Various Works

  universally, a syllogism always results relating the minor to the

  major term, e.g. if A belongs to all or some B, and B belongs to no C:

  for if the premisses are converted it is necessary that C does not

  belong to some A. Similarly also in the other figures: a syllogism

  always results by means of conversion. It is evident also that the

  substitution of an indefinite for a particular affirmative will effect

  the same syllogism in all the figures.

  It is clear too that all the imperfect syllogisms are made perfect

  by means of the first figure. For all are brought to a conclusion

  either ostensively or per impossibile. In both ways the first figure

  is formed: if they are made perfect ostensively, because (as we saw)

  all are brought to a conclusion by means of conversion, and conversion

  produces the first figure: if they are proved per impossibile, because

  on the assumption of the false statement the syllogism comes about

  by means of the first figure, e.g. in the last figure, if A and B

  belong to all C, it follows that A belongs to some B: for if A

  belonged to no B, and B belongs to all C, A would belong to no C:

  but (as we stated) it belongs to all C. Similarly also with the rest.

  It is possible also to reduce all syllogisms to the universal

  syllogisms in the first figure. Those in the second figure are clearly

  made perfect by these, though not all in the same way; the universal

  syllogisms are made perfect by converting the negative premiss, each

  of the particular syllogisms by reductio ad impossibile. In the

  first figure particular syllogisms are indeed made perfect by

  themselves, but it is possible also to prove them by means of the

  second figure, reducing them ad impossibile, e.g. if A belongs to

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

  belonged to no C, and belongs to all B, then B will belong to no C:

  this we know by means of the second figure. Similarly also

  demonstration will be possible in the case of the negative. For if A

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

  for if it belonged to all C, and belongs to no B, then B will belong

  to no C: and this (as we saw) is the middle figure. Consequently,

  since all syllogisms in the middle figure can be reduced to

  universal syllogisms in the first figure, and since particular

  syllogisms in the first figure can be reduced to syllogisms in the

  middle figure, it is clear that particular syllogisms can be reduced

  to universal syllogisms in the first figure. Syllogisms in the third

  figure, if the terms are universal, are directly made perfect by means

  of those syllogisms; but, when one of the premisses is particular,

  by means of the particular syllogisms in the first figure: and these

  (we have seen) may be reduced to the universal syllogisms in the first

  figure: consequently also the particular syllogisms in the third

  figure may be so reduced. It is clear then that all syllogisms may

  be reduced to the universal syllogisms in the first figure.

  We have stated then how syllogisms which prove that something

  belongs or does not belong to something else are constituted, both how

  syllogisms of the same figure are constituted in themselves, and how

  syllogisms of different figures are related to one another.

  8

  Since there is a difference according as something belongs,

  necessarily belongs, or may belong to something else (for many

  things belong indeed, but not necessarily, others neither

  necessarily nor indeed at all, but it is possible for them to belong),

  it is clear that there will be different syllogisms to prove each of

  these relations, and syllogisms with differently related terms, one

  syllogism concluding from what is necessary, another from what is, a

  third from what is possible.

  There is hardly any difference between syllogisms from necessary

  premisses and syllogisms from premisses which merely assert. When

  the terms are put in the same way, then, whether something belongs

  or necessarily belongs (or does not belong) to something else, a

  syllogism will or will not result alike in both cases, the only

  difference being the addition of the expression 'necessarily' to the

  terms. For the negative statement is convertible alike in both

  cases, and we should give the same account of the expressions 'to be

  contained in something as in a whole' and 'to be predicated of all

  of something'. With the exceptions to be made below, the conclusion

  will be proved to be necessary by means of conversion, in the same

  manner as in the case of simple predication. But in the middle

  figure when the universal statement is affirmative, and the particular

  negative, and again in the third figure when the universal is

  affirmative and the particular negative, the demonstration will not

  take the same form, but it is necessary by the 'exposition' of a

  part of the subject of the particular negative proposition, to which

  the predicate does not belong, to make the syllogism in reference to

  this: with terms so chosen the conclusion will necessarily follow. But

  if the relation is necessary in respect of the part taken, it must

  hold of some of that term in which this part is included: for the part

  taken is just some of that. And each of the resulting syllogisms is in

  the appropriate figure.

