animal, we shall be able to infer character from physical features. For if there is an affection which belongs properly to an individual genus, e.g. courage to lions, it is necessary that [15] there should be a sign of it; for ex hypothesi body and soul are affected together. Suppose this sign is the possession of large extremities: this may belong to other genera also though not universally. For the sign is proper in the sense that it is proper to the whole genus, though not proper to it alone, according to our usual manner of speaking. This then will be found in other genera too, and man may be [20] brave, and some other genera of animal as well. They will then have the sign; for ex hypothesi there is one sign for one affection. If then this is so, and we can collect signs of this sort in these animals which have only one affection proper to them, and each affection has its sign, since it is necessary that it should have a single sign, we [25] shall then be able to infer character from physical features. But if the genus as a whole has two properties, e.g. if the lion is both brave and generous, how shall we know which of the signs which are its proper concomitants is the sign of which affection? Perhaps if both belong to some other genus though not to the whole of it, and if, in those genera in which each is found though not in the whole of their members, some members possess one of the affections and not the other: e.g. if a man is brave but not generous, but possesses, of the two signs, this one, it is clear [30] that this is the sign of courage in the lion also.

To judge character from physical features, then, is possible in the first figure if the middle term is convertible with the first extreme, but is wider than the third term and not convertible with it: e.g. let A stand for courage, B for large extremities, [35] and C for lion. B then belongs to everything to which C belongs, but also to others. But A belongs to everything to which B belongs, and to nothing besides, but is convertible with B: otherwise, there would not be one sign for one affection.