Various Works Page 49

  stands as we said above). But it turns out in these also that we use

  for the demonstration the very thing that is being proved: for C is

  proved of B, and B of by assuming that C is said of and C is proved of

  A through these premisses, so that we use the conclusion for the

  demonstration.

  In negative syllogisms reciprocal proof is as follows. Let B

  belong to all C, and A to none of the Bs: we conclude that A belongs

  to none of the Cs. If again it is necessary to prove that A belongs to

  none of the Bs (which was previously assumed) A must belong to no C,

  and C to all B: thus the previous premiss is reversed. If it is

  necessary to prove that B belongs to C, the proposition AB must no

  longer be converted as before: for the premiss 'B belongs to no A'

  is identical with the premiss 'A belongs to no B'. But we must

  assume that B belongs to all of that to none of which longs. Let A

  belong to none of the Cs (which was the previous conclusion) and

  assume that B belongs to all of that to none of which A belongs. It is

  necessary then that B should belong to all C. Consequently each of the

  three propositions has been made a conclusion, and this is circular

  demonstration, to assume the conclusion and the converse of one of the

  premisses, and deduce the remaining premiss.

  In particular syllogisms it is not possible to demonstrate the

  universal premiss through the other propositions, but the particular

  premiss can be demonstrated. Clearly it is impossible to demonstrate

  the universal premiss: 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 premiss.

  Further a syllogism cannot be made at all if the other premiss is

  converted: for the result is that both premisses are particular. But

  the particular premiss may be proved. Suppose that A has been proved

  of some C through B. If then it is assumed that B belongs to all A and

  the conclusion is retained, B will belong to some C: for we obtain the

  first figure and A is middle. But if the syllogism is negative, it

  is not possible to prove the universal premiss, for the reason given

  above. But it is possible to prove the particular premiss, if the

  proposition AB is converted as in the universal syllogism, i.e 'B

  belongs to some of that to some of which A does not belong': otherwise

  no syllogism results because the particular premiss is negative.

  6

  In the second figure it is not possible to prove an affirmative

  proposition in this way, but a negative proposition may be proved.

  An affirmative proposition is not proved because both premisses of the

  new syllogism are not affirmative (for the conclusion is negative) but

  an affirmative proposition is (as we saw) proved from premisses

  which are both affirmative. The negative is proved as follows. Let A

  belong to all B, and to no C: we conclude that B belongs to no C. If

  then it is assumed that B belongs to all A, it is necessary that A

  should belong to no C: for we get the second figure, with B as middle.

  But if the premiss AB was negative, and the other affirmative, we

  shall have the first figure. For C belongs to all A and B to no C,

  consequently B belongs to no A: neither then does A belong to B.

  Through the conclusion, therefore, and one premiss, we get no

  syllogism, but if another premiss is assumed in addition, a

  syllogism will be possible. But if the syllogism not universal, the

  universal premiss cannot be proved, for the same reason as we gave

  above, but the particular premiss can be proved whenever the universal

  statement is affirmative. Let A belong to all B, and not to all C: the

  conclusion is BC. If then it is assumed that B belongs to all A, but

  not to all C, A will not belong to some C, B being middle. But if

  the universal premiss is negative, the premiss AC will not be

  demonstrated by the conversion of AB: for it turns out that either

  both or one of the premisses is negative; consequently a syllogism

  will not be possible. But the proof will proceed as in the universal

  syllogisms, 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 premisses are taken universally, it

  is not possible to prove them reciprocally: for that which is

  universal is proved through statements 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 premiss. But if one premiss is universal, the other

  particular, proof of the latter will sometimes be possible,

  sometimes not. When both the premisses assumed 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

  all C and B to some C: the conclusion is the statement AB. If then

  it is assumed that C belongs to all A, it has been proved that C

  belongs to 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 syllogism no longer results from the conclusion and the

  other premiss. But if B belongs to all C, and A to some C, it will

  be possible to prove the proposition AC, when it is assumed that C

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

  to some B, it is necessary that A should belong to some C, B being

  middle. And whenever one premiss is affirmative the other negative,

  and the affirmative is universal, the other premiss can be proved. Let

  B belong to all 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 all B, it is necessary that A should not belong to some

