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  syllogism is possible, whether the other premiss is affirmative or

  negative. Nor can a conclusion be drawn when both premisses are

  indefinite, whether affirmative or negative, or particular. The

  proof is the same and by the same terms.

  19

  If one of the premisses is necessary, the other problematic, then if

  the negative is necessary a syllogistic conclusion can be drawn, not

  merely a negative problematic but also a negative assertoric

  conclusion; but if the affirmative premiss is necessary, no conclusion

  is possible. Suppose that A necessarily belongs to no B, but may

  belong to all C. If the negative premiss is converted B will belong to

  no A: but A ex hypothesi is capable of belonging to all C: so once

  more a conclusion is drawn by the first figure that B may belong to no

  C. But at the same time it is clear that B will not belong to any C.

  For assume that it does: then if A cannot belong to any B, and B

  belongs to some of the Cs, A cannot belong to some of the Cs: but ex

  hypothesi it may belong to all. A similar proof can be given if the

  minor premiss is negative. Again let the affirmative proposition be

  necessary, and the other problematic; i.e. suppose that A may belong

  to no B, but necessarily belongs to all C. When the terms are arranged

  in this way, no syllogism is possible. For (1) it sometimes turns

  out that B necessarily does not belong to C. Let A be white, B man,

  C swan. White then necessarily belongs to swan, but may belong to no

  man; and man necessarily belongs to no swan; Clearly then we cannot

  draw a problematic conclusion; for that which is necessary is

  admittedly distinct from that which is possible. (2) Nor again can

  we draw a necessary conclusion: for that presupposes that both

  premisses are necessary, or at any rate the negative premiss. (3)

  Further it is possible also, when the terms are so arranged, that B

  should belong to C: for nothing prevents C falling under B, A being

  possible for all B, and necessarily belonging to C; e.g. if C stands

  for 'awake', B for 'animal', A for 'motion'. For motion necessarily

  belongs to what is awake, and is possible for every animal: and

  everything that is awake is animal. Clearly then the conclusion cannot

  be the negative assertion, if the relation must be positive when the

  terms are related as above. Nor can the opposite affirmations be

  established: consequently no syllogism is possible. A similar proof is

  possible if the major premiss is affirmative.

  But if the premisses are similar in quality, when they are

  negative a syllogism can always be formed by converting the

  problematic premiss into its complementary affirmative as before.

  Suppose A necessarily does not belong to B, and possibly may not

  belong to C: if the premisses are converted B belongs to no A, and A

  may possibly belong to all C: thus we have the first figure. Similarly

  if the minor premiss is negative. But if the premisses are affirmative

  there cannot be a syllogism. Clearly the conclusion cannot be a

  negative assertoric or a negative necessary proposition because no

  negative premiss has been laid down either in the assertoric or in the

  necessary mode. Nor can the conclusion be a problematic negative

  proposition. For if the terms are so related, there are cases in which

  B necessarily will not belong to C; e.g. suppose that A is white, B

  swan, C man. Nor can the opposite affirmations be established, since

  we have shown a case in which B necessarily does not belong to C. A

  syllogism then is not possible at all.

  Similar relations will obtain in particular syllogisms. For whenever

  the negative proposition is universal and necessary, a syllogism

  will always be possible to prove both a problematic and a negative

  assertoric proposition (the proof proceeds by conversion); but when

  the affirmative proposition is universal and necessary, no syllogistic

  conclusion can be drawn. This can be proved in the same way as for

  universal propositions, and by the same terms. Nor is a syllogistic

  conclusion possible when both premisses are affirmative: this also may

  be proved as above. But when both premisses are negative, and the

  premiss that definitely disconnects two terms is universal and

  necessary, though nothing follows necessarily from the premisses as

  they are stated, a conclusion can be drawn as above if the problematic

  premiss is converted into its complementary affirmative. But if both

  are indefinite or particular, no syllogism can be formed. The same

  proof will serve, and the same terms.

  It is clear then from what has been said that if the universal and

  negative premiss is necessary, a syllogism is always possible, proving

  not merely a negative problematic, but also a negative assertoric

  proposition; but if the affirmative premiss is necessary no conclusion

  can be drawn. It is clear too that a syllogism is possible or not

  under the same conditions whether the mode of the premisses is

  assertoric or necessary. And it is clear that all the syllogisms are

  imperfect, and are completed by means of the figures mentioned.

