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
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