Various Works Page 50

  each figure when the conclusion is converted; when a result contrary

  to the premiss, and when a result contradictory to the premiss, is

  obtained. It is clear that in the first figure the syllogisms are

  formed through the middle and the last figures, and the premiss

  which concerns the minor extreme is alway refuted through the middle

  figure, the premiss which concerns the major through the last

  figure. In the second figure syllogisms proceed through the first

  and the last figures, and the premiss which concerns the minor extreme

  is always refuted through the first figure, the premiss which concerns

  the major extreme through the last. In the third figure the refutation

  proceeds through the first and the middle figures; the premiss which

  concerns the major is always refuted through the first figure, the

  premiss which concerns the minor through the middle figure.

  11

  It is clear then what conversion is, how it is effected in each

  figure, and what syllogism results. The syllogism per impossibile is

  proved when the contradictory of the conclusion stated and another

  premiss is assumed; it can be made in all the figures. For it

  resembles conversion, differing only in this: conversion takes place

  after a syllogism has been formed and both the premisses have been

  taken, but a reduction to the impossible takes place not because the

  contradictory has been agreed to already, but because it is clear that

  it is true. The terms are alike in both, and the premisses of both are

  taken in the same way. For example if A belongs to all B, C being

  middle, then if it is supposed that A does not belong to all B or

  belongs to no B, but to all C (which was admitted to be true), it

  follows that C belongs to no B or not to all B. But this is

  impossible: consequently the supposition is false: its contradictory

  then is true. Similarly in the other figures: for whatever moods admit

  of conversion admit also of the reduction per impossibile.

  All the problems can be proved per impossibile in all the figures,

  excepting the universal affirmative, which is proved in the middle and

  third figures, but not in the first. Suppose that A belongs not to all

  B, or to no B, and take besides another premiss concerning either of

  the terms, viz. that C belongs to all A, or that B belongs to all D;

  thus we get the first figure. If then it is supposed that A does not

  belong to all B, no syllogism results whichever term the assumed

  premiss concerns; but if it is supposed that A belongs to no B, when

  the premiss BD is assumed as well we shall prove syllogistically

  what is false, but not the problem proposed. For if A belongs to no B,

  and B belongs to all D, A belongs to no D. Let this be impossible:

  it is false then A belongs to no B. But the universal affirmative is

  not necessarily true if the universal negative is false. But if the

  premiss CA is assumed as well, no syllogism results, nor does it do so

  when it is supposed that A does not belong to all B. Consequently it

  is clear that the universal affirmative cannot be proved in the

  first figure per impossibile.

  But the particular affirmative and the universal and particular

  negatives can all be proved. Suppose that A belongs to no B, and let

  it have been assumed that B belongs to all or to some C. Then it is

  necessary that A should belong to no C or not to all C. But this is

  impossible (for let it be true and clear that A belongs to all C):

  consequently if this is false, it is necessary that A should belong to

  some B. But if the other premiss assumed relates to A, no syllogism

  will be possible. Nor can a conclusion be drawn when the contrary of

  the conclusion is supposed, e.g. that A does not belong to some B.

  Clearly then we must suppose the contradictory.

  Again suppose that A belongs to some B, and let it have been assumed

  that C belongs to all A. It is necessary then that C should belong

  to some B. But let this be impossible, so that the supposition is

  false: in that case it is true that A belongs to no B. We may

  proceed in the same way if the proposition CA has been taken as

  negative. But if the premiss assumed concerns B, no syllogism will

  be possible. If the contrary is supposed, we shall have a syllogism

  and an impossible conclusion, but the problem in hand is not proved.

  Suppose that A belongs to all B, and let it have been assumed that C

  belongs to all A. It is necessary then that C should belong to all

  B. But this is impossible, so that it is false that A belongs to all

  B. But we have not yet shown it to be necessary that A belongs to no

  B, if it does not belong to all B. Similarly if the other premiss

  taken concerns B; we shall have a syllogism and a conclusion which

  is impossible, but the hypothesis is not refuted. Therefore it is

  the contradictory that we must suppose.

