possible, whether the major premiss is positive or negative,
indefinite or particular: e.g. if some B is or is not A, and all C
is B. As an example of a positive relation between the extremes take
the terms good, state, wisdom: of a negative relation, good, state,
ignorance. Again if no C is B, but some B is or is not A or not
every B is A, there cannot be a syllogism. Take the terms white,
horse, swan: white, horse, raven. The same terms may be taken also
if the premiss BA is indefinite.
Nor when the major premiss is universal, whether affirmative or
negative, and the minor premiss is negative and particular, can
there be a syllogism, whether the minor premiss be indefinite or
particular: e.g. if all B is A and some C is not B, or if not all C is
B. For the major term may be predicable both of all and of none of the
minor, to some of which the middle term cannot be attributed.
Suppose the terms are animal, man, white: next take some of the
white things of which man is not predicated-swan and snow: animal is
predicated of all of the one, but of none of the other. Consequently
there cannot be a syllogism. Again let no B be A, but let some C not
be B. Take the terms inanimate, man, white: then take some white
things of which man is not predicated-swan and snow: the term
inanimate is predicated of all of the one, of none of the other.
Further since it is indefinite to say some C is not B, and it is
true that some C is not B, whether no C is B, or not all C is B, and
since if terms are assumed such that no C is B, no syllogism follows
(this has already been stated) it is clear that this arrangement of
terms will not afford a syllogism: otherwise one would have been
possible with a universal negative minor premiss. A similar proof
may also be given if the universal premiss is negative.
Nor can there in any way be a syllogism if both the relations of
subject and predicate are particular, either positively or negatively,
or the one negative and the other affirmative, or one indefinite and
the other definite, or both indefinite. Terms common to all the
above are animal, white, horse: animal, white, stone.
It is clear then from what has been said that if there is a
syllogism in this figure with a particular conclusion, the terms
must be related as we have stated: if they are related otherwise, no
syllogism is possible anyhow. It is evident also that all the
syllogisms in this figure are perfect (for they are all completed by
means of the premisses originally taken) and that all conclusions
are proved by this figure, viz. universal and particular,
affirmative and negative. Such a figure I call the first.
5
Whenever the same thing belongs to all of one subject, and to none
of another, or to all of each subject or to none of either, I call
such a figure the second; by middle term in it I mean that which is
predicated of both subjects, by extremes the terms of which this is
said, by major extreme that which lies near the middle, by minor
that which is further away from the middle. The middle term stands
outside the extremes, and is first in position. A syllogism cannot
be perfect anyhow in this figure, but it may be valid whether the
terms are related universally or not.
If then the terms are related universally a syllogism will be
possible, whenever the middle belongs to all of one subject and to
none of another (it does not matter which has the negative
relation), but in no other way. Let M be predicated of no N, but of
all O. Since, then, the negative relation is convertible, N will
belong to no M: but M was assumed to belong to all O: consequently N
will belong to no O. This has already been proved. Again if M
belongs to all N, but to no O, then N will belong to no O. For if M
belongs to no O, O belongs to no M: but M (as was said) belongs to all
N: O then will belong to no N: for the first figure has again been
formed. But since the negative relation is convertible, N will
belong to no O. Thus it will be the same syllogism that proves both
conclusions.
impossibile.
It is clear then that a syllogism is formed when the terms are so
related, but not a perfect syllogism; for necessity is not perfectly
established merely from the original premisses; others also are
needed.
But if M is predicated of every N and O, there cannot be a
syllogism. Terms to illustrate a positive relation between the
extremes are substance, animal, man; a negative relation, substance,
animal, number-substance being the middle term.
Nor is a syllogism possible when M is predicated neither of any N
nor of any O. Terms to illustrate a positive relation are line,
animal, man: a negative relation, line, animal, stone.
It is clear then that if a syllogism is formed when the terms are
universally related, the terms must be related as we stated at the
outset: for if they are otherwise related no necessary consequence
follows.
