Various Works Page 41

  problematic premiss is converted, a syllogism will be possible, as

  before. Let A belong to all B, and let B possibly belong to no C. If

  the terms are arranged thus, nothing necessarily follows: but if the

  proposition BC is converted and it is assumed that B is possible for

  all C, a syllogism results as before: for the terms are in the same

  relative positions. Likewise if both the relations are negative, if

  the major premiss states that A does not belong to B, and the minor

  premiss indicates that B may possibly belong to no C. Through the

  premisses actually taken nothing necessary results in any way; but

  if the problematic premiss is converted, we shall have a syllogism.

  Suppose that A belongs to no B, and B may possibly belong to no C.

  Through these comes nothing necessary. But if B is assumed to be

  possible for all C (and this is true) and if the premiss AB remains as

  before, we shall again have the same syllogism. But if it be assumed

  that B does not belong to any C, instead of possibly not belonging,

  there cannot be a syllogism anyhow, whether the premiss AB is negative

  or affirmative. As common instances of a necessary and positive

  relation we may take the terms white-animal-snow: of a necessary and

  negative relation, white-animal-pitch. Clearly then if the terms are

  universal, and one of the premisses is assertoric, the other

  problematic, whenever the minor premiss is problematic a syllogism

  always results, only sometimes it results from the premisses that

  are taken, sometimes it requires the conversion of one premiss. We

  have stated when each of these happens and the reason why. But if

  one of the relations is universal, the other particular, then whenever

  the major premiss is universal and problematic, whether affirmative or

  negative, and the particular is affirmative and assertoric, there will

  be a perfect syllogism, just as when the terms are universal. The

  demonstration is the same as before. But whenever the major premiss is

  universal, but assertoric, not problematic, and the minor is

  particular and problematic, whether both premisses are negative or

  affirmative, or one is negative, the other affirmative, in all cases

  there will be an imperfect syllogism. Only some of them will be proved

  per impossibile, others by the conversion of the problematic

  premiss, as has been shown above. And a syllogism will be possible

  by means of conversion when the major premiss is universal and

  assertoric, whether positive or negative, and the minor particular,

  negative, and problematic, e.g. if A belongs to all B or to no B,

  and B may possibly not belong to some C. For if the premiss BC is

  converted in respect of possibility, a syllogism results. But whenever

  the particular premiss is assertoric and negative, there cannot be a

  syllogism. As instances of the positive relation we may take the terms

  white-animal-snow; of the negative, white-animal-pitch. For the

  demonstration must be made through the indefinite nature of the

  particular premiss. But if the minor premiss is universal, and the

  major particular, whether either premiss is negative or affirmative,

  problematic or assertoric, nohow is a syllogism possible. Nor is a

  syllogism possible when the premisses are particular or indefinite,

  whether problematic or assertoric, or the one problematic, the other

  assertoric. The demonstration is the same as above. As instances of

  the necessary and positive relation we may take the terms

  animal-white-man; of the necessary and negative relation,

  animal-white-garment. It is evident then that if the major premiss

  is universal, a syllogism always results, but if the minor is

  universal nothing at all can ever be proved.

  16

  Whenever one premiss is necessary, the other problematic, there will

  be a syllogism when the terms are related as before; and a perfect

  syllogism when the minor premiss is necessary. If the premisses are

  affirmative the conclusion will be problematic, not assertoric,

  whether the premisses are universal or not: but if one is affirmative,

  the other negative, when the affirmative is necessary the conclusion

  will be problematic, not negative assertoric; but when the negative is

  necessary the conclusion will be problematic negative, and

  assertoric negative, whether the premisses are universal or not.

  Possibility in the conclusion must be understood in the same manner as

  before. There cannot be an inference to the necessary negative

  proposition: for 'not necessarily to belong' is different from

  'necessarily not to belong'.

  If the premisses are affirmative, clearly the conclusion which

  follows is not necessary. Suppose A necessarily belongs to all B,

  and let B be possible for all C. We shall have an imperfect

  syllogism to prove that A may belong to all C. That it is imperfect is

  clear from the proof: for it will be proved in the same manner as

  above. Again, let A be possible for all B, and let B necessarily

  belong to all C. We shall then have a syllogism to prove that A may

  belong to all C, not that A does belong to all C: and it is perfect,

  not imperfect: for it is completed directly through the original

  premisses.