  To prove that A does not belong to all B, we must suppose that it

  belongs to all B: for if A belongs to all B, and C to all A, then C

  belongs to all B; so that if this is impossible, the hypothesis is

  false. Similarly if the other premiss assumed concerns B. The same

  results if the original proposition CA was negative: for thus also

  we get a syllogism. But if the negative proposition concerns B,

  nothing is proved. If the hypothesis is that A belongs not to all

  but to some B, it is not proved that A belongs not to all B, but

  that it belongs to no B. For if A belongs to some B, and C to all A,

  then C will belong to some B. If then this is impossible, it is

  false that A belongs to some B; consequently it is true that A belongs

  to no B. But if this is proved, the truth is refuted as well; for

  the original conclusion was that A belongs to some B, and does not

  belong to some B. Further the impossible does not result from the

  hypothesis: for then the hypothesis would be false, since it is

  impossible to draw a false conclusion from true premisses: but in fact

  it is true: for A belongs to some B. Consequently we must not

  suppose that A belongs to some B, but that it belongs to all B.

  Similarly if we should be proving that A does not belong to some B:

  for if 'not to belong to some' and 'to belong not to all' have the

  same meaning, the demonstration of both will be identical.

  It is clear then that not the contrary but the contradictory ought

  to be supposed in all the syllogisms. For thus we shall have necessity

  of inference, and the claim we make is one that will be generally

  accepted. For if of everything one or other of two contradictory

  statements holds good, then if it is proved that the negation does not

  hold, the affirmation must be true. Again if it is not admitted that

  the affirmation is true, the claim that the negation is true will be

  generally accepted. But in neither way does it suit to maintain the

  contrary: for it is not necessary that if the universal negative is

  false, the universal affirmative should be true, nor is it generally

  accepted that if the one is false the other is true.

  12

  It is clear then that in the first figure all problems except the
>
  universal affirmative are proved per impossibile. But in the middle

  and the last figures this also is proved. Suppose that A does not

  belong to all B, and let it have been assumed that A belongs to all C.

  If then A belongs not to all B, but to all C, C will not belong to all

  B. But this is impossible (for suppose it to be clear that C belongs

  to all B): consequently the hypothesis is false. It is true then

  that A belongs to all B. But if the contrary is supposed, we shall

  have a syllogism and a result which is impossible: but the problem

  in hand is not proved. For if A belongs to no B, and to all C, C

  will belong to no B. This is impossible; so that it is false that A

  belongs to no B. But though this is false, it does not follow that

  it is true that A belongs to all B.

  When A belongs to some B, suppose that A belongs to no B, and let

  A belong to all C. It is necessary then that C should belong to no

  B. Consequently, if this is impossible, A must belong to some B. But

  if it is supposed that A does not belong to some B, we shall have

  the same results as in the first figure.

  Again suppose that A belongs to some B, and let A belong to no C. It

  is necessary then that C should not belong to some B. But originally

  it belonged to all B, consequently the hypothesis is false: A then

  will belong to no B.

  When A does not belong to an B, suppose it does belong to all B, and

  to no C. It is necessary then that C should belong to no B. But this

  is impossible: so that it is true that A does not belong to all B.

  It is clear then that all the syllogisms can be formed in the middle

  figure.

  13

  Similarly they can all be formed in the last figure. Suppose that

  A does not belong to some B, but C belongs to all B: then A does not

  belong to some C. If then this is impossible, it is false that A

  does not belong to some B; so that it is true that A belongs to all B.

  But if it is supposed that A belongs to no B, we shall have a

  syllogism and a conclusion which is impossible: but the problem in

  hand is not proved: for if the contrary is supposed, we shall have the

  same results as before.

  But to prove that A belongs to some B, this hypothesis must be made.

  If A belongs to no B, and C to some B, A will belong not to all C.

  If then this is false, it is true that A belongs to some B.

  When A belongs to no B, suppose A belongs to some B, and let it have

  been assumed that C belongs to all B. Then it is necessary that A

  should belong to some C. But ex hypothesi it belongs to no C, so

  that it is false that A belongs to some B. But if it is supposed

  that A belongs to all B, the problem is not proved.

  But this hypothesis must be made if we are prove that A belongs

  not to all B. For if A belongs to all B and C to some B, then A

  belongs to some C. But this we assumed not to be so, so it is false

  that A belongs to all B. But in that case it is true that A belongs

  not to all B. If however it is assumed that A belongs to some B, we

  shall have the same result as before.

  It is clear then that in all the syllogisms which proceed per

  impossibile the contradictory must be assumed. And it is plain that in

  the middle figure an affirmative conclusion, and in the last figure

  a universal conclusion, are proved in a way.

