The Complete Works of Aristotle
Page 16
BOOK II
1 · We have already explained the number of the figures, the character and number of the propositions, when and how a deduction is formed; further what we [53a1] must look for when refuting and establishing propositions, and how we should investigate a given problem in any branch of inquiry, also by what means we shall obtain principles appropriate to each subject. Since some deductions are universal, [5] others particular, all the universal deductions give more than one result, and of particular deductions the affirmative yield more than one, the negative yield only the stated conclusion. For all propositions are convertible save only the particular negative; and the conclusion states one thing about another. Consequently all other deductions yield more than one conclusion, e.g. if A has been proved to belong to [10] every or to some B, then B must belong to some A; and if A has been proved to belong to no B, then B belongs to no A. This is a different conclusion from the former. But if A does not belong to some B, it is not necessary that B should not belong to some A; for it may belong to every A.
This then is the reason common to all deductions whether universal or [15] particular. But it is possible to give another reason concerning those which are universal. For all the things that are subordinate to the middle term or to the conclusion may be proved by the same deduction, if the former are placed in the middle, the latter in the conclusion; e.g. if the conclusion AB is proved through C, then A must be said of all of whatever is subordinate to B or C; for if D is in B as in a [20] whole, and B is in A, then D will be in A. Again if E is in C as in a whole, and C is in A, then E will be in A. Similarly if the deduction is negative. In the second figure it [25] will be possible to deduce only that which is subordinate to the conclusion, e.g. if A belong to no B and to every C; we conclude that B belongs to no C. If then D is subordinate to C, clearly B does not belong to it. But that B does not belong to what is subordinate to A, is not clear by means of the deduction. And yet B does not belong to E, if E is subordinate to A. But while it has been proved through the [30] deduction that B belongs to no C, it has been assumed without proof that B does not belong to A, consequently it does not result through the deduction that B does not belong to E.
But in particular deductions there will be no necessity of inferring what is subordinate to the conclusion (for a deduction does not result when this is [35] particular), but whatever is subordinate to the middle term may be inferred, not however through the deduction, e.g. if A belongs to every B and B to some C. Nothing can be inferred about that which is subordinate to C; something can be inferred about that which is subordinate to B, but not through the preceding deduction. Similarly in the other figures: that which is subordinate to the conclusion [40] cannot be proved; the other subordinate can be proved, only not through the deduction, just as in the universal deductions what is subordinate to the middle term [53b1] is proved (as we saw) from a proposition which is not demonstrated; consequently either a conclusion is not possible there or else it is possible here too.
2 · It is possible for the premisses of the deduction to be true, or to be false, or [5] to be the one true, the other false. The conclusion is either true or false necessarily. From true premisses it is not possible to draw a false conclusion; but a true conclusion may be drawn from false premisses—true however only in respect to the fact, not to the reason. The reason cannot be established from false premisses: why this is so will be explained in the sequel.18 [10]
First then that it is not possible to draw a false conclusion from true premisses, is made clear by this consideration. If it is necessary that B should be when A is, it is necessary that A should not be when B is not. If then A is true, B must be true: [15] otherwise it will turn out that the same thing both is and is not at the same time. But this is impossible. (Let it not, because A is laid down as a single term, be supposed that it is possible, when a single fact is given, that something should necessarily result. For that is not possible. For what results necessarily is the conclusion, and [20] the means by which this comes about are at the least three terms, and two relations or propositions. If then it is true that A belongs to all that to which B belongs, and that B belongs to all that to which C belongs, it is necessary that A should belong to all that to which C belongs, and this cannot be false; for then the same thing will belong and not belong at the same time. So A is posited as one thing, being two premisses taken together.) The same holds good of negative deductions: it is not [25] possible to prove a false conclusion from truths.
