Various Works Page 47

is not-white or all are not-white is false. Similarly also 'every

  animal is not-white' is not the negation of 'every animal is white'

  (for both are false): the proper negation is 'every animal is not

  white'. Since it is clear that 'it is not-white' and 'it is not white'

  mean different things, and one is an affirmation, the other a

  denial, it is evident that the method of proving each cannot be the

  same, e.g. that whatever is an animal is not white or may not be

  white, and that it is true to call it not-white; for this means that

  it is not-white. But we may prove that it is true to call it white

  or not-white in the same way for both are proved constructively by

  means of the first figure. For the expression 'it is true' stands on a

  similar footing to 'it is'. For the negation of 'it is true to call it

  white' is not 'it is true to call it not-white' but 'it is not true to

  call it white'. If then it is to be true to say that whatever is a man

  is musical or is not-musical, we must assume that whatever is an

  animal either is musical or is not-musical; and the proof has been

  made. That whatever is a man is not musical is proved destructively in

  the three ways mentioned.

  In general whenever A and B are such that they cannot belong at

  the same time to the same thing, and one of the two necessarily

  belongs to everything, and again C and D are related in the same

  way, and A follows C but the relation cannot be reversed, then D

  must follow B and the relation cannot be reversed. And A and D may

  belong to the same thing, but B and C cannot. First it is clear from

  the following consideration that D follows B. For since either C or

  D necessarily belongs to everything; and since C cannot belong to that

  to which B belongs, because it carries A along with it and A and B

  cannot belong to the same thing; it is clear that D must follow B.

  Again since C does not reciprocate with but A, but C or D belongs to

  everything, it is possible that A and D should belong to the same

  thing. But B and C cannot belong to the same thing, because A

  follows C; and so something impossible results. It is clear then

  that B does not reciprocate with D either, since it is possible that D

  and A should belong at the same time to the same thing.

  It results sometimes even in such an arrangement of terms that one

  is deceived through not apprehending the opposites rightly, one of

  which must belong to everything, e.g. we may reason that 'if A and B

  cannot belong at the same time to the same thing, but it is

  necessary that one of them should belong to whatever the other does

  not belong to: and again C and D are related in the same way, and

  follows everything which C follows: it will result that B belongs

  necessarily to everything to which D belongs': but this is false.

  'Assume that F stands for the negation of A and B, and again that H

  stands for the negation of C and D. It is necessary then that either A

  or F should belong to everything: for either the affirmation or the

  denial must belong. And again either C or H must belong to everything:

  for they are related as affirmation and denial. And ex hypothesi A

  belongs to everything ever thing to which C belongs. Therefore H

  belongs to everything to which F belongs. Again since either F or B

  belongs to everything, and similarly either H or D, and since H

  follows F, B must follow D: for we know this. If then A follows C, B

  must follow D'. But this is false: for as we proved the sequence is

  reversed in terms so constituted. The fallacy arises because perhaps

  it is not necessary that A or F should belong to everything, or that F

  or B should belong to everything: for F is not the denial of A. For

  not good is the negation of good: and not-good is not identical with

  'neither good nor not-good'. Similarly also with C and D. For two

  negations have been assumed in respect to one term.

  Book II

  1

  WE have already explained the number of the figures, the character

  and number of the premisses, when and how a syllogism is formed;

  further what we must look for when a 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 syllogisms are universal,

  others particular, all the universal syllogisms give more than one

  result, and of particular syllogisms 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 definite thing about another definite thing.

  Consequently all syllogisms save the particular negative yield more

  than one conclusion, e.g. if A has been proved to to all 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 possibly belong to all A.

  This then is the reason common to all syllogisms whether universal

  or 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

  syllogism, if the former are placed in the middle, the latter in the

  conclusion; e.g. if the conclusion AB is proved through C, whatever is

  subordinate to B or C must accept the predicate A: for if D is

  included in B as in a whole, and B is included in A, then D will be

  included in A. Again if E is included in C as in a whole, and C is

  included in A, then E will be included in A. Similarly if the

  syllogism is negative. In the second figure it will be possible to

  infer only that which is subordinate to the conclusion, e.g. if A

  belongs to no B and to all 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 syllogism. And yet B does not belong to E, if E is

  subordinate to A. But while it has been proved through the syllogism

  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

  syllogism that B does not belong to E.

