The Politics of Aristotle

by Aristotle

  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.

  Similarly if the deduction is negative. Suppose it has been proved that A does not belong to some B, BC being affirmative, AC being negative; for it was thus that, as we saw, a deduction could be made. Whenever then the contrary of the [30] conclusion is assumed a deduction will not be possible. For if A belongs to some B, and B to every C, no deduction is possible (as we saw) about A and C. Nor, if A belongs to some B, and to no C, was a deduction possible concerning B and C. Therefore the propositions are not refuted. But when the opposite of the conclusion is assumed, they are refuted. For if A belongs to every B, and B to C, A belongs to every C; but A was supposed originally to belong to no C. Again if A belongs to [35] every B, and to no C, then B belongs to no C; but it was supposed to belong to every C. A similar proof is possible if the propositions are not universal. For AC becomes universal and negative, the other premiss particular and affirmative. If then A belongs to every B, and B to some C, it results that A belongs to some C; but it was [40] supposed to belong to no C. Again if A belongs to every B, and to no C, then B belongs to no C; but it was assumed to belong to some C. If A belongs to some B and [61a1] B to some C, no deduction results; nor yet if A belongs to some B, and to no C. Thus in the former case the propositions are refuted, in the latter they are not.

  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 othe
r 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 f
igure.

  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]