But from what is false a true conclusion may be drawn, whether both the premisses are false or only one (provided that this is not either of the premisses indifferently, but the second, if it is taken as wholly false; but if it is not taken as [30] wholly false, it does not matter which of the two is false). Let A belong to the whole of C, but to no B, neither let B belong to C. This is possible, e.g. animal belongs to no stone, nor stone to any man. If then A is taken to belong to every B and B to every C, A will belong to every C; consequently though both the premisses are false the [35] conclusion is true; for every man is an animal. Similarly with the negative. For it is possible that neither A nor B should belong to any C, although A belongs to every B, e.g. if the same terms are taken and man is put as middle; for neither animal nor man belongs to any stone, but animal belongs to every man. Consequently if one [40] term is taken to belong to none of that to which it does belong, and the other term is taken to belong to all of that to which it does not belong, though both the premisses [54a1] are false the conclusion will be true. A similar proof may be given if each premiss is partially false.
But if one only of the premisses is false, when the first premiss is wholly false, i.e. AB, the conclusion will not be true, but if BC is wholly false, a true conclusion [5] will be possible. I mean by wholly false the contrary of the truth, e.g. if what belongs to none is assumed to belong to all, or if what belongs to all is assumed to belong to none. Let A belong to no B, and B to every C. If then the proposition BC which I take is true, and AB is wholly false, viz. that A belongs to every B, it is impossible [10] that the conclusion should be true; for A belonged to none of the Cs, since A belonged to nothing to which B belonged, and B belonged to every C. Similarly there cannot be a true conclusion if A belongs to every B, and B to every C, but while the true proposition BC is assumed, the wholly false AB is also assumed, viz. that A belongs to nothing to which B belongs—here the conclusion must be false. For A [15] will belong to every C, since A belongs to everything to which B belongs, and B to every C. It is clear then that when the first premiss is wholly false, whether affirmative or negative, and the other premiss is true, the conclusion cannot be true.
But if the premiss is not wholly false, a true conclusion is possible. For if A belongs to every C and to some B, and if B belongs to every C, e.g. animal to every [20] swan and to some white thing, and white to every swan, then if we assume that A belongs to every B, and B to every C, A will belong to every C truly; for every swan is an animal. Similarly if AB is negative. For it is possible that A should belong to some B and to no C, and that B should belong to every C, e.g. animal to some white [25] thing, but to no snow, and white to all snow. If then one should assume that A belongs to no B, and B to every C, then A will belong to no C.
But if the proposition AB, which is assumed, is wholly true, and BC is wholly false, a true deduction will be possible; for nothing prevents A belonging to every B [30] and to every C, though B belongs to no C, e.g. these being species of the same genus which are not subordinate one to the other—for animal belongs both to horse and to man, but horse to no man. If then it is assumed that A belongs to every B and B to every C, the conclusion will be true, although the proposition BC is wholly false. [35] Similarly if the proposition AB is negative. For it is possible that A should belong neither to any B nor to any C, and that B should not belong to any C, e.g. a genus to species of another genus—for animal belongs neither to music nor to medicine, nor does music belong to the medicine. If then it is assumed
that A belongs to no B, and [54b1] B to every C, the conclusion will be true.
And if BC is not wholly false but in part only, even so that conclusion may be true. For nothing prevents A belonging to the whole of B and of C, while B belongs [5] to some C, e.g. a genus to its species and difference—for animal belongs to every man and to every footed thing, and man to some footed things though not to all. If then it is assumed that A belongs to every B, and B to every C, A will belong to every C; and this ex hypothesi is true. Similarly if the proposition AB is negative. For it is [10] possible that A should neither belong to any B nor to any C, though B belongs to some C, e.g. a genus to the species of another genus and its difference for animal neither belongs to any wisdom nor to any speculative science, but wisdom belongs to some speculative sciences. If then it should be assumed that A belongs to no B, and B to every C, A will belong to no C; and this ex hypothesi is true. [15]
In particular deductions it is possible when the first proposition is wholly false, and the other true, that the conclusion should be true; also when the first is false in part, and the other true; and when the first is true, and the particular is false; and [20] when both are false. For nothing prevents A belonging to no B, but to some C, and B to some C, e.g. animal belongs to no snow, but to some white thing, and snow to some white thing. If then snow is taken as middle, and animal as first term, and it is assumed that A belongs to the whole of B, and B to some C, then AB is wholly false, [25] BC true, and the conclusion true. Similarly if the proposition AB is negative; for it is possible that A should belong to the whole of B, but not to some C, although B belongs to some C, e.g. animal belongs to every man, but does not follow some white, [30] but man belongs to some white; consequently if man be taken as middle term and it is assumed that A belongs to no B but B belongs to some C, the conclusion will be true although the proposition AB is wholly false.