  C, B being middle. But when the negative premiss is universal, the

  other premiss is not 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 B. If then it is assumed that C belongs to some

  of that to some of which does not belong, it is necessary that C

  should belong to some of the Bs. In no other way is it possible by

  converting the universal premiss to prove the other: for in no other

  way can a syllogism 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 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 syllogism 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 middle

  figure those syllogisms which are not made through those figures

  themselves either are not of the nature of circular proof or are

  imperfect.

  8

  To convert a syllogism means to alter the conclusion and make

  another syllogism to prove that either the extreme cannot belong to

  the middle or the middle to the last term. For it is necessary, if the

  conclusion has been changed into its opposite and one of the premisses

  stands, that the other premiss should be destroyed. For if it should

  stand, the conclusion also must stand. It makes a difference whether

  the conclusion is converted into its contradictory or into its

  contrary. For the same syllogism does not result whichever form the

  conversion takes. This will be made clear by the sequel. By

  contradictory opposition I mean the opposition of 'to all' to 'not

  to all', and of 'to some' to 'to none'; by contrary opposition I

  mean the opposition of 'to all' to 'to none', and of 'to some' to 'not

  to some'. Suppose that A been proved of C, through B as middle term.

  If then it should be assumed that A belongs to no C, but to all B, B

  will belong to no C. And if A belongs to no C, and B to all C, A

  will belong, not to no B at all, but not to all B. For (as we saw) the

  universal is not proved through the last figure. In a word it is not

  possible to refute universally by conversion the premiss which

  concerns the major extreme: for the refutation always proceeds through

  the third since it is necessary to take both premisses in reference to

  the minor extreme. Similarly if the syllogism is negative. Suppose

  it has been proved that A belongs to no C through B. Then if it is

  assumed that A belongs to all C, and to no B, B will belong to none of

  the Cs. And if A and B belong to all C, A will belong to some B: but

  in the original premiss it belonged to no B.

  If the conclusion is converted into its contradictory, the

  syllogisms will be contradictory and not universal. For one premiss is

  particular, so that the conclusion also will be particular. Let the

  syllogism be affirmative, and let it be converted as stated. Then if A

  belongs not to all C, but to all B, B will belong not to all C. And if

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

  all B. Similarly if the syllogism 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 all C, as was

  originally assumed, A will belong to some B.

  In particular syllogisms when the conclusion is converted into its

  contradictory, both premisses may be refuted, but when it is converted

  into its contrary, neither. For the result is no longer, as in the

  universal syllogisms, refutation in which the conclusion reached by O,

  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 C, and B to some C, A will not belong to some B: and if A

  belongs to no C, but to all B, B will belong to no C. Thus both

  premisses 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

  to all 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 premiss AB cannot be

  affected by a syllogism at all: for if A does not belong to some of

  the Cs, but B belongs to some of the Cs, neither of the premisses is

  universal. Similarly if the syllogism is negative: for if it should be

  assumed that A belongs to all C, both premisses are refuted: but if

  the assumption is that A belongs to some C, neither premiss is

  refuted. The proof is the same as before.

  9

  In the second figure it is not possible to refute the premiss

  which concerns the major extreme by establishing something contrary to

  it, whichever form the conversion of the conclusion may take. For

  the conclusion of the refutation will always be in the third figure,

  and in this figure (as we saw) there is no universal syllogism. The

  other premiss can be refuted in a manner similar to the conversion:

  I mean, if the conclusion of the first syllogism is converted into its

  contrary, the conclusion of the refutation will be the contrary of the

  minor premiss of the first, if into its contradictory, the

  contradictory. Let A belong to all B and to no C: conclusion BC. If

  then it is assumed that B belongs to all C, and the proposition AB

  stands, A will belong to all C, since the first figure is produced. If

  B belongs to all C, and A to no C, then A belongs not to all B: the

  figure is the last. But if the conclusion BC is converted into its

  contradictory, the premiss AB will be refuted as before, the

  premiss, AC by its contradictory. For if B belongs to some C, and A to

  no C, then A will not belong to some B. Again if B belongs to some

  C, and A to all B, A will belong to some C, so that the syllogism

  results in the contradictory of the minor premiss. A similar proof can

  be given if the premisses are transposed in respect of their quality.

