Various Works Page 38

  possible, whether the major premiss is positive or negative,

  indefinite or particular: e.g. if some B is or is not A, and all C

  is B. As an example of a positive relation between the extremes take

  the terms good, state, wisdom: of a negative relation, good, state,

  ignorance. Again if no C is B, but some B is or is not A or not

  every B is A, there cannot be a syllogism. Take the terms white,

  horse, swan: white, horse, raven. The same terms may be taken also

  if the premiss BA is indefinite.

  Nor when the major premiss is universal, whether affirmative or

  negative, and the minor premiss is negative and particular, can

  there be a syllogism, whether the minor premiss be indefinite or

  particular: e.g. if all B is A and some C is not B, or if not all C is

  B. For the major term may be predicable both of all and of none of the

  minor, to some of which the middle term cannot be attributed.

  Suppose the terms are animal, man, white: next take some of the

  white things of which man is not predicated-swan and snow: animal is

  predicated of all of the one, but of none of the other. Consequently

  there cannot be a syllogism. Again let no B be A, but let some C not

  be B. Take the terms inanimate, man, white: then take some white

  things of which man is not predicated-swan and snow: the term

  inanimate is predicated of all of the one, of none of the other.

  Further since it is indefinite to say some C is not B, and it is

  true that some C is not B, whether no C is B, or not all C is B, and

  since if terms are assumed such that no C is B, no syllogism follows

  (this has already been stated) it is clear that this arrangement of

  terms will not afford a syllogism: otherwise one would have been

  possible with a universal negative minor premiss. A similar proof

  may also be given if the universal premiss is negative.

  Nor can there in any way be a syllogism if both the relations of

  subject and predicate are particular, either positively or negatively,

  or the one negative and the other affirmative, or one indefinite and

  the other definite, or both indefinite. Terms common to all the

  above are animal, white, horse: animal, white, stone.

  It is clear then from what has been said that if there is a

  syllogism in this figure with a particular conclusion, the terms

  must be related as we have stated: if they are related otherwise, no

  syllogism is possible anyhow. It is evident also that all the

  syllogisms in this figure are perfect (for they are all completed by

  means of the premisses originally taken) and that all conclusions

  are proved by this figure, viz. universal and particular,

  affirmative and negative. Such a figure I call the first.

  5

  Whenever the same thing belongs to all of one subject, and to none

  of another, or to all of each subject or to none of either, I call

  such a figure the second; by middle term in it I mean that which is

  predicated of both subjects, by extremes the terms of which this is

  said, by major extreme that which lies near the middle, by minor

  that which is further away from the middle. The middle term stands

  outside the extremes, and is first in position. A syllogism cannot

  be perfect anyhow in this figure, but it may be valid whether the

  terms are related universally or not.

  If then the terms are related universally a syllogism will be

  possible, whenever the middle belongs to all of one subject and to

  none of another (it does not matter which has the negative

  relation), but in no other way. Let M be predicated of no N, but of

  all O. Since, then, the negative relation is convertible, N will

  belong to no M: but M was assumed to belong to all O: consequently N

  will belong to no O. This has already been proved. Again if M

  belongs to all N, but to no O, then N will belong to no O. For if M

  belongs to no O, O belongs to no M: but M (as was said) belongs to all

  N: O then will belong to no N: for the first figure has again been

  formed. But since the negative relation is convertible, N will

  belong to no O. Thus it will be the same syllogism that proves both

  conclusions.

  It is possible to prove these results also by reductio ad

  impossibile.

  It is clear then that a syllogism is formed when the terms are so

  related, but not a perfect syllogism; for necessity is not perfectly

  established merely from the original premisses; others also are

  needed.

  But if M is predicated of every N and O, there cannot be a

  syllogism. Terms to illustrate a positive relation between the

  extremes are substance, animal, man; a negative relation, substance,

  animal, number-substance being the middle term.

  Nor is a syllogism possible when M is predicated neither of any N

  nor of any O. Terms to illustrate a positive relation are line,

  animal, man: a negative relation, line, animal, stone.

  It is clear then that if a syllogism is formed when the terms are

  universally related, the terms must be related as we stated at the

  outset: for if they are otherwise related no necessary consequence

  follows.