  But in particular syllogisms there will be no necessity of inferring

  what is subordinate to the conclusion (for a syllogism does not result

  when this premiss is particular), but whatever is subordinate to the

  middle term may be inferred, not however through the syllogism, e.g.

  if A belongs to all 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 syllogism.

  Similarly in the other figures. That which is subordinate to the

  conclusion cannot be proved; the other subordinate can be proved, only

  not through the syllogism, just as in the universal syllogisms what
is

  subordinate to the middle term is proved (as we saw) from a premiss

  which is not demonstrated: consequently either a conclusion is not

  possible in the case of universal syllogisms or else it is possible

  also in the case of particular syllogisms.

  2

  It is possible for the premisses of the syllogism to be true, or

  to be false, or 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.

  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: 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 the

  means by which this comes about are at the least three terms, and

  two relations of subject and predicate or premisses. 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 syllogisms: it is not possible to prove a false

  conclusion from true premisses.

  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, if it is taken as wholly false:

  but if the premiss is not taken as wholly false, it does not matter

  which of the two is false. (1) Let A belong to the whole of C, but

  to none of the Bs, 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 all B and B to all C, A will belong to all C;

  consequently though both the premisses are false the 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 all 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 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 are false the conclusion will be true. (2) A similar proof

  may be given if each premiss is partially false.

  (3) But if one only of the premisses is false, when the first

  premiss is wholly false, e.g. AB, the conclusion will not be true, but

  if the premiss BC is wholly false, a true conclusion 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 all C. If

  then the premiss BC which I take is true, and the premiss AB is wholly

  false, viz. that A belongs to all B, it is impossible 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 all C.

  Similarly there cannot be a true conclusion if A belongs to all B, and

  B to all C, but while the true premiss BC is assumed, the wholly false

  premiss AB is also assumed, viz. that A belongs to nothing to which

  B belongs: here the conclusion must be false. For A will belong to all

  C, since A belongs to everything to which B belongs, and B to all 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.

  (4) But if the premiss is not wholly false, a true conclusion is

  possible. For if A belongs to all C and to some B, and if B belongs to

  all C, e.g. animal to every swan and to some white thing, and white to

  every swan, then if we take as premisses that A belongs to all B,

  and B to all C, A will belong to all C truly: for every swan is an

  animal. Similarly if the statement AB is negative. For it is

  possible that A should belong to some B and to no C, and that B should

  belong to all C, e.g. animal to some white thing, but to no snow,

  and white to all snow. If then one should assume that A belongs to

  no B, and B to all C, then will belong to no C.

  (5) But if the premiss AB, which is assumed, is wholly true, and the

  premiss BC is wholly false, a true syllogism will be possible: for

  nothing prevents A belonging to all B and to all 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

  all B and B to all C, the conclusion will be true, although the

  premiss BC is wholly false. Similarly if the premiss 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 the art of

  healing, nor does music belong to the art of healing. If then it is

  assumed that A belongs to no B, and B to all C, the conclusion will be

  true.

  (6) And if the premiss BC is not wholly false but in part only, even

  so the conclusion may be true. For nothing prevents A belonging to the

  whole of B and of C, while B belongs 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 all B, and B to all C, A will belong

  to all C: and this ex hypothesi is true. Similarly if the premiss AB

  is negative. For it is 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 instance of 'speculative', but wisdom

  belongs to some instance of 'speculative'. If then it should be

  assumed that A belongs to no B, and B to all C, will belong to no C:

  and this ex hypothesi is true.

  In particular syllogisms it is possible when the first premiss is

  wholly false, and the other true, that the conclusion should be

  true; also when the first premiss is false in part, and the other

  true; and when th
e first is true, and the particular is false; and

  when both are false. (7) 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 the premiss BC is

  wholly false, the premiss BC true, and the conclusion true.

  Similarly if the premiss 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, 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 premiss AB

  is wholly false. (If the premiss AB is false in part, the conclusion

  may be true. For nothing prevents A belonging both to 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 all B, and B to some C, the a premiss AB

  will be partially false, the premiss BC will be true, and the

  conclusion true. Similarly if the premiss AB is negative. For the same

  terms will serve, and in the same positions, to prove the point.

  (9) Again if the premiss AB is true, and the premiss 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 all B,

  and B to some C, the conclusion will be true, although the statement

  BC is false. Similarly if the premiss 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 accident of its own species: for animal belongs to no number and

  not to some white things, and number belongs to nothing white. If then