It is clear then from what has been said that if the universal and negative proposition is necessary, a deduction is always possible, proving not merely that it [40] can not belong but also that it does not; but if the affirmative is necessary no conclusion can be drawn. It is clear too that a deduction is possible or not under the [39a1] same conditions whether simple or necessary. And it is clear that all the deductions are imperfect, and are completed by means of the figures mentioned.
[5] 20 · In the last figure a deduction is possible whether both or only one of the propositions is possible. When the propositions indicate possibility the conclusion will be possible; and also when one indicates possibility, the other belonging. But when the other is necessary, if it is affirmative the conclusion will be neither [10] necessary nor simple; but if it is negative there will be a deduction that it does not belong, as above. In these also we must understand the expression ‘possible’ in the conclusion in the same way as before.
[15] First let them be possible and suppose that both A and B may belong to every C. Since then the affirmative is convertible into a particular, and B may belong to every C, it follows that C may belong to some B. So, if A is possible for every C, and C is possible for some B, then A must be possible for some B. For we have got the [20] first figure. And if A may belong to no C, but B may belong to every C, it follows that A may not belong to some B; for we shall have the first figure again by conversion. But if both should be negative no necessary consequence will follow [25] from them as they are stated, but if the propositions are converted there will be a deduction as before. For if A and B may not belong to C, if ‘may belong’ is substituted we shall again have the first figure by means of conversion. But if one of the terms is universal, the other particular, a deduction will be possible, or not, [30] under the same arrangement of the terms as in the case of simple propositions. Suppose that A may belong to every C, and B to some C. We shall have the first figure again if the particular proposition is converted. For if A is possible for every [35] C, and C for some B, then A is possible for some B. Similarly if BC is universal. Likewise also if AC is negative, and BC affirmative; for we shall again have the first figure by conversion. But if both should be negative—the one universal and the [39b1] other particular—although no conclusion will follow from them as they are put, it will follow if they are converted, as above. But when both are indefinite or particular, no deduction can be formed; for A must belong both to every B and to no B. To illustrate the affirmative relation take the terms animal, man, white; to [5] illustrate the negative, take the terms horse, man, white, white being the middle term.
21 · If one of the propositions indicates belonging, the other possibility, the conclusion will be that it is possible, not that it belongs; and a deduction will be [10] possible under the same arrangement of the terms as before. First let them be affirmative: suppose that A belongs to every C, and B may belong to every C. If BC is converted, we shall have the first figure, and the conclusion that A may belong to some B. For when one of the propositions in the first figure indicates possibility, the [15] conclusion also (as we saw) is possible. Similarly if BC indicates belonging, AC possibility; or if AC is negative, BC affirmative, no matter which of the two is simple; in both cases the conclusion will be possible; for the first figure is obtained once more, and it has been proved that if one proposition indicates possibility in that [20] figure the conclusion also will be possible. But if the negative relates to the minor extreme, or if both are negative, no conclusion can be drawn from them as they stand, but if they are converted a deduction is obtained as before. [25]
If one of the propositions is universal, the other particular, then when both are affirmative, or when the universal is negative, the particular affirmative, we shall have the same sort of deductions; for all are completed by means of the first figure. So it is clear that the deduction will be not that it belongs but that it is possible. But [30] if the affirmative is universal, the negative particular, the proof will proceed by a reductio ad impossibile. Suppose that B belongs to every C, and A may not belong to some C: it follows that A may not belong to some B. For if A necessarily belongs [35] to every B, and B (as has been assumed) belongs to every C, A will necessarily belong to every C; for this has been proved before. But it was assumed that A may not belong to some C.
Whenever both are indefinite or particular, no deduction will be possible. The [40a1] demonstration is the same as before, and proceeds by means of the same terms.
22 · If one of the propositions is necessary, the other possible, when the [5] terms are affirmative a possible conclusion can always be drawn; when one is affirmative, the other negative, if the affirmative is necessary a possible negative can be inferred; but if the negative is necessary both a possible and a simple negative conclusion are possible. But a necessary negative conclusion will not be [10] possible, any more than in the other figures.
Suppose first that the terms are affirmative, i.e. that A necessarily belongs to every C, and B may belong to every C. Since then A must belong to every C, and C may belong to some B, it follows that A may (not does) belong to some B; for so it [15] resulted in the first figure. A similar proof may be given if BC is necessary, and AC is possible. Again suppose one is affirmative, the other negative, the affirmative being necessary, i.e. suppose A may belong to no C, but B necessarily belongs to [20] every C. We shall have the first figure once more; and—since the negative proposition indicates possibility—it is clear that the conclusion will be possible; for when the propositions stand thus in the first figure, the conclusion (as we found) is possible. But if the negative proposition is necessary, the conclusion will be not only [25] that A may not belong to some B but also that it does not belong to some B. For suppose that A necessarily does not belong to C, but B may belong to every C. If the affirmative BC is converted, we shall have the first figure, and the negative proposition is necessary. But when the propositions stood thus, it resulted that A [30] might not belong to some C, and that it did not belong to some C; consequently here it follows that A does not belong to some B. But when the negative relates to the minor extreme, if it is possible we shall have a deduction by altering the proposition, [35] as before; but if it is necessary no deduction can be formed. For A both necessarily belongs to every B, and cannot belong to any B. To illustrate the former take the terms sleep, sleeping horse, man; to illustrate the latter take the terms sleep, waking horse, man.