  If the middle term is related universally to one of the extremes,

  a particular negative syllogism must result whenever the middle term

  is related universally to the major whether positively or

  negatively, and particularly to the minor and in a manner opposite

  to that of the universal statement: by 'an opposite manner' I mean, if

  the universal statement is negative, the particular is affirmative: if

  the universal is affirmative, the particular is negative. For if M

  belongs to no N, but to some O, it is necessary that N does not belong

  to some O. For since the negative statement is convertible, N will

  belong to no M: but M was admitted to belong to some O: therefore N

  will not belong to some O: for the result is reached by means of the

  first figure. Again if M belongs to all N, but not to some O, it is

  necessary that N does not belong to some O: for if N belongs to all O,

  and M is predicated also of all N, M must belong to all O: but we

  assumed that M does not belong to some O. And if M belongs to all N

  but not to all O, we shall conclude that N does not belong to all O:

  the proof is the same as the above. But if M is predicated of all O,

  but not of all N, there will be no syllogism. Take the terms animal,

  substance, raven; animal, white, raven. Nor will there be a conclusion

  when M is predicated of no O, but of some N. Terms to illustrate a

  positive relation between the extremes are animal, substance, unit:

  a negative relation, animal, substance, science.

  If then the universal statement is opposed to the particular, we

  have stated when a syllogism will be possible and when not: but if the

  premisses are similar in form, I mean both negative or both

  affirmative, a syllogism will not be possible anyhow. First let them

  be negative, and let the major premiss be universal, e.g. let M belong

  to no N, and not to some O. It is possible then for N to belong either

  to all O or to no O. Terms to illustrate the negative relation are

  black,
snow, animal. But it is not possible to find terms of which the

  extremes are related positively and universally, if M belongs to

  some O, and does not belong to some O. For if N belonged to all O, but

  M to no N, then M would belong to no O: but we assumed that it belongs

  to some O. In this way then it is not admissible to take terms: our

  point must be proved from the indefinite nature of the particular

  statement. For since it is true that M does not belong to some O, even

  if it belongs to no O, and since if it belongs to no O a syllogism

  is (as we have seen) not possible, clearly it will not be possible now

  either.

  Again let the premisses be affirmative, and let the major premiss as

  before be universal, e.g. let M belong to all N and to some O. It is

  possible then for N to belong to all O or to no O. Terms to illustrate

  the negative relation are white, swan, stone. But it is not possible

  to take terms to illustrate the universal affirmative relation, for

  the reason already stated: the point must be proved from the

  indefinite nature of the particular statement. But if the minor

  premiss is universal, and M belongs to no O, and not to some N, it

  is possible for N to belong either to all O or to no O. Terms for

  the positive relation are white, animal, raven: for the negative

  relation, white, stone, raven. If the premisses are affirmative, terms

  for the negative relation are white, animal, snow; for the positive

  relation, white, animal, swan. Evidently then, whenever the

  premisses are similar in form, and one is universal, the other

  particular, a syllogism can, not be formed anyhow. Nor is one possible

  if the middle term belongs to some of each of the extremes, or does

  not belong to some of either, or belongs to some of the one, not to

  some of the other, or belongs to neither universally, or is related to

  them indefinitely. Common terms for all the above are white, animal,

  man: white, animal, inanimate.

  It is clear then from what has been said that if the terms are related

  to one another in the way stated, a syllogism results of necessity;

  and if there is a syllogism, the terms must be so related. But it is

  evident also that all the syllogisms in this figure are imperfect: for

  all are made perfect by certain supplementary statements, which either

  are contained in the terms of necessity or are assumed as

  hypotheses, i.e. when we prove per impossibile. And it is evident that

  an affirmative conclusion is not attained by means of this figure, but

  all are negative, whether universal or particular.

  6

  But if one term belongs to all, and another to none, of a third,

  or if both belong to all, or to none, of it, I call such a figure

  the third; by middle term in it I mean that of which both the

  predicates are predicated, by extremes I mean the predicates, by the

  major extreme that which is further from the middle, by the minor that

  which is nearer to it. The middle term stands outside the extremes,

  and is last in position. A syllogism cannot be perfect in this

  figure either, but it may be valid whether the terms are related

  universally or not to the middle term.

  If they are universal, whenever both P and R belong to S, it follows

  that P will necessarily belong to some R. For, since the affirmative

  statement is convertible, S will belong to some R: consequently

  since P belongs to all S, and S to some R, P must belong to some R:

  for a syllogism in the first figure is produced. It is possible to

  demonstrate this also per impossibile and by exposition. For if both P

  and R belong to all S, should one of the Ss, e.g. N, be taken, both

  P and R will belong to this, and thus P will belong to some R.