  But if the premisses are not similar in quality, suppose first

  that the negative premiss is necessary, and let necessarily A not be

  possible for any B, but let B be possible for all C. It is necessary

  then that A belongs to no C. For suppose A to belong to all C or to

  some C. Now we assumed that A is not possible for any B. Since then

  the negative proposition is convertible, B is not possible for any

  A. But A is supposed to belong to all C or to some C. Consequently B

  will not be possible for any C or for all C. But it was originally

  laid down that B is possible for all C. And it is clear that the

  possibility of belonging can be inferred, since the fact of not

  belonging is inferred. Again, let the affirmative premiss be

  necessary, and let A possibly not belong to any B, and let B

  necessarily belong to all C. The syllogism will be perfect, but it

  will establish a problematic negative, not an assertoric negative. For

  the major premiss was problematic, and further it is not possible to

  prove the assertoric conclusion per impossibile. For if it were

  supposed that A belongs to some C, and it is laid down that A possibly

  does not belong to any B, no impossible relation between B and C

  follows from these premisses. But if the minor premiss is negative,

  when it is problematic a syllogism is possible by conversion, as

  above; but when it is necessary no syllogism can be formed. Nor

  again when both premisses are negative, and the minor is necessary.

  The same terms as before serve both for the positive

  relation-white-animal-snow, and for the negative

  relation-white-animal-pitch.

  The same relation will obtain in particular syllogisms. Whenever the

  negative proposition is necessary, the conclusion will be negative

  assertoric: e.g. if it is not possible that A should belong to any

  B, but B may belong to some of the Cs, it is nece
ssary that A should

  not belong to some of the Cs. For if A belongs to all C, but cannot

  belong to any B, neither can B belong to any A. So if A belongs to all

  C, to none of the Cs can B belong. But it was laid down that B may

  belong to some C. But when the particular affirmative in the

  negative syllogism, e.g. BC the minor premiss, or the universal

  proposition in the affirmative syllogism, e.g. AB the major premiss,

  is necessary, there will not be an assertoric conclusion. The

  demonstration is the same as before. But if the minor premiss is

  universal, and problematic, whether affirmative or negative, and the

  major premiss is particular and necessary, there cannot be a

  syllogism. Premisses of this kind are possible both where the relation

  is positive and necessary, e.g. animal-white-man, and where it is

  necessary and negative, e.g. animal-white-garment. But when the

  universal is necessary, the particular problematic, if the universal

  is negative we may take the terms animal-white-raven to illustrate the

  positive relation, or animal-white-pitch to illustrate the negative;

  and if the universal is affirmative we may take the terms

  animal-white-swan to illustrate the positive relation, and

  animal-white-snow to illustrate the negative and necessary relation.

  Nor again is a syllogism possible when the premisses are indefinite,

  or both particular. Terms applicable in either case to illustrate

  the positive relation are animal-white-man: to illustrate the

  negative, animal-white-inanimate. For the relation of animal to some

  white, and of white to some inanimate, is both necessary and

  positive and necessary and negative. Similarly if the relation is

  problematic: so the terms may be used for all cases.

  Clearly then from what has been said a syllogism results or not from

  similar relations of the terms whether we are dealing with simple

  existence or necessity, with this exception, that if the negative

  premiss is assertoric the conclusion is problematic, but if the

  negative premiss is necessary the conclusion is both problematic and

  negative assertoric. [It is clear also that all the syllogisms are

  imperfect and are perfected by means of the figures above mentioned.]

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  In the second figure whenever both premisses are problematic, no

  syllogism is possible, whether the premisses are affirmative or

  negative, universal or particular. But when one premiss is assertoric,

  the other problematic, if the affirmative is assertoric no syllogism

  is possible, but if the universal negative is assertoric a

  conclusion can always be drawn. Similarly when one premiss is

  necessary, the other problematic. Here also we must understand the

  term 'possible' in the conclusion, in the same sense as before.

  First we must point out that the negative problematic proposition is

  not convertible, e.g. if A may belong to no B, it does not follow that

  B may belong to no A. For suppose it to follow and assume that B may

  belong to no A. Since then problematic affirmations are convertible

  with negations, whether they are contraries or contradictories, and

  since B may belong to no A, it is clear that B may belong to all A.

  But this is false: for if all this can be that, it does not follow

  that all that can be this: consequently the negative proposition is

  not convertible. Further, these propositions are not incompatible,

  'A may belong to no B', 'B necessarily does not belong to some of

  the As'; e.g. it is possible that no man should be white (for it is

  also possible that every man should be white), but it is not true to

  say that it is possible that no white thing should be a man: for

  many white things are necessarily not men, and the necessary (as we

  saw) other than the possible.

