syllogism must be made in one or other of these figures. The
argument is the same if several middle terms should be necessary to
establish the relation to B; for the figure will be the same whether
there is one middle term or many.
It is clear then that the ostensive syllogisms are effected by means
of the aforesaid figures; these considerations will show that
reductiones ad also are effected in the same way. For all who effect
an argument per impossibile infer syllogistically what is false, and
prove the original conclusion hypothetically when something impossible
results from the assumption of its contradictory; e.g. that the
diagonal of the square is incommensurate with the side, because odd
numbers are equal to evens if it is supposed to be commensurate. One
infers syllogistically that odd numbers come out equal to evens, and
one proves hypothetically the incommensurability of the diagonal,
since a falsehood results through contradicting this. For this we
found to be reasoning per impossibile, viz. proving something
impossible by means of an hypothesis conceded at the beginning.
Consequently, since the falsehood is established in reductions ad
impossibile by an ostensive syllogism, and the original conclusion
is proved hypothetically, and we have already stated that ostensive
syllogisms are effected by means of these figures, it is evident
that syllogisms per impossibile also will be made through these
figures. Likewise all the other hypothetical syllogisms: for in
every case the syllogism leads up to the proposition that is
substituted for the original thesis; but the original thesis is
reached by means of a concession or some other hypothesis. But if this
is true, every demonstration and every syllogism must be formed by
means of the three figures mentioned above. But when this has been
shown it is clear that every syllogism is perfected by means of the
first figure and is reducible to the universal syllogisms in this
figure.
24
Further in every syllogism one of the premisses must be affirmative,
and universality must be present: unless one of the premisses is
universal either a syllogism will not be possible, or it will not
refer to the subject proposed, or the original position will be
begged. Suppose we have to prove that pleasure in music is good. If
one should claim as a premiss that pleasure is good without adding
'all', no syllogism will be possible; if one should claim that some
pleasure is good, then if it is different from pleasure in music, it
is not relevant to the subject proposed; if it is this very
pleasure, one is assuming that which was proposed at the outset to
be proved. This is more obvious in geometrical proofs, e.g. that the
angles at the base of an isosceles triangle are equal. Suppose the
lines A and B have been drawn to the centre. If then one should assume
that the angle AC is equal to the angle BD, without claiming generally
that angles of semicircles are equal; and again if one should assume
that the angle C is equal to the angle D, without the additional
assumption that every angle of a segment is equal to every other angle
of the same segment; and further if one should assume that when
equal angles are taken from the whole angles, which are themselves
equal, the remainders E and F are equal, he will beg the thing to be
proved, unless he also states that when equals are taken from equals
the remainders are equal.
It is clear then that in every syllogism there must be a universal
premiss, and that a universal statement is proved only when all the
premisses are universal, while a particular statement is proved both
from two universal premisses and from one only: consequently if the
conclusion is universal, the premisses also must be universal, but
if the premisses are universal it is possible that the conclusion
may not be universal. And it is clear also that in every syllogism
either both or one of the premisses must be like the conclusion. I
mean not only in being affirmative or negative, but also in being
necessary, pure, problematic. We must consider also the other forms of
predication.
It is clear also when a syllogism in general can be made and when it
cannot; and when a valid, when a perfect syllogism can be formed;
and that if a syllogism is formed the terms must be arranged in one of
the ways that have been mentioned.
25
It is clear too that every demonstration will proceed through
three terms and no more, unless the same conclusion is established
by different pairs of propositions; e.g. the conclusion E may be
established through the propositions A and B, and through the
propositions C and D, or through the propositions A and B, or A and C,
or B and C. For nothing prevents there being several middles for the
same terms. But in that case there is not one but several
syllogisms. Or again when each of the propositions A and B is obtained
by syllogistic inference, e.g. by means of D and E, and again B by
means of F and G. Or one may be obtained by syllogistic, the other
by inductive inference. But thus also the syllogisms are many; for the
conclusions are many, e.g. A and B and C. But if this can be called
one syllogism, not many, the same conclusion may be reached by more
than three terms in this way, but it cannot be reached as C is
established by means of A and B. Suppose that the proposition E is
inferred from the premisses A, B, C, and D. It is necessary then
that of these one should be related to another as whole to part: for
it has already been proved that if a syllogism is formed some of its
terms must be related in this way. Suppose then that A stands in
this relation to B. Some conclusion then follows from them. It must
either be E or one or other of C and D, or something other than these.
