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
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.]
17
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