  20

  In the last figure a syllogism is possible whether both or only

  one of the premisses is problematic. When the premisses are

  problematic the conclusion will be problematic; and also when one

  premiss is problematic, the other assertoric. But when the other

  premiss is necessary, if it is affirmative the conclusion will be

  neither necessary or assertoric; but if it is negative the syllogism

  will result in a negative assertoric proposition, as above. In these

  also we must understand the expression 'possible' in the conclusion in

  the same way as before.

  First let the premisses be problematic and suppose that both A and B

  may possibly belong to every C. Since then the affirmative proposition

  is convertible into a particular, and B may possibly belong to every

  C, it follows that C may possibly belong to some B. So, if A is

  possible for every C, and C is possible for some of the Bs, then A

  is possible for some of the Bs. For we have got the first figure.

  And A if may possibly belong to no C, but B may possibly belong to all

  C, it follows that A may possibly not belong to some B: for we shall

  have the first figure again by conversion. But if both premisses

  should be negative no necessary consequence will follow from them as

  they are stated, but if the premisses are converted into their

  corresponding affirmatives there will be a syllogism as before. For if

  A and B may possibly not belong to C, if 'may possibly belong' is

  substituted we shall again have the first figure by means of

  conversion. But if one of the premisses is universal, the other

  particular, a syllogism will be possible, or not, under the

  arrangement of the terms as in the case of assertoric propositions.

  Suppose that A may possibly belong to all C, and B to some C. We shall

  have the first figure again if the particular premiss is converted.

  For if A is possible for all C, and C for some of the Bs, then A is

  possible for some of the Bs. Sim
ilarly if the proposition BC is

  universal. Likewise also if the proposition AC is negative, and the

  proposition BC affirmative: for we shall again have the first figure

  by conversion. But if both premisses should be negative-the one

  universal and the other particular-although no syllogistic

  conclusion will follow from the premisses as they are put, it will

  follow if they are converted, as above. But when both premisses are

  indefinite or particular, no syllogism can be formed: for A must

  belong sometimes to all B and sometimes to no B. To illustrate the

  affirmative relation take the terms animal-man-white; to illustrate

  the negative, take the terms horse-man-white--white being the middle

  term.

  21

  If one premiss is pure, the other problematic, the conclusion will

  be problematic, not pure; and a syllogism will be possible under the

  same arrangement of the terms as before. First let the premisses be

  affirmative: suppose that A belongs to all C, and B may possibly

  belong to all C. If the proposition BC is converted, we shall have the

  first figure, and the conclusion that A may possibly belong to some of

  the Bs. For when one of the premisses in the first figure is

  problematic, the conclusion also (as we saw) is problematic. Similarly

  if the proposition BC is pure, AC problematic; or if AC is negative,

  BC affirmative, no matter which of the two is pure; in both cases

  the conclusion will be problematic: for the first figure is obtained

  once more, and it has been proved that if one premiss is problematic

  in that figure the conclusion also will be problematic. But if the

  minor premiss BC is negative, or if both premisses are negative, no

  syllogistic conclusion can be drawn from the premisses as they

  stand, but if they are converted a syllogism is obtained as before.

  If one of the premisses is universal, the other particular, then

  when both are affirmative, or when the universal is negative, the

  particular affirmative, we shall have the same sort of syllogisms: for

  all are completed by means of the first figure. So it is clear that we

  shall have not a pure but a problematic syllogistic conclusion. But if

  the affirmative premiss is universal, the negative particular, the

  proof will proceed by a reductio ad impossibile. Suppose that B

  belongs to all C, and A may possibly not belong to some C: it

  follows that may possibly not belong to some B. For if A necessarily

  belongs to all B, and B (as has been assumed) belongs to all C, A will

  necessarily belong to all C: for this has been proved before. But it

  was assumed at the outset that A may possibly not belong to some C.

  Whenever both premisses are indefinite or particular, no syllogism

  will be possible. The demonstration is the same as was given in the

  case of universal premisses, and proceeds by means of the same terms.

  22

  If one of the premisses is necessary, the other problematic, when

  the premisses are affirmative a problematic affirmative conclusion can

  always be drawn; when one proposition is affirmative, the other

  negative, if the affirmative is necessary a problematic negative can

  be inferred; but if the negative proposition is necessary both a

  problematic and a pure negative conclusion are possible. But a

  necessary negative conclusion will not be possible, any more than in

  the other figures. Suppose first that the premisses are affirmative,

  i.e. that A necessarily belongs to all C, and B may possibly belong to

  all C. Since then A must belong to all C, and C may belong to some

  B, it follows that A may (not does) belong to some B: for so it

  resulted in the first figure. A similar proof may be given if the

  proposition BC is necessary, and AC is problematic. Again suppose

  one proposition is affirmative, the other negative, the affirmative

  being necessary: i.e. suppose A may possibly belong to no C, but B

  necessarily belongs to all C. We shall have the first figure once

  more: and-since the negative premiss is problematic-it is clear that

  the conclusion will be problematic: for when the premisses stand

  thus in the first figure, the conclusion (as we found) is problematic.