  9

  It happens sometimes also that when one premiss is necessary the

  conclusion is necessary, not however when either premiss is necessary,

  but only when the major is, e.g. if A is taken as necessarily

  belonging or not belonging to B, but B is taken as simply belonging to

  C: for if the premisses are taken in this way, A will necessarily

  belong or not belong to C. For since necessarily belongs, or does

  not belong, to every B, and since C is one of the Bs, it is clear that

  for C also the positive or the negative relation to A will hold

  necessarily. But if the major premiss is not necessary, but the

  minor is necessary, the conclusion will not be necessary. For if it

  were, it would result both through the first figure and through the

  third that A belongs necessarily to some B. But this is false; for B

  may be such that it is possible that A should belong to none of it.

  Further, an example also makes it clear that the conclusion not be

  necessary, e.g. if A were movement, B animal, C man: man is an

  animal necessarily, but an animal does not move necessarily, nor

  does man. Similarly also if the major premiss is negative; for the

  proof is the same.

  In particular syllogisms, if the universal premiss is necessary,

  then the conclusion will be necessary; but if the particular, the

  conclusion will not be necessary, whether the universal premiss is

  negative or affirmative. First let the universal be necessary, and let

  A belong to all B necessarily, but let B simply belong to some C: it

  is necessary then that A belongs to some C necessarily: for C falls

  under B, and A was assumed to belong necessarily to all B. Similarly

  also if the syllogism should be negative: for the proof will be the
/>
  same. But if the particular premiss is necessary, the conclusion

  will not be necessary: for from the denial of such a conclusion

  nothing impossible results, just as it does not in the universal

  syllogisms. The same is true of negative syllogisms. Try the terms

  movement, animal, white.

  10

  In the second figure, if the negative premiss is necessary, then the

  conclusion will be necessary, but if the affirmative, not necessary.

  First let the negative be necessary; let A be possible of no B, and

  simply belong to C. Since then the negative statement is

  convertible, B is possible of no A. But A belongs to all C;

  consequently B is possible of no C. For C falls under A. The same

  result would be obtained if the minor premiss were negative: for if

  A is possible be of no C, C is possible of no A: but A belongs to

  all B, consequently C is possible of none of the Bs: for again we have

  obtained the first figure. Neither then is B possible of C: for

  conversion is possible without modifying the relation.

  But if the affirmative premiss is necessary, the conclusion will not

  be necessary. Let A belong to all B necessarily, but to no C simply.

  If then the negative premiss is converted, the first figure results.

  But it has been proved in the case of the first figure that if the

  negative major premiss is not necessary the conclusion will not be

  necessary either. Therefore the same result will obtain here. Further,

  if the conclusion is necessary, it follows that C necessarily does not

  belong to some A. For if B necessarily belongs to no C, C will

  necessarily belong to no B. But B at any rate must belong to some A,

  if it is true (as was assumed) that A necessarily belongs to all B.

  Consequently it is necessary that C does not belong to some A. But

  nothing prevents such an A being taken that it is possible for C to

  belong to all of it. Further one might show by an exposition of

  terms that the conclusion is not necessary without qualification,

  though it is a necessary conclusion from the premisses. For example

  let A be animal, B man, C white, and let the premisses be assumed to

  correspond to what we had before: it is possible that animal should

  belong to nothing white. Man then will not belong to anything white,

  but not necessarily: for it is possible for man to be born white,

  not however so long as animal belongs to nothing white. Consequently

  under these conditions the conclusion will be necessary, but it is not

  necessary without qualification.

  Similar results will obtain also in particular syllogisms. For

  whenever the negative premiss is both universal and necessary, then

  the conclusion will be necessary: but whenever the affirmative premiss

  is universal, the negative particular, the conclusion will not be

  necessary. First then let the negative premiss be both universal and

  necessary: let it be possible for no B that A should belong to it, and

  let A simply belong to some C. Since the negative statement is

  convertible, it will be possible for no A that B should belong to

  it: but A belongs to some C; consequently B necessarily does not

  belong to some of the Cs. Again let the affirmative premiss be both

  universal and necessary, and let the major premiss be affirmative.

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

  it is clear that B will not belong to some C, but not necessarily. For

  the same terms can be used to demonstrate the point, which were used

  in the universal syllogisms. Nor again, if the negative statement is

  necessary but particular, will the conclusion be necessary. The

  point can be demonstrated by means of the same terms.