If the proposition AB is false in part, the conclusion may be true. For nothing [35] prevents A belonging both to some B and to some C, and B belonging to some C, e.g. animal to something beautiful and to something great, and beautiful belonging to something great. If then A is assumed to belong to every B, and B to some C, the [55a1] proposition AB will be partially false, BC will be true, and the conclusion true. Similarly if the proposition AB is negative. For the same terms will serve, and in the same positions, to prove the point.
[5] Again if AB is true, and BC is false, the conclusion may be true. For nothing prevents A belonging to the whole of B and to some C, while B belongs to no C, e.g. animal to every swan and to some black things, though swan belongs to no black thing. Consequently if it should be assumed that A belongs to every B, and B to [10] some C, the conclusion will be true, although BC is false. Similarly if the proposition AB is negative. For it is possible that A should belong to no B, and not to some C, while B belongs to no C, e.g. a genus to the species of another genus and to the [15] accident of its own species—for animal belongs to no number and to some white things, and number belongs to nothing white. If then number is taken as middle, and it is assumed that A belongs to no B, and B to some C, then A will not belong to some C, which ex hypothesi is true. And the proposition AB is true, BC false.
[20] Also if AB is partially false, and BC is false too, the conclusion may be true. For nothing prevents A belonging to some B and to some C, though B belongs to no C, e.g. if B is the contrary of C, and both are accidents of the same genus—for animal belongs to some white things and to some black things, but white belongs to [25] no black thing. If then it is assumed that A belongs to every B, and B to some C, the conclusion will be true. Similarly if AB is negative; for the same terms arranged in the same way will serve for the proof.
Also though both premisses are false the conclusion may be true. For it is [30] possible that A may belong to no B and to some C, while B belongs to no C, e.g. a genus in relation to the species of another genus, and to the accident of its own species for animal belongs to no number, but to some white things, and number to nothing white. If then it is assumed that A belongs to every B and B to some C, the [35] conclusion will be true, though both propositions are false. Similarly also if AB is negative. For nothing prevents A belonging to the whole of B, and not to some C, while B belongs to no C, e.g. animal belongs to every swan, and not to some black things, and swan belongs to nothing black. Consequently if it is assumed that A [55b1] belongs to no B, and B to some C, then A does not belong to some C. The conclusion then is true, but the propositions are false.
3 · In the middle figure it is possible in every way to reach a true conclusion through false premisses, whether the deductions are universal or particular, viz. [5] when both propositions are wholly false; when each is partially false; when one is true, the other [wholly] false (it does not matter which of the two premisses is false). [if both premisses are partially false; if one is quite true, the other partially false; if [10] one is wholly false, the other partially true.]19 For if A belongs to no B and to every C, e.g. animal to no stone and to every horse, then if the propositions are stated contrariwise and it is assumed that A belongs to every B and to no C, though the propositions are wholly false they will yield a true conclusion. Similarly if A belongs to every B and to no C; for we shall have the same deduction. [15]
Again if one premiss is wholly false, the other wholly true; for nothing prevents A belonging to every B and to every C, though B belongs to no C, e.g. a genus to its co-ordinate species. For animal belongs to every horse and man, and no man is a [20] horse. If then it is assumed that animal belongs to all of the one, and none of the other, the one premiss will be wholly false, the other wholly true, and the conclusion will be true whichever term the negative statement concerns.