Similar results will obtain if one of the terms is related universally to the [40b1] middle, the other in part. If both are affirmative, the conclusion will be possible, not simple; and also when one is negative, the other affirmative, the latter being necessary. But when the negative is necessary, the conclusion also will be a simple [5] negative; for the same kind of proof can be given whether the terms are universal or not. For the deductions must be made perfect by means of the first figure, so that a result which follows in the first figure follows also in the third. But when the negative is universal and relates to the minor extreme, if it is possible a deduction [10] can be formed by means of conversion; but if it is necessary a deduction is not possible. The proof will follow the same course as for the universal deductions; and the same terms may be used.
It is clear then in this figure also when and how a deduction can be formed, and when the conclusion is possible, and when it is simple. It is evident also that all [15] deductions in this figure are imperfect, and that they are made perfect by means of the first figure.
23 · It is clear from what has been said that the deductions in these figures are made perfect by means of the universal deductions in the first figure and are [20] reduced to them. That every deduction without qualification can be so treated, will be clear presently, when it has been proved that every deduction is formed through one or other of these figures.
It is necessary that every demonstration and every deduction should prove either that something belongs or that it does not, and this either universally or in
[25] part, and further either probatively or hypothetically. One sort of hypothetical proof is the reductio ad impossibile. Let us speak first of probative deductions; for after it has been proved in their case, the truth of our contention will be clear with regard to those which are proved per impossibile, and in general hypothetically.
[30] If then one wants to deduce that A belongs or does not belong to B, one must assume something of something. If now A should be assumed of B, the proposition originally in question will have been assumed. But if A should be assumed of C, but C should not be assumed of anything, nor anything of it, nor anything else of A, no [35] deduction will be possible. For nothing necessarily follows from the assumption of some one thing concerning some one thing. Thus we must take another proposition as well. If then A be assumed of something else, or something else of A, or something different of C, nothing prevents a deduction being formed, but it will not be in relation to B through the propositions taken. Nor when C belongs to something [41a1] else, and that to something else and so on, no connexion however being made with B, will a deduction be possible in relation to B. For in general we stated that no deduction can establish the attribution of one thing to another, unless some middle term is taken, which is somehow related to each by way of predication. For a deduction in general is made out of propositions, and a deduction referring to this [5] out of propositions with the same reference, and a deduction relating this to that proceeds through propositions which relate this to that. But it is impossible to take a proposition in reference to B, if we neither affirm nor deny anything of it; or again to take a proposition relating A to B, if we take nothing common, but affirm or deny peculiar attributes of each. So we must take a middle term relating to both, which [10] will connect the predications, if we are to have a deduction relating this to that. If then we must take something common in relation to both, and this is possible in three ways (either by predicating A of C, and C of B, or C of both, or both of C), and [15] these are the figures of which we have spoken, it is clear that every deduction must be made in one or other of these figures. The argument is the same if several middle terms should be necessary to establish the relation to B; for the figure will be the same whether there is one middle term or many. [20]
It is clear then that probative deductions are effected by means of the aforesaid figures; the following considerations will show that reductiones ad impossibile also are effected in the same way. For all who effect an argument per impossibile deduce what is false, and prove the original conclusion hypothetically when something impossible results from the assumption of its contradictory; e.g. that the [25] diagonal of the square is incommensurate with the side, because odd numbers are equal to evens if it is supposed to be commensurate. One deduces that odd numbers come out equal to evens, and one proves hypothetically the incommensurability of the diagonal, since a falsehood results from its contradictory. For this we found to [30] be deducing per impossibile, viz. proving something impossible by means of an hypothesis conceded at the beginning. Consequently, since the falsehood is established in reductions ad impossibile by a probative deduction, and the original conclusion is proved hypothetically, and we have already stated that probative [35] deductions are effected by means of these figures, it is evident that deductions per impossibile also will be made through these figures. Likewise all the other hypothetical deductions; for in every case the deduction leads up to the substituted proposition; but the original thesis is reached by means of a concession or some other hypothesis. But if this is true, every demonstration and every deduction must [41b1] be formed by means of the three figures mentioned above. But when this has been shown it is clear that every deduction is perfected by means of the first figure and is reducible to the universal deductions in this figure. [5]
24 · Further in every deduction one of the terms must be affirmative, and universality must be present: unless one of the premisses is universal either a deduction will not be possible, or it will not refer to the subject proposed, or the original position will be begged. Suppose we have to prove that pleasure in music is [10] good. If one should claim that pleasure is good without adding ‘every’, no deduction will be possible; if one should claim that some pleasure is good, then if it is different from pleasure in music, it is not relevant to the subject proposed; if it is this very pleasure, one is assuming that which was originally proposed. This is more obvious in geometrical proofs, e.g. that the angles at the base of an isosceles triangle are [15] equal. Suppose the lines A and B have been drawn to the centre. If then one should assume that the angle AC is equal to the angle BD, without claiming generally that angles of semicircles are equal; and again if one should assume that the angle C is equal to the angle D, without the additional assumption that every angle of a segment is equal to every other angle of the same segment; and further if one should assume that when equal angles are taken from the whole angles, which [20] are themselves equal, the remainders E and F are equal, he will beg the original position, unless he also assumes that when equals are taken from equals the remainders are equal.