  If R belongs to all S, and P to no S, there will be a syllogism to

  prove that P will necessarily not belong to some R. This may be

  demonstrated in the same way as before by converting the premiss RS.

  It might be proved also per impossibile, as in the former cases. But

  if R belongs to no S, P to all S, there will be no syllogism. Terms

  for the positive relation are animal, horse, man: for the negative

  relation animal, inanimate, man.

  Nor can there be a syllogism when both terms are asserted of no S.

  Terms for the positive relation are animal, horse, inanimate; for

  the negative relation man, horse, inanimate-inanimate being the middle

  term.

  It is clear then in this figure also when a syllogism will be

  possible and when not, if the terms are related universally. For

  whenever both the terms are affirmative, there will be a syllogism

  to prove that one extreme belongs to some of the other; but when

  they are negative, no syllogism will be possible. But when one is

  negative, the other affirmative, if the major is negative, the minor

  affirmative, there will be a syllogism to prove that the one extreme

  does not belong to some of the other: but if the relation is reversed,

  no syllogism will be possible. If one term is related universally to

  the middle, the other in part only, when both are affirmative there

  must be a syllogism, no matter which of the premisses is universal.

  For if R belongs to all S, P to some S, P must belong to some R. For

  since the affirmative statement is convertible S will belong to some

  P: consequently since R belongs to all S, and S to some P, R must also

  belong to some P: therefore P must belong to some R.

  Again if R belongs to some S, and P to all S, P must belong to

  some R. This may be demonstrated in the same way as the preceding. And

  it is possible to demonstrate it also per impossibile and by

  exposition, as in the former cases. But if one term is affirmative,

  the other negative, and if the affirmative is universal, a syllogism

  will be possible whenever the minor term is affirmative. For if R

  belongs to all S, but P does not belong to some S, it is necessary

  that P does not belong to some R. For if P belongs to all R, and R

  belongs to all S, then P will belong to all S: but we assumed that

  it did not. Proof is possible also without reduction ad impossibile,

  if one of the Ss be taken to which P does not belong.

  But whenever the major is affirmative, no syllogism will be

  possible, e.g. if P belongs to all S and R does not belong to some

  S. Terms for the universal affirmative relation are animate, man,

  animal. For the universal negative relation it is not possible to

  get terms, if R belongs to some S, and does not belong to some S.

  For if P belongs to all S, and R to some S, then P will belong to some

  R: but we assumed that it belongs to no R. We must put the matter as

  before.' Since the expression 'it does not belong to some' is

  indefinite, it may be used truly of that also which belongs to none.

  But if R belongs to no S, no syllogism is possible, as has been shown.

  Clearly then no syllogism will be possible here.

  But if the negative term is universal, whenever the major is

  negative and the minor affirmative there will be a syllogism. For if P

  belongs to no S, and R belongs to some S, P will not belong to some R:

  for we shall have
the first figure again, if the premiss RS is

  converted.

  But when the minor is negative, there will be no syllogism. Terms

  for the positive relation are animal, man, wild: for the negative

  relation, animal, science, wild-the middle in both being the term

  wild.

  Nor is a syllogism possible when both are stated in the negative,

  but one is universal, the other particular. When the minor is

  related universally to the middle, take the terms animal, science,

  wild; animal, man, wild. When the major is related universally to

  the middle, take as terms for a negative relation raven, snow,

  white. For a positive relation terms cannot be found, if R belongs

  to some S, and does not belong to some S. For if P belongs to all R,

  and R to some S, then P belongs to some S: but we assumed that it

  belongs to no S. Our point, then, must be proved from the indefinite

  nature of the particular statement.

  Nor is a syllogism possible anyhow, if each of the extremes

  belongs to some of the middle or does not belong, or one belongs and

  the other does not to some of the middle, or one belongs to some of

  the middle, the other not to all, or if the premisses are

  indefinite. Common terms for all are animal, man, white: animal,

  inanimate, white.

  It is clear then in this figure also when a syllogism will be

  possible, and when not; and that if the terms are as stated, a

  syllogism results of necessity, and if there is a syllogism, the terms

  must be so related. It is clear also that all the syllogisms in this

  figure are imperfect (for all are made perfect by certain

  supplementary assumptions), and that it will not be possible to

  reach a universal conclusion by means of this figure, whether negative

  or affirmative.

  7

  It is evident also that in all the figures, whenever a proper

  syllogism does not result, if both the terms are affirmative or

  negative nothing necessary follows at all, but if one is

  affirmative, the other negative, and if the negative is stated