by Aristotle
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 sequel17 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 rela
ted 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