  But if the negative premiss is necessary, the conclusion will be not

  only that A may possibly not belong to some B but also that it does

  not belong to some B. For suppose that A necessarily does not belong

  to C, but B may belong to all C. If the affirmative proposition BC

  is converted, we shall have the first figure, and the negative premiss

  is necessary. But when the premisses stood thus, it resulted that A

  might possibly not belong to some C, and that it did not belong to

  some C; consequently here it follows that A does not belong to some B.

  But when the minor premiss is negative, if it is problematic we

  shall have a syllogism by altering the premiss into its

  complementary affirmative, as before; but if it is necessary no

  syllogism can be formed. For A sometimes necessarily belongs to all B,

  and sometimes cannot possibly belong to any B. To illustrate the

  former take the terms sleep-sleeping horse-man; to illustrate the

  latter take the terms sleep-waking horse-man.

  Similar results will obtain if one of the terms is related

  universally to the middle, the other in part. If both premisses are

  affirmative, the conclusion will be problematic, not pure; and also

  when one premiss is negative, the other affirmative, the latter

  being necessary. But when the negative premiss is necessary, the

  conclusion also will be a pure negative proposition; for the same kind

  of proof can be given whether the terms are universal or not. For

  the syllogisms must be made perfect by means of the first figure, so

  that a result which follows in the first figure follows also in the

  third. But when the minor premiss is negative and universal, if it

  is problematic a syllogism can be formed by means of conversion; but

  if it is necessary a syllogism is not possible. The proof will

  follow the same course as where the premisses are universal; and the

  same terms may be used.

  It is clear then in this figure also when and how a syllogism can be

  formed, and when the conclusion is problematic, and when it is pure.

  It is evident also that all syllogisms in this figure are imperfect,

  and that they are made perfect by means of the first figure.

  23

  It is clear from what has been said that the syllogisms in these

  figures are made perfect by means of universal syllogisms in the first

  figure and are reduced to them. That every syllogism without

  qualification can be so treated, will be clear presently, when it

  has been proved that every syllogism is formed through one or other of

  these figures.

  It is necessary that every demonstration and every syllogism

  should prove either that something belongs or that it does not, and

  this either universally or in part, and further either ostensively

  or hypothetically. One sort of hypothetical proof is the reductio ad

  impossibile. Let us speak first of ostensive syllogisms: for after

  these have been poin
ted out the truth of our contention will be

  clear with regard to those which are proved per impossibile, and in

  general hypothetically.

  If then one wants to prove syllogistically A of B, either as an

  attribute of it or as not an attribute of it, one must assert

  something of something else. If now A should be asserted of B, the

  proposition originally in question will have been assumed. But if A

  should be asserted of C, but C should not be asserted of anything, nor

  anything of it, nor anything else of A, no syllogism will be possible.

  For nothing necessarily follows from the assertion of some one thing

  concerning some other single thing. Thus we must take another

  premiss as well. If then A be asserted of something else, or something

  else of A, or something different of C, nothing prevents a syllogism

  being formed, but it will not be in relation to B through the

  premisses taken. Nor when C belongs to something else, and that to

  something else and so on, no connexion however being made with B, will

  a syllogism be possible concerning A in its relation to B. For in

  general we stated that no syllogism can establish the attribution of

  one thing to another, unless some middle term is taken, which is

  somehow related to each by way of predication. For the syllogism in

  general is made out of premisses, and a syllogism referring to this

  out of premisses with the same reference, and a syllogism relating

  this to that proceeds through premisses which relate this to that. But

  it is impossible to take a premiss in reference to B, if we neither

  affirm nor deny anything of it; or again to take a premiss relating

  A to B, if we take nothing common, but affirm or deny peculiar

  attributes of each. So we must take something midway between the

  two, which will connect the predications, if we are to have a

  syllogism relating this to that. If then we must take something common

  in relation to both, and this is possible in three ways (either by

  predicating A of C, and C of B, or C of both, or both of C), and these

  are the figures of which we have spoken, it is clear that every