qualification, we must put as middle term 'that which is', but if we
add the qualification 'that it is good', the middle term must be 'that
which is something'. Let A stand for 'knowledge that it is something',
B stand for 'something', and C stand for 'good'. It is true to
predicate A of B: for ex hypothesi there is a science of that which is
something, that it is something. B too is true of C: for that which
C represents is something. Consequently A is true of C: there will
then be knowledge of the good, that it is good: for ex hypothesi the
term 'something' indicates the thing's special nature. But if
'being' were taken as middle and 'being' simply were joined to the
extreme, not 'being something', we should not have had a syllogism
proving that there is knowledge of the good, that it is good, but that
it is; e.g. let A stand for knowledge that it is, B for being, C for
good. Clearly then in syllogisms which are thus limited we must take
the terms in the way stated.
39
We ought also to exchange terms which have the same value, word
for word, and phrase for phrase, and word and phrase, and always
take a word in preference to a phrase: for thus the setting out of the
terms will be easier. For example if it makes no difference whether we
say that the supposable is not the genus of the opinable or that the
opinable is not identical with a particular kind of supposable (for
what is meant is the same in both statements), it is better to take as
the terms the supposable and the opinable in preference to the
phrase suggested.
40
Since the expressions 'pleasure is good' and 'pleasure is the
good' are not identical, we must not set out the terms in the same
way; but if the syllogism is to prove that pleasure is the good, the
term must be 'the good', but if the object is to prove that pleasure
is good, the term will be 'good'. Similarly in all other cases.
41
It is not the same, either in fact or in speech, that A belongs to
all of that to which B belongs, and that A belongs to all of that to
all of which B belongs: for nothing prevents B from belonging to C,
though not to all C: e.g. let B stand for beautiful, and C for
white. If beauty belongs to something white, it is true to say that
beauty belongs to that which is white; but not perhaps to everything
that is white. If then A belongs to B, but not to everything of
which B is predicated, then whether B belongs to all C or merely
belongs to C, it is not necessary that A should belong, I do not say
to all C, but even to C at all. But if A belongs to everything of
which B is truly stated, it will follow that A can be said of all of
that of all of which B is said. If however A is said of that of all of
which B may be said, nothing prevents B belonging to C, and yet A
not belonging to all C or to any C at all. If then we take three terms
it is clear that the expression 'A is said of all of which B is
said' means this, 'A is said of all the things of which B is said'.
And if B is said of all of a third term, so also is A: but if B is not
said of all of the third term, there is no necessity that A should
be said of all of it.
We must not suppose that something absurd results through setting
out the terms: for we do not use the existence of this particular
thing, but imitate the geometrician who says that 'this line a foot
long' or 'this straight line' or 'this line without breadth' exists
although it does not, but does not use the diagrams in the sense
that he reasons from them. For in general, if two things are not
related as whole to part and part to whole, the prover does not
prove from them, and so no syllogism a is formed. We (I mean the
learner) use the process of setting out terms like perception by
sense, not as though it were impossible to demonstrate without these
illustrative terms, as it is to demonstrate without the premisses of
the syllogism.
42
We should not forget that in the same syllogism not all
conclusions are reached through one figure, but one through one
figure, another through another. Clearly then we must analyse
arguments in accordance with this. Since not every problem is proved
in every figure, but certain problems in each figure, it is clear from
the conclusion in what figure the premisses should be sought.
43
In reference to those arguments aiming at a definition which have
been directed to prove some part of the definition, we must take as
a term the point to which the argument has been directed, not the
whole definition: for so we shall be less likely to be disturbed by
the length of the term: e.g. if a man proves that water is a drinkable
liquid, we must take as terms drinkable and water.
44
Further we must not try to reduce hypothetical syllogisms; for
with the given premisses it is not possible to reduce them. For they
have not been proved by syllogism, but assented to by agreement. For
instance if a man should suppose that unless there is one faculty of
contraries, there cannot be one science, and should then argue that
not every faculty is of contraries, e.g. of what is healthy and what
is sickly: for the same thing will then be at the same time healthy
and sickly. He has shown that there is not one faculty of all
contraries, but he has not proved that there is not a science. And yet
one must agree. But the agreement does not come from a syllogism,
but from an hypothesis. This argument cannot be reduced: but the proof
that there is not a single faculty can. The latter argument perhaps
was a syllogism: but the former was an hypothesis.