(1) If it is E the syllogism will have A and B for its sole
premisses. But if C and D are so related that one is whole, the
other part, some conclusion will follow from them also; and it must be
either E, or one or other of the propositions A and B, or something
other than these. And if it is (i) E, or (ii) A or B, either (i) the
syllogisms will be more than one, or (ii) the same thing happens to be
inferred by means of several terms only in the sense which we saw to
be possible. But if (iii) the conclusion is other than E or A or B,
the syllogisms will be many, and unconnected with one another. But
if C is not so related to D as to make a syllogism, the propositions
will have been assumed to no purpose, unless for the sake of induction
or of obscuring the argument or something of the sort.
(2) But if from the propositions A and B there follows not E but
some other conclusion, and if from C and D either A or B follows or
something else, then there are several syllogisms, and they do not
establish the conclusion proposed: for we assumed that the syllogism
proved E. And if no conclusion follows from C and D, it turns out that
these propositions have been assumed to
no purpose, and the
syllogism does not prove the original proposition.
So it is clear that every demonstration and every syllogism will
proceed through three terms only.
This being evident, it is clear that a syllogistic conclusion
follows from two premisses and not from more than two. For the three
terms make two premisses, unless a new premiss is assumed, as was said
at the beginning, to perfect the syllogisms. It is clear therefore
that in whatever syllogistic argument the premisses through which
the main conclusion follows (for some of the preceding conclusions
must be premisses) are not even in number, this argument either has
not been drawn syllogistically or it has assumed more than was
necessary to establish its thesis.
If then syllogisms are taken with respect to their main premisses,
every syllogism will consist of an even number of premisses and an odd
number of terms (for the terms exceed the premisses by one), and the
conclusions will be half the number of the premisses. But whenever a
conclusion is reached by means of prosyllogisms or by means of several
continuous middle terms, e.g. the proposition AB by means of the
middle terms C and D, the number of the terms will similarly exceed
that of the premisses by one (for the extra term must either be
added outside or inserted: but in either case it follows that the
relations of predication are one fewer than the terms related), and
the premisses will be equal in number to the relations of predication.
The premisses however will not always be even, the terms odd; but they
will alternate-when the premisses are even, the terms must be odd;
when the terms are even, the premisses must be odd: for along with one
term one premiss is added, if a term is added from any quarter.
Consequently since the premisses were (as we saw) even, and the
terms odd, we must make them alternately even and odd at each
addition. But the conclusions will not follow the same arrangement
either in respect to the terms or to the premisses. For if one term is
added, conclusions will be added less by one than the pre-existing
terms: for the conclusion is drawn not in relation to the single
term last added, but in relation to all the rest, e.g. if to ABC the
term D is added, two conclusions are thereby added, one in relation to
A, the other in relation to B. Similarly with any further additions.
And similarly too if the term is inserted in the middle: for in
relation to one term only, a syllogism will not be constructed.
Consequently the conclusions will be much more numerous than the terms
or the premisses.
26
Since we understand the subjects with which syllogisms are
concerned, what sort of conclusion is established in each figure,
and in how many moods this is done, it is evident to us both what sort
of problem is difficult and what sort is easy to prove. For that which
is concluded in many figures and through many moods is easier; that
which is concluded in few figures and through few moods is more
difficult to attempt. The universal affirmative is proved by means
of the first figure only and by this in only one mood; the universal
negative is proved both through the first figure and through the
second, through the first in one mood, through the second in two.
The particular affirmative is proved through the first and through the
last figure, in one mood through the first, in three moods through the
last. The particular negative is proved in all the figures, but once
in the first, in two moods in the second, in three moods in the third.