It is clear then that in every deduction there must be a universal, and that a universal is proved only when all the terms are universal, while a particular is [25] proved in both cases; consequently if the conclusion is universal, the terms also must be universal, but if the terms are universal it is possible that the conclusion may not be universal. And it is clear also that in every deduction either both or one of the propositions must be like the conclusion. I mean not only in being affirmative or [30] negative, but also in being necessary, simple, or possible. We must consider also the other forms of predication.
It is clear also when a deduction in general can be made and when it cannot; and when a potential, when a perfect deduction can be formed; and that if a deduction is formed the terms must be arranged in one of the ways that have been [35] mentioned.
25 · It is clear too that every demonstration will proceed through three terms and no more, unless the same conclusion is established by different pairs of propositions; e.g. E may be established through A and B, and through C and D, or through A and B, or A and C and D. For nothing prevents there being several middles for the same terms. But in that case there is not one but several deductions. [42a1] Or again when each of A and B is obtained by deduction, e.g. A by means of D and E, and again B by means of F and G. Or one may be obtained by deduction, the other by induction. But thus also the deductions are many; for the conclusions are [5] many, e.g. A and B and C.
But if this can be called one deduction, not many, the same conclusion may be reached by several terms in this way, but it cannot be reached as C is established by means of A and B. Suppose that E is inferred from A, B, C, and D. It is necessary [10] then that of these one should be related to another as whole to part; for it has already been proved that if a deduction is formed some of its terms must be related in this way. Suppose then that A stands in this relation to B. Some conclusion then follows from them. It must either be E or one or other of C and D, or something other than these.
[15] If it is E the deduction will have A and B for its sole premisses. But if C and D are so related that one is whole, the other part, some conclusion will follow from them also; and it must be either E, or one or other of A and B, or something other than these. And if it is E, or A or B, either the deductions will be more than one, or the same thing happens to be inferred by means of several terms in the sense which we saw to be possible. But if the conclusion is other than E or A or B, the deductions [20] will be many, and unconnected with one another. But if C is not related to D as to make a deduction, the propositions will have been assumed to no purpose, unless for the sake of induction or of obscuring the argument or something of the sort.
But if from A and B there follows not E but some other conclusion, and if from [25] C and D either A or B follows or something else, then there are several deductions, and they do not establish the conclusion proposed; for we assumed that the deduction proved E. And if no co
nclusion follows from C and D, it turns out that these propositions have been assumed to no purpose, and the deduction does not prove the original proposition.
So it is clear that every demonstration and every deduction will proceed [30] through three terms only.
This being evident, it is clear that a conclusion follows from two propositions and not from more than two for the three terms make two propositions unless a new proposition is assumed, as was said at the beginning, to perfect the deductions. It is [35] clear therefore that in whatever deductive argument the propositions through which the main conclusion follows (for some of the preceding conclusions must be propositions) are not even in number, this argument either has not been deduced or it has assumed more than was necessary to establish its thesis.
If then deductions are taken with respect to their main propositions, every [42b1] deduction will consist of an even number of propositions and an odd number of terms (for the terms exceed the propositions by one), and the conclusions will be half the number of the propositions. But whenever a conclusion is reached by means of preliminary deductions or by means of several continuous middle terms, e.g. AB [5] by means of C and D, the number of the terms will similarly exceed that of the propositions by one (for the extra term must either be added outside or inserted; but in either case it follows that the relations of predication are one fewer than the terms related, and the propositions will be equal in number to the relations of predication). [10] The propositions however will not always be even, the terms odd; but they will alternate—when the propositions are even, the terms must be odd; when the terms are even, the propositions must be odd; for along with one term one proposition is added, if a term is added from any quarter. Consequently since the propositions were (as we saw) even, and the terms odd, we must make them alternately even and [15] odd at each addition. But the conclusions will not follow the same arrangement either in respect to the terms or to the propositions. For if one term is added, conclusions will be added less by one than the pre-existing terms; for the conclusion is drawn not in relation to the single term last added, but in relation to all the rest, [20] e.g. if to ABC the term D is added, two conclusions are thereby added, one in relation to A, the other in relation to B. Similarly with any further additions. And similarly too if the term is inserted in the middle; for a deduction will not be effected in relation to one term only. Consequently the conclusions will be much more [25] numerous than the terms or the propositions.
The Complete Works of Aristotle Page 12