  If the syllogism is particular, when the conclusion is converted

  into its contrary neither premiss can be refuted, as also happened

  in the first figure,' if the conclusion is converted into its

  contradictory, both premisses can be refuted. Suppose that A belongs

  to no B, and to some C: the conclusion is BC. If then it is assumed

  that B belongs to some C, and the statement AB stands, the

  conclusion will be that A does not belong to some C. But the

  original statement has not been refuted: for it is possible that A

  should belong to some C and also not to some C. Again if B belongs

  to some C and A to some C, no syllogism will be possible: for

  neither of the premisses taken is universal. Consequently the

  proposition AB is not refuted. But if the conclusion is converted into

  its contradictory, both premisses can be refuted. For if B belongs

  to all C, and A to no B, A will belong to no C: but it was assumed

  to belong to some C. Again if B belongs to all C and A to some C, A

  will belong to some B. The same proof can be given if the universal

  statement is affirmative.

  10

  In the third figure when the conclusion is converted into its

  contrary, neither of the premisses can be refuted in any of the

  syllogisms, but when the conclusion is converted into its

  contradictory, both premisses may be refuted and in all the moods.

  Suppose it has been proved that A belongs to some B, C being taken

  as middle, and the premisses being universal. If then it is assumed

  that A does not belong to some B, but B belongs to all C, no syllogism

  is formed about A and C. Nor if A does not belong to some B, but

  belongs to all C, will a syllogism be possible about B and C. A

  similar proof can be given if
the premisses are not universal. For

  either both premisses arrived at by the conversion must be particular,

  or the universal premiss must refer to the minor extreme. But we found

  that no syllogism is possible thus either in the first or in the

  middle figure. But if the conclusion is converted into its

  contradictory, both the premisses can be refuted. For if A belongs

  to no B, and B to all C, then A belongs to no C: again if A belongs to

  no B, and to all C, B belongs to no C. And similarly if one of the

  premisses is not universal. For if A belongs to no B, and B to some C,

  A will not belong to some C: if A belongs to no B, and to C, B will

  belong to no C.

  Similarly if the original syllogism is negative. Suppose it has been

  proved that A does not belong to some B, BC being affirmative, AC

  being negative: for it was thus that, as we saw, a syllogism could

  be made. Whenever then the contrary of the conclusion is assumed a

  syllogism will not be possible. For if A belongs to some B, and B to

  all C, no syllogism is possible (as we saw) about A and C. Nor, if A

  belongs to some B, and to no C, was a syllogism possible concerning

  B and C. Therefore the premisses are not refuted. But when the

  contradictory of the conclusion is assumed, they are refuted. For if A

  belongs to all B, and B to C, A belongs to all C: but A was supposed

  originally to belong to no C. Again if A belongs to all B, and to no

  C, then B belongs to no C: but it was supposed to belong to all C. A

  similar proof is possible if the premisses are not universal. For AC

  becomes universal and negative, the other premiss particular and

  affirmative. If then A belongs to all B, and B to some C, it results

  that A belongs to some C: but it was supposed to belong to no C. Again

  if A belongs to all B, and to no C, then B belongs to no C: but it was

  assumed to belong to some C. If A belongs to some B and B to some C,

  no syllogism results: nor yet if A belongs to some B, and to no C.

  Thus in one way the premisses are refuted, in the other way they are

  not.

  From what has been said it is clear how a syllogism results in