If the middle term is related universally to one of the extremes,
a particular negative syllogism must result whenever the middle term
is related universally to the major whether positively or
negatively, and particularly to the minor and in a manner opposite
to that of the universal statement: by 'an opposite manner' I mean, if
the universal statement is negative, the particular is affirmative: if
the universal is affirmative, the particular is negative. For if M
belongs to no N, but to some O, it is necessary that N does not belong
to some O. For since the negative statement is convertible, N will
belong to no M: but M was admitted to belong to some O: therefore N
will not belong to some O: for the result is reached by means of the
first figure. Again if M belongs to all N, but not to some O, it is
necessary that N does not belong to some O: for if N belongs to all O,
and M is predicated also of all N, M must belong to all O: but we
assumed that M does not belong to some O. And if M belongs to all N
but not to all O, we shall conclude that N does not belong to all O:
the proof is the same as the above. But if M is predicated of all O,
but not of all N, there will be no syllogism. Take the terms animal,
substance, raven; animal, white, raven. Nor will there be a conclusion
when M is predicated of no O, but of some N. Terms to illustrate a
positive relation between the extremes are animal, substance, unit:
a negative relation, animal, substance, science.
If then the universal statement is opposed to the particular, we
have stated when a syllogism will be possible and when not: but if the
premisses are similar in form, I mean both negative or both
affirmative, a syllogism will not be possible anyhow. First let them
be negative, and let the major premiss be universal, e.g. let M belong
to no N, and not to some O. It is possible then for N to belong either
to all O or to no O. Terms to illustrate the negative relation are
black,
snow, animal. But it is not possible to find terms of which the
extremes are related positively and universally, if M belongs to
some O, and does not belong to some O. For if N belonged to all O, but
M to no N, then M would belong to no O: but we assumed that it belongs
to some O. In this way then it is not admissible to take terms: our
point must be proved from the indefinite nature of the particular
statement. For since it is true that M does not belong to some O, even
if it belongs to no O, and since if it belongs to no O a syllogism
is (as we have seen) not possible, clearly it will not be possible now
either.
Again let the premisses be affirmative, and let the major premiss as
before be universal, e.g. let M belong to all N and to some O. It is
possible then for N to belong to all O or to no O. Terms to illustrate
the negative relation are white, swan, stone. But it is not possible
to take terms to illustrate the universal affirmative relation, for
the reason already stated: the point must be proved from the
indefinite nature of the particular statement. But if the minor
premiss is universal, and M belongs to no O, and not to some N, it
is possible for N to belong either to all O or to no O. Terms for
the positive relation are white, animal, raven: for the negative
relation, white, stone, raven. If the premisses are affirmative, terms
for the negative relation are white, animal, snow; for the positive
relation, white, animal, swan. Evidently then, whenever the
premisses are similar in form, and one is universal, the other
particular, a syllogism can, not be formed anyhow. Nor is one possible
if the middle term belongs to some of each of the extremes, or does
not belong to some of either, or belongs to some of the one, not to
some of the other, or belongs to neither universally, or is related to
them indefinitely. Common terms for all the above are white, animal,
man: white, animal, inanimate.
It is clear then from what has been said that if the terms are related
to one another in the way stated, a syllogism results of necessity;
and if there is a syllogism, the terms must be so related. But it is
evident also that all the syllogisms in this figure are imperfect: for
all are made perfect by certain supplementary statements, which either
are contained in the terms of necessity or are assumed as
hypotheses, i.e. when we prove per impossibile. And it is evident that
an affirmative conclusion is not attained by means of this figure, but
all are negative, whether universal or particular.
6
But if one term belongs to all, and another to none, of a third,
or if both belong to all, or to none, of it, I call such a figure
the third; by middle term in it I mean that of which both the
predicates are predicated, by extremes I mean the predicates, by the
major extreme that which is further from the middle, by the minor that
which is nearer to it. The middle term stands outside the extremes,
and is last in position. A syllogism cannot be perfect in this
figure either, but it may be valid whether the terms are related
universally or not to the middle term.
If they are universal, whenever both P and R belong to S, it follows
that P will necessarily belong to some R. For, since the affirmative
statement is convertible, S will belong to some R: consequently
since P belongs to all S, and S to some R, P must belong to some R:
for a syllogism in the first figure is produced. It is possible to
demonstrate this also per impossibile and by exposition. For if both P
and R belong to all S, should one of the Ss, e.g. N, be taken, both
P and R will belong to this, and thus P will belong to some R.
