by Aristotle
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.
Similarly if BC is false, AC true, the conclusion may be true. The same terms will serve for the proof. Also if both are affirmative, the conclusion may be true. For [57a1] nothing prevents B from following every C, and A from not belonging to C at all, though A belongs to some B, e.g. animal belongs to every swan, black to no swan, [5] and black to some animals. Consequently if it is assumed that A and B belong to every C, BC is wholly true, AC is wholly false, and the conclusion is true. Similarly if AC is true: the proof can be made through the same terms.
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.