  14

  Demonstration per impossibile differs from ostensive proof in that

  it posits what it wishes to refute by reduction to a statement

  admitted to be false; whereas ostensive proof starts from admitted

  positions. Both, indeed, take two premisses that are admitted, but the

  latter takes the premisses from which the syllogism starts, the former

  takes one of these, along with the contradictory of the original

  conclusion. Also in the ostensive proof it is not necessary that the

  conclusion should be known, nor that one should suppose beforehand

  that it is true or not: in the other it is necessary to suppose

  beforehand that it is not true. It makes no difference whether the

  conclusion is affirmative or negative; the method is the same in

  both cases. Everything which is concluded ostensively can be proved

  per impossibile, and that which is proved per impossibile can be

  proved ostensively, through the same terms. Whenever the syllogism

  is formed in the first figure, the truth will be found in the middle

  or the last figure, if negative in the middle, if affirmative in the

  last. Whenever the syllogism is formed in the middle figure, the truth

  will be found in the first, whatever the problem may be. Whenever

  the syllogism is formed in the last figure, the truth will be found in

  the first and middle figures, if affirmative in first, if negative

  in the middle. Suppose that A has been proved to belong to no B, or

  not to all B, through the first figure. Then the hypothesis must

  have been that A belongs to some B, and the original premisses that

  C belongs to all A and to no B. For thus the syllogism was made and

  the impossible conclusion reached. But this is the middle figure, if C

  belongs to all A and to no B. And it is clear from these premisses

  that A belongs to no B. Similarly if has been proved not to belong

  to all B. For the hypothesis is that A belongs to all B; and the

  original premisses are that C belongs to all A but not to all B.

  Similarly too, if the premiss CA should be negative: for thus also

  we have the middle figure. Again suppose it has been proved that A

  belongs to some B. The hypothesis here is that is that A belongs to no

  B; and the original premisses that B belongs to all C, and A either to

  all or to some C: for in this way we shall get what is impossible. But

  if A and B belong to all C, we have the last figure. And it is clear

  from these premisses that A must belong to some B. Similarly if B or A

  should be assumed to belong to some C.

  Again suppose it has been proved in the middle figure that A belongs

  to all B. Then the hypothesis must have been that A belongs not to all

  B, and the original premisses that A belongs to all C, and C to all B:

  for thus we shall get what is impossible. But if A belongs to all C,

  and C to all B, we have the first figure. Similarly if it has been

  proved that A belongs to some B: for the hypothesis then must have

  been that A belongs to no B, and the original premisses that A belongs

  to all C, and C to some B. If the syllogism is negative, the

  hypothesis must have been that A belongs to some B, and the original

  premisses that A belongs to no C, and C to all B, so that the first

  figure results. If the syllogism is not universal, but proof has

  been given that A does not belong to some B, we may infer in the

  same way. The hypothesis is that A belongs to all B, the original

  premisses that A belongs to no C, and C belongs to some B: for thus we

  get the first figure.

  Again suppose it has been proved in the third figure that A

  belongs to all B. Then the hypothesis must have been that A belongs

  not to all B, and the original premisses that C belongs to all B,

  and A belongs to all C; for thus we shall
get what is impossible.

  And the original premisses form the first figure. Similarly if the

  demonstration establishes a particular proposition: the hypothesis

  then must have been that A belongs to no B, and the original premisses

  that C belongs to some B, and A to all C. If the syllogism is

  negative, the hypothesis must have been that A belongs to some B,

  and the original premisses that C belongs to no A and to all B, and

  this is the middle figure. Similarly if the demonstration is not

  universal. The hypothesis will then be that A belongs to all B, the

  premisses that C belongs to no A and to some B: and this is the middle

  figure.

  It is clear then that it is possible through the same terms to prove

  each of the problems ostensively as well. Similarly it will be

  possible if the syllogisms are ostensive to reduce them ad impossibile

  in the terms which have been taken, whenever the contradictory of

  the conclusion of the ostensive syllogism is taken as a premiss. For

  the syllogisms become identical with those which are obtained by means

  of conversion, so that we obtain immediately the figures through which

  each problem will be solved. It is clear then that every thesis can be

  proved in both ways, i.e. per impossibile and ostensively, and it is

  not possible to separate one method from the other.

  15

  In what figure it is possible to draw a conclusion from premisses

  which are opposed, and in what figure this is not possible, will be

  made clear in this way. Verbally four kinds of opposition are

  possible, viz. universal affirmative to universal negative,

  universal affirmative to particular negative, particular affirmative

  to universal negative, and particular affirmative to particular

  negative: but really there are only three: for the particular

  affirmative is only verbally opposed to the particular negative. Of

  the genuine opposites I call those which are universal contraries, the

  universal affirmative and the universal negative, e.g. 'every

  science is good', 'no science is good'; the others I call