If R belongs to all S, and P to no S, there will be a syllogism to
prove that P will necessarily not belong to some R. This may be
demonstrated in the same way as before by converting the premiss RS.
It might be proved also per impossibile, as in the former cases. But
if R belongs to no S, P to all S, there will be no syllogism. Terms
for the positive relation are animal, horse, man: for the negative
relation animal, inanimate, man.
Nor can there be a syllogism when both terms are asserted of no S.
Terms for the positive relation are animal, horse, inanimate; for
the negative relation man, horse, inanimate-inanimate being the middle
term.
It is clear then in this figure also when a syllogism will be
possible and when not, if the terms are related universally. For
whenever both the terms are affirmative, there will be a syllogism
to prove that one extreme belongs to some of the other; but when
they are negative, no syllogism will be possible. But when one is
negative, the other affirmative, if the major is negative, the minor
affirmative, there will be a syllogism to prove that the one extreme
does not belong to some of the other: but if the relation is reversed,
no syllogism will be possible. If one term is related universally to
the middle, the other in part only, when both are affirmative there
must be a syllogism, no matter which of the premisses is universal.
For if R belongs to all S, P to some S, P must belong to some R. For
since the affirmative statement is convertible S will belong to some
P: consequently since R belongs to all S, and S to some P, R must also
belong to some P: therefore P must belong to some R.
Again if R belongs to some S, and P to all S, P must belong to
some R. This may be demonstrated in the same way as the preceding. And
it is possible to demonstrate it also per impossibile and by
exposition, as in the former cases. But if one term is affirmative,
the other negative, and if the affirmative is universal, a syllogism
will be possible whenever the minor term is affirmative. For if R
belongs to all S, but P does not belong to some S, it is necessary
that P does not belong to some R. For if P belongs to all R, and R
belongs to all S, then P will belong to all S: but we assumed that
it did not. Proof is possible also without reduction ad impossibile,
if one of the Ss be taken to which P does not belong.
But whenever the major is affirmative, no syllogism will be
possible, e.g. if P belongs to all S and R does not belong to some
S. Terms for the universal affirmative relation are animate, man,
animal. For the universal negative relation it is not possible to
get terms, if R belongs to some S, and does not belong to some S.
For if P belongs to all S, and R to some S, then P will belong to some
R: but we assumed that it belongs to no R. We must put the matter as
before.' Since the expression 'it does not belong to some' is
indefinite, it may be used truly of that also which belongs to none.
But if R belongs to no S, no syllogism is possible, as has been shown.
Clearly then no syllogism will be possible here.
But if the negative term is universal, whenever the major is
negative and the minor affirmative there will be a syllogism. For if P
belongs to no S, and R belongs to some S, P will not belong to some R:
for we shall have
the first figure again, if the premiss RS is
converted.
But when the minor is negative, there will be no syllogism. Terms
for the positive relation are animal, man, wild: for the negative
relation, animal, science, wild-the middle in both being the term
wild.
Nor is a syllogism possible when both are stated in the negative,
but one is universal, the other particular. When the minor is
related universally to the middle, take the terms animal, science,
wild; animal, man, wild. When the major is related universally to
the middle, take as terms for a negative relation raven, snow,
white. For a positive relation terms cannot be found, if R belongs
to some S, and does not belong to some S. For if P belongs to all R,
and R to some S, then P belongs to some S: but we assumed that it
belongs to no S. Our point, then, must be proved from the indefinite
nature of the particular statement.
Nor is a syllogism possible anyhow, if each of the extremes
belongs to some of the middle or does not belong, or one belongs and
the other does not to some of the middle, or one belongs to some of
the middle, the other not to all, or if the premisses are
indefinite. Common terms for all are animal, man, white: animal,
inanimate, white.
It is clear then in this figure also when a syllogism will be
possible, and when not; and that if the terms are as stated, a
syllogism results of necessity, and if there is a syllogism, the terms
must be so related. It is clear also that all the syllogisms in this
figure are imperfect (for all are made perfect by certain
supplementary assumptions), and that it will not be possible to
reach a universal conclusion by means of this figure, whether negative
or affirmative.
7
It is evident also that in all the figures, whenever a proper
syllogism does not result, if both the terms are affirmative or
negative nothing necessary follows at all, but if one is
affirmative, the other negative, and if the negative is stated