It is clear then that the universal affirmative is most difficult to
establish, most easy to overthrow. In general, universals are easier
game for the destroyer than particulars: for whether the predicate
belongs to none or not to some, they are destroyed: and the particular
negative is proved in all the figures, the universal negative in
two. Similarly with universal negatives: the original statement is
destroyed, whether the predicate belongs to all or to some: and this
we found possible in two figures. But particular statements can be
refuted in one way only-by proving that the predicate belongs either
to all or to none. But particular statements are easier to
establish: for proof is possible in more figures and through more
moods. And in general we must not forget that it is possible to refute
statements by means of one another, I mean, universal statements by
means of particular, and particular statements by means of
universal: but it is not possible to establish universal statements by
means of particular, though it is possible to establish particular
statements by means of universal. At the same time it is evident
that it is easier to refute than to establish.
The manner in which every syllogism is produced, the number of the
terms and premisses through which it proceeds, the relation of the
premisses to one another, the character of the problem proved in
each figure, and the number of the figures appropriate to each
problem, all these matters are clear from what has been said.
We must now state how we may ourselves always have a supply of
syllogisms in reference to the problem proposed and by what road we
may reach the principles relative to the problem: for perhaps we ought
not only to investigate the construction of syllogisms, but also to
have the power of making them.
Of all the things which exist some are such that they cannot be
predicated of anything else truly and universally, e.g. Cleon and
Callias, i.e. the individual and sensible, but other things may be
predicated of them (for each of these is both man and animal); and
some things are themselves predicated of others, but nothing prior
is predicated of them; and some are predicated of others, and yet
others of them, e.g. man of Callias and animal of man. It is clear
then that some things are naturally not stated of anything: for as a
rule each sensible thing is such that it cannot be predicated of
anything, save incidentally: for we sometimes say that that white
object is Socrates, or that that which approaches is Callias. We shall
explain in another place that there is an upward limit also to the
process of predicating: for the present we must assume this. Of
these ultimate predicates it is not possible to demonstrate another
predicate, save as a matter of opinion, but these may be predicated of
other things. Neither can individuals be predicated of other things,
though other things can be predicated of them. Whatever lies between
these limits can be spoken of in both ways: they may be stated of
others, and others stated of them. And as a rule arguments and
inquiries are concerned with these things. We must select the
premisses suitable to each problem in this manner: first we must lay
down the subject and the definitions and the properties of the
thing; next we must lay down those attributes which follow the
thing, and again those which the thing follows, and those which cannot
belong to it. But those to which it cannot belong need not be
selected, because the negative statement implied above is convertible.
Of the attributes which follow we must distinguish those which fall
within the definition, those which are predicated as properties, and
those which are predicated as accidents, and of the latter those which
apparently and those which really belong. The larger the supply a
man has of these, the more quickly will he reach a conclusion; and
in proportion as he apprehends those which are truer, the more
cogently will he demonstrate. But he must select not those which
follow some particular but those which follow the thing as a whole,
e.g. not what follows a particular man but what follows every man: for
the syllogism proceeds through universal premisses. If the statement
is indefinite, it is uncertain whether the premiss is universal, but
if the statement is definite, the matter is clear. Similarly one
must select those attributes which the subject follows as wholes,
for the reason given. But that which follows one must not suppose to
follow as a whole, e.g. that every animal follows man or every science
music, but only that it follows, without qualification, and indeed
we state it in a proposition: for the other statement is useless and
impossible, e.g. that every man is every animal or justice is all
good. But that which something follows receives the mark 'every'.
Whenever the subject, for which we must obtain the attributes that
follow, is contained by something else, what follows or does not
follow the highest term universally must not be selected in dealing
with the subordinate term (for these attributes have been taken in
dealing with the superior term; for what follows animal also follows
man, and what does not belong to animal does not belong to man); but
we must choose those attributes which are peculiar to each subject.
For some things are peculiar to the species as distinct from the
genus; for species being distinct there must be attributes peculiar to
each. Nor must we take as things which the superior term follows,