  11

  In the last figure when the terms are related universally to the

  middle, and both premisses are affirmative, if one of the two is

  necessary, then the conclusion will be necessary. But if one is

  negative, the other affirmative, whenever the negative is necessary

  the conclusion also will be necessary, but whenever the affirmative is

  necessary the conclusion will not be necessary. First let both the

  premisses be affirmative, and let A and B belong to all C, and let

  AC be necessary. Since then B belongs to all C, C also will belong

  to some B, because the universal is convertible into the particular:

  consequently if A belongs necessarily to all C, and C belongs to

  some B, it is necessary that A should belong to some B also. For B

  is under C. The first figure then is formed. A similar proof will be

  given also if BC is necessary. For C is convertible with some A:

  consequently if B belongs necessarily to all C, it will belong

  necessarily also to some A.

  Again let AC be negative, BC affirmative, and let the negative

  premiss be necessary. Since then C is convertible with some B, but A

  necessarily belongs to no C, A will necessarily not belong to some B

  either: for B is under C. But if the affirmative is necessary, the

  conclusion will not be necessary. For suppose BC is affirmative and

  necessary, while AC is negative and not necessary. Since then the

  affirmative is convertible, C also will belong to some B

  necessarily: consequently if A belongs to none of the Cs, while C

  belongs to some of the Bs, A will not belong to some of the Bs-but not

  of necessity; for it has been proved, in the case of the first figure,

  that if the negative premiss is not necessary, neither will the

  conclusion be necessary. Further, the point may be made clear by

  considering the terms. Let the term A be 'good', let that which B

  signifies be 'animal', let the term C be 'horse'. It is possible

  then that the term good should belong to no horse, and it is necessary

  that the term animal should belong to every horse: but it is not

  necessary that some animal should not be good, since it is possible

  for every animal to be good. Or if that is not possible, take as the

  term 'awake' or 'asleep': for every animal can accept these.

  If, then, the premisses are universal, we have stated when the

  conclusion will be necessary. But if one premiss is universal, the

  other particular, and if both are affirmative, whenever the

  universal is necessary the conclusion also must be necessary. The

  demonstration is the same as before; for the particular affirmative

  also is convertible. If then it is necessary that B should belong to

  all C, and A falls under C, it is necessary that B should belong to

  some A. But if B must belong to some A, then A must belong to some

  B: for conversion is possible. Similarly also if AC should be

  necessary and universal: for B falls under C. But if the particular

  premiss is necessary, the conclusion will not be necessary. Let the

  premiss BC be both particular and necessary, and let A belong to all

  C, not however necessarily. If the proposition BC is converted the

  first figure is formed, and the universal premiss is not necessary,

  but the particular is necessary. But when the premisses were thus, the

  conclusion (as we proved was not necessary: consequently it is not

  here either. Further, the point is clear if we look at the te
rms.

  Let A be waking, B biped, and C animal. It is necessary that B

  should belong to some C, but it is possible for A to belong to C,

  and that A should belong to B is not necessary. For there is no

  necessity that some biped should be asleep or awake. Similarly and

  by means of the same terms proof can be made, should the proposition

  AC be both particular and necessary.

  But if one premiss is affirmative, the other negative, whenever

  the universal is both negative and necessary the conclusion also

  will be necessary. For if it is not possible that A should belong to

  any C, but B belongs to some C, it is necessary that A should not

  belong to some B. But whenever the affirmative proposition is

  necessary, whether universal or particular, or the negative is

  particular, the conclusion will not be necessary. The proof of this by

  reduction will be the same as before; but if terms are wanted, when

  the universal affirmative is necessary, take the terms

  'waking'-'animal'-'man', 'man' being middle, and when the

  affirmative is particular and necessary, take the terms

  'waking'-'animal'-'white': for it is necessary that animal should

  belong to some white thing, but it is possible that waking should

  belong to none, and it is not necessary that waking should not

  belong to some animal. But when the negative proposition being

  particular is necessary, take the terms 'biped', 'moving', 'animal',

  'animal' being middle.

  12

  It is clear then that a simple conclusion is not reached unless both

  premisses are simple assertions, but a necessary conclusion is

  possible although one only of the premisses is necessary. But in

  both cases, whether the syllogisms are affirmative or negative, it

  is necessary that one premiss should be similar to the conclusion. I

  mean by 'similar', if the conclusion is a simple assertion, the

  premiss must be simple; if the conclusion is necessary, the premiss

  must be necessary. Consequently this also is clear, that the

  conclusion will be neither necessary nor simple unless a necessary