The same holds good of arguments which are brought to a conclusion
per impossibile. These cannot be analysed either; but the reduction to
what is impossible can be analysed since it is proved by syllogism,
though the rest of the argument cannot, because the conclusion is
reached from an hypothesis. But these differ from the previous
arguments: for in the former a preliminary agreement must be reached
if one is to accept the conclusion; e.g. an agreement that if there is
proved to be one faculty of contraries, then contraries fall under the
same science; whereas in the latter, even if no preliminary
agreement has been made, men still accept the reasoning, because the
falsity is patent, e.g. the falsity of what follows from the
assumption that the diagonal is commensurate, viz. that then odd
numbers are equal to evens.
Many other arguments are brought to a conclusion by the help of an
hypothesis; these we ought to consider and mark out clearly. We
shall describe in the sequel their differences, and the various ways
in which hypothetical arguments are formed: but at present this much
must be clear, that it is not possible to resolve such arguments
into the figures. And we have explained the reason.
45
Whatever problems are proved in more than one figure, if they have
been established in one figure by syllogism, can be red
uced to another
figure, e.g. a negative syllogism in the first figure can be reduced
to the second, and a syllogism in the middle figure to the first,
not all however but some only. The point will be clear in the
sequel. If A belongs to no B, and B to all C, then A belongs to no
C. Thus the first figure; but if the negative statement is
converted, we shall have the middle figure. For B belongs to no A, and
to all C. Similarly if the syllogism is not universal but
particular, e.g. if A belongs to no B, and B to some C. Convert the
negative statement and you will have the middle figure.
The universal syllogisms in the second figure can be reduced to
the first, but only one of the two particular syllogisms. Let A belong
to no B and to all C. Convert the negative statement, and you will
have the first figure. For B will belong to no A and A to all C. But
if the affirmative statement concerns B, and the negative C, C must be
made first term. For C belongs to no A, and A to all B: therefore C
belongs to no B. B then belongs to no C: for the negative statement is
convertible.
But if the syllogism is particular, whenever the negative
statement concerns the major extreme, reduction to the first figure
will be possible, e.g. if A belongs to no B and to some C: convert the
negative statement and you will have the first figure. For B will
belong to no A and A to some C. But when the affirmative statement
concerns the major extreme, no resolution will be possible, e.g. if
A belongs to all B, but not to all C: for the statement AB does not
admit of conversion, nor would there be a syllogism if it did.
Again syllogisms in the third figure cannot all be resolved into the
first, though all syllogisms in the first figure can be resolved
into the third. Let A belong to all B and B to some C. Since the
particular affirmative is convertible, C will belong to some B: but
A belonged to all B: so that the third figure is formed. Similarly
if the syllogism is negative: for the particular affirmative is
convertible: therefore A will belong to no B, and to some C.
Of the syllogisms in the last figure one only cannot be resolved
into the first, viz. when the negative statement is not universal: all
the rest can be resolved. Let A and B be affirmed of all C: then C can
be converted partially with either A or B: C then belongs to some B.
Consequently we shall get the first figure, if A belongs to all C, and
C to some of the Bs. If A belongs to all C and B to some C, the
argument is the same: for B is convertible in reference to C. But if B
belongs to all C and A to some C, the first term must be B: for B
belongs to all C, and C to some A, therefore B belongs to some A.
But since the particular statement is convertible, A will belong to
some B. If the syllogism is negative, when the terms are universal
we must take them in a similar way. Let B belong to all C, and A to no
C: then C will belong to some B, and A to no C; and so C will be
middle term. Similarly if the negative statement is universal, the
affirmative particular: for A will belong to no C, and C to some of
the Bs. But if the negative statement is particular, no resolution
will be possible, e.g. if B belongs to all C, and A not belong to some
C: convert the statement BC and both premisses will be particular.
It is clear that in order to resolve the figures into one another
the premiss which concerns the minor extreme must be converted in both
the figures: for when this premiss is altered, the transition to the
other figure is made.