Also if one premiss is partially false, the other wholly true. For it is possible that A should belong to some B and to every C, though B belongs to no C, e.g. [25] animal to some white things and to every raven, though white belongs to no raven. If then it is assumed that A belongs to no B, but to the whole of C, the proposition AB is partially false, AC wholly true, and the conclusion true. Similarly if the negative is transposed: the proof can be made by means of the same terms. Also if the [30] affirmative proposition is partially false, the negative wholly true, a true conclusion is possible. For nothing prevents A belonging to some B, but not to C as a whole, while B belongs to no C, e.g. animal belongs to some white things, but to no pitch, and white belongs to no pitch. Consequently if it is assumed that A belongs to the [35] whole of B, but to no C, AB is partially false, AC is wholly true, and the conclusion is true.
And if both the propositions are partially false, the conclusion may be true. For it is possible that A should belong to some B and to some C, and B to no C, e.g. animal to some white things and to some black things, though white belongs to [56a1] nothing black. If then it is assumed that A belongs to every B and to no C, both propositions are partially false, but the conclusion is true. Similarly, if the negative is transposed, the proof can be made by means of the same terms.
It is clear also that the same holds for particular deductions. For nothing [5] prevents A belonging to every B and to some C, though B does not belong to some C, e.g. animal to every man and to some white things, though man will not belong to some white things. If then it is stated that A belongs to no B and to some C, the universal proposition is wholly false, the particular is true, and the conclusion is [10] true. Similarly if AB is affirmative; for it is possible that A should belong to no B, and not to some C, though B does not belong to some C, e.g. animal belongs to nothing inanimate, and to some white things, and inanimate will not belong to some [15] white things. If then it is stated that A belongs to B and not to some C, the AB which is universal is wholly false, AC is true, and the conclusion is true. Also a true conclusion is possible when the universal is true, and the particular is false. For nothing prevents A following neither B nor C at all, while B does not belong to some [20] C, e.g. animal belongs to no number nor to anything inanimate, and number doe
s not follow some inanimate things. If then it is stated that A belongs to no B and to some C, the conclusion will be true, and the universal proposition true, but the particular false. Similarly if the premiss which is stated universally affirmative. For [25] it is possible that A should belong both to B and to C as wholes, though B does not follow some C, e.g. a genus in relation to its species and difference—for animal follows every man and footed things as a whole, but man does not follow every footed thing. Consequently if it is assumed that A belongs to the whole of B, but [30] does not belong to some C, the universal proposition is true, the particular false, and the conclusion true.
It is clear too that though both propositions are false they may yield a true conclusion, since it is possible that A should belong both to B and to C as wholes, [35] though B does not follow some C. For if it is assumed that A belongs to no B and to some C, the propositions are both false, but the conclusion is true. Similarly if the universal proposition is affirmative and the particular negative. For it is possible that A should follow no B and every C, though B does not belong to some C, e.g. animal follows no science but every man, though science does not follow every man. [56b1] If then A is assumed to belong to the whole of B, and not to follow some C, the propositions are false but the conclusion is true.
[5] 4 · In the last figure a true conclusion may come through what is false, alike when both are wholly false, when each is partly false, when one is wholly true, the other false, when one is partly false, the other wholly true, and vice versa, and in every other way in which it is possible to alter the propositions. For nothing prevents [10] neither A nor B from belonging to any C, while A belongs to some B, e.g. neither man nor footed follows anything inanimate, though man belongs to some footed things. If then it is assumed that A and B belong to every C, the propositions will be wholly false, but the conclusion true. Similarly if one is negative, the other [15] affirmative. For it is possible that B should belong to no C, but A to every C, and that A should not belong to B, e.g. black belongs to no swan, animal to every swan, and animal not to everything black. Consequently if it is assumed that B belongs to every C, and A to no C, A will not belong to some B; and the conclusion is true, [20] though the propositions are false.