  Moreover it is not possible to prove the convertibility of these

  propositions by a reductio ad absurdum, i.e. by claiming assent to the

  following argument: 'since it is false that B may belong to no A, it

  is true that it cannot belong to no A, for the one statement is the

  contradictory of the other. But if this is so, it is true that B

  necessarily belongs to some of the As: consequently A necessarily

  belongs to some of the Bs. But this is impossible.' The argument

  cannot be admitted, for it does not follow that some A is

  necessarily B, if it is not possible that no A should be B. For the

  latter expression is used in two senses, one if A some is

  necessarily B, another if some A is necessarily not B. For it is not

  true to say that that which necessarily does not belong to some of the

  As may possibly not belong to any A, just as it is not true to say

  that what necessarily belongs to some A may possibly belong to all

  A. If any one then should claim that because it is not possible for

  C to belong to all D, it necessarily does not belong to some D, he

  would make a false assumption: for it does belong to all D, but

  because in some cases it belongs necessarily, therefore we say that it

  is not possible for it to belong to all. Hence both the propositions

  'A necessarily belongs to some B' and 'A necessarily does not belong

  to some B' are opposed to the proposition 'A belongs to all B'.

  Similarly also they are opposed to the proposition 'A may belong to no

  B'. It is clear then that in relation to what is possible and not

  possible, in the sense originally defined, we must assume, not that

  A necessarily belongs to some B, but that A necessarily does not

  belong to some B. But if this is assumed, no absurdity results:

  consequently no syllogism. It is clear from what has been said that

  the negative proposition is not convertible.

  This being proved, suppose it possible that A may belong to no B and

  to all C. By means of conversion no syllogism will result: for the

  major premiss, as has been said, is not convertible. Nor can a proof

  be obtained by a reductio ad absurdum: for if it is assumed that B can

  belong to all C, no false consequence results: for A may belong both

  to all C and to no C. In general, if there is a syllogism, it is clear

  that its conclusion will be problematic because neither of the

  premisses is assertoric; and this must be either affirmative or

  negative. But neither is possible. Suppose the conclusion is

  affirmative: it will be proved by an example that the predicate cannot

  belong to the subject. Suppose the conclusion is negative: it will

  be proved that it is not problematic but necessary. Let A be white,

  B man, C horse. It is possible then for A to belong to all of the

  one and to none of the other. But it is not possible for B to belong

  nor not to belong to C. That it is not possible for it to belong, is

  clear. For no horse is a man. Neither is it possible for it not to

  belong. For it is necessary that no horse should be a man, but the

  necessary we found to be different from the possible. No syllogism

  then results. A similar proof can be given if the major premiss is

  negative, the minor affirmative, or if both are affirmative or

  negative. The demonstration can be made by means of the same terms.

  And whenever
one premiss is universal, the other particular, or both

  are particular or indefinite, or in whatever other way the premisses

  can be altered, the proof will always proceed through the same

  terms. Clearly then, if both the premisses are problematic, no

  syllogism results.

  18

  But if one premiss is assertoric, the other problematic, if the

  affirmative is assertoric and the negative problematic no syllogism

  will be possible, whether the premisses are universal or particular.

  The proof is the same as above, and by means of the same terms. But

  when the affirmative premiss is problematic, and the negative

  assertoric, we shall have a syllogism. Suppose A belongs to no B,

  but can belong to all C. If the negative proposition is converted, B

  will belong to no A. But ex hypothesi can belong to all C: so a

  syllogism is made, proving by means of the first figure that B may

  belong to no C. Similarly also if the minor premiss is negative. But

  if both premisses are negative, one being assertoric, the other

  problematic, nothing follows necessarily from these premisses as

  they stand, but if the problematic premiss is converted into its

  complementary affirmative a syllogism is formed to prove that B may

  belong to no C, as before: for we shall again have the first figure.

  But if both premisses are affirmative, no syllogism will be

  possible. This arrangement of terms is possible both when the relation

  is positive, e.g. health, animal, man, and when it is negative, e.g.

  health, horse, man.

  The same will hold good if the syllogisms are particular. Whenever

  the affirmative proposition is assertoric, whether universal or

  particular, no syllogism is possible (this is proved similarly and

  by the same examples as above), but when the negative proposition is

  assertoric, a conclusion can be drawn by means of conversion, as

  before. Again if both the relations are negative, and the assertoric

  proposition is universal, although no conclusion follows from the

  actual premisses, a syllogism can be obtained by converting the

  problematic premiss into its complementary affirmative as before.

  But if the negative proposition is assertoric, but particular, no