One of the syllogisms in the middle figure can, the other cannot, be
resolved into the third figure. Whenever the universal statement is
negative, resolution is possible. For if A belongs to no B and to some
C, both B and C alike are convertible in relation to A, so that B
belongs to no A and C to some A. A therefore is middle term. But
when A belongs to all B, and not to some C, resolution will not be
possible: for neither of the premisses is universal after conversion.
Syllogisms in the third figure can be resolved into the middle
figure, whenever the negative statement is universal, e.g. if A
belongs to no C, and B to some or all C. For C then will belong to
no A and to some B. But if the negative statement is particular, no
resolution will be possible: for the particular negative does not
admit of conversion.
It is clear then that the same syllogisms cannot be resolved in
these figures which could not be resolved into the first figure, and
that when syllogisms are reduced to the first figure these alone are
confirmed by reduction to what is impossible.
It is clear from what we have said how we ought to reduce
syllogisms, and that the figures may be resolved into one another.
46
In establishing or refuting, it makes some difference whether we
suppose the expressions 'not to be this' and 'to be not-this' are
identical or different in meaning, e.g. 'not to be white' and 'to be
not-white'. For they do not mean the same thing, nor is 'to be
not-white' the negation of 'to be white', but 'not to be white'. The
reason for this is as follows. The relation of 'he can walk' to 'he
can not-walk' is similar to the relation of 'it is white' to 'it is
not-white'; so is that of 'he knows what is good' to 'he knows what is
not-good'. For there is no difference between the expressions 'he
knows what is good' and 'he is knowing what is good', or 'he can walk'
and 'he is able to walk': therefore there is no difference between
their contraries 'he cannot walk'-'he is not able to walk'. If then
'he is not able to walk' means the same as 'he is able not to walk',
capacity to walk and incapacity to walk will belong at the same time
to the same person (for the same man can both walk and not-walk, and
is possessed of knowledge of what is good and of what is not-good),
but an affirmation and a denial which are opposed to one another do
not belong at the same time to the same thing. As then 'not to know
what is good' is not the same as 'to know what is not good', so 'to be
not-good' is not the same as 'not to be good'. For when two pairs
correspond, if the one pair are different from one another, the
other pair also must be different. Nor is 'to be not-equal' the same
as 'not to be equal': for there is something underlying the one,
viz. that which is not-equal, and this is the unequal, but there is
nothing underlying the other. Wherefore not everything is either equal
or unequal, but everything is equal or is not equal. Further the
expressions 'it is a not-white log' and 'it is not a white log' do not
imply one another's truth. For if 'it is a not-white log', it must
be a log: but that which is not a white log need not be a log at
all. Therefore it is clear that 'it is not-good' is not the denial
of 'it is good'. If then every single statement may truly be said to
be either an affirmation or a negation, if it is not a negation
clearly it must in a sense be an affirmation. But every affirmation
'it is not not-good'. The relation of these statements to one
another is as follows. Let A stand for 'to be good', B for 'not to
be good', let C stand for 'to be not-good' and be placed under B,
and let D stand for not to be not-good' and be placed under A. Then
either A or B will belong to everything, but they will never belong to
the same thing; and either C or D will belong to everything, but
they will never belong to the same thing. And B must belong to
everything to which C belongs. For if it is true to say 'it is a
not-white', it is true also to say 'it is not white': for it is
impossible that a thing should simultaneously be white and be
not-white, or be a not-white log and be a white log; consequently if
the affirmation does not belong, the denial must belong. But C does
not always belong to B: for what is not a log at all, cannot be a
not-white log either. On the other hand D belongs to everything to
which A belongs. For either C or D belongs to everything to which A
belongs. But since a thing cannot be simultaneously not-white and
white, D must belong to everything to which A belongs. For of that
which is white it is true to say that it is not not-white. But A is
not true of all D. For of that which is not a log at all it is not
true to say A, viz. that it is a white log. Consequently D is true,
but A is not true, i.e. that it is a white log. It is clear also
that A and C cannot together belong to the same thing, and that B
and D may possibly belong to the same thing.
Privative terms are similarly related positive ter terms respect
of this arrangement. Let A stand for 'equal', B for 'not equal', C for
'unequal', D for 'not unequal'.
In many things also, to some of which something belongs which does
not belong to others, the negation may be true in a similar way,
viz. that all are not white or that each is not white, while that each
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