by Aristotle
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