The Politics of Aristotle

by Aristotle

  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.

  Similarly if he should assume that B belongs to C, this being as uncertain as [20] the question whether A belongs to C, the point at issue is not yet begged, but no demonstration is made. If however A and B are identical either because they are convertible or because A follows B, then the point at issue is begged for the same reason as before. For we have explained the meaning of begging the point at issue, [25] viz. proving by means of itself that which is not clear by means of itself.

  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 b
elieved 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.