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Prior Analytics
By Aristotle
Translated by A. J. Jenkinson
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BOOK I
Part 1
We must first state the subject of our inquiry and the faculty to
which it belongs: its subject is demonstration and the faculty that
carries it out demonstrative science. We must next define a premiss,
a term, and a syllogism, and the nature of a perfect and of an imperfect
syllogism; and after that, the inclusion or noninclusion of one term
in another as in a whole, and what we mean by predicating one term
of all, or none, of another.
A premiss then is a sentence affirming or denying one thing of another.
This is either universal or particular or indefinite. By universal
I mean the statement that something belongs to all or none of something
else; by particular that it belongs to some or not to some or not
to all; by indefinite that it does or does not belong, without any
mark to show whether it is universal or particular, e.g. 'contraries
are subjects of the same science', or 'pleasure is not good'. The
demonstrative premiss differs from the dialectical, because the demonstrative
premiss is the assertion of one of two contradictory statements (the
demonstrator does not ask for his premiss, but lays it down), whereas
the dialectical premiss depends on the adversary's choice between
two contradictories. But this will make no difference to the production
of a syllogism in either case; for both the demonstrator and the dialectician
argue syllogistically after stating that something does or does not
belong to something else. Therefore a syllogistic premiss without
qualification will be an affirmation or denial of something concerning
something else in the way we have described; it will be demonstrative,
if it is true and obtained through the first principles of its science;
while a dialectical premiss is the giving of a choice between two
contradictories, when a man is proceeding by question, but when he
is syllogizing it is the assertion of that which is apparent and generally
admitted, as has been said in the Topics. The nature then of a premiss
and the difference between syllogistic, demonstrative, and dialectical
premisses, may be taken as sufficiently defined by us in relation
to our present need, but will be stated accurately in the sequel.
I call that a term into which the premiss is resolved, i.e. both the
predicate and that of which it is predicated, 'being' being added
and 'not being' removed, or vice versa.
A syllogism is discourse in which, certain things being stated, something
other than what is stated follows of necessity from their being so.
I mean by the last phrase that they produce the consequence, and by
this, that no further term is required from without in order to make
the consequence necessary.
I call that a perfect syllogism which needs nothing other than what
has been stated to make plain what necessarily follows; a syllogism
is imperfect, if it needs either one or more propositions, which are
indeed the necessary consequences of the terms set down, but have
not been expressly stated as premisses.
That one term should be included in another as in a whole is the same
as for the other to be predicated of all of the first. And we say
that one term is predicated of all of another, whenever no instance
of the subject can be found of which the other term cannot be asserted:
'to be predicated of none' must be understood in the same way.
Part 2
Every premiss states that something either is or must be or may be
the attribute of something else; of premisses of these three kinds
some are affirmative, others negative, in respect of each of the three
modes of attribution; again some affirmative and negative premisses
are universal, others particular, others indefinite. It is necessary
then that in universal attribution the terms of the negative premiss
should be convertible, e.g. if no pleasure is good, then no good will
be pleasure; the terms of the affirmative must be convertible, not
however, universally, but in part, e.g. if every pleasure,is good,
some good must be pleasure; the particular affirmative must convert
in part (for if some pleasure is good, then some good will be pleasure);
but the particular negative need not convert, for if some animal is
not man, it does not follow that some man is not animal.
First then take a universal negative with the terms A and B. If no
B is A, neither can any A be B. For if some A (say C) were B, it would
not be true that no B is A; for C is a B. But if every B is A then
some A is B. For if no A were B, then no B could be A. But we assumed
that every B is A. Similarly too, if the premiss is particular. For
if some B is A, then some of the As must be B. For if none were, then
no B would be A. But if some B is not A, there is no necessity that
some of the As should not be B; e.g. let B stand for animal and A
for man. Not every animal is a man; but every man is an animal.
Part 3
The same manner of conversion will hold good also in respect of necessary
premisses. The universal negative converts universally; each of the
affirmatives converts into a particular. If it is necessary that no
B is A, it is necessary also that no A is B. For if it is possible
that some A is B, it would be possible also that some B is A. If all
or some B is A of necessity, it is necessary also that some A is B:
for if there were no necessity, neither would some of the Bs be A
necessarily. But the particular negative does not convert, for the
same reason which we have already stated.
In respect of possible premisses, since possibility is used in several
senses (for we say that what is necessary and what is not necessary
and what is potential is possible), affirmative statements will all
convert in a manner similar to those described. For if it is possible
that all or some B is A, it will be possible that some A is B. For
if that were not possible, then no B could possibly be A. This has
been already proved. But in negative statements the case is different.
Whatever is said to be possible, either because B necessarily is A,
or because B is not necessarily A, admits of conversion like other
negative statements, e.g. if one should say, it is possible that man
is not horse, or that no garment is white. For in the former case
the one term necessarily does not belong to the other; in the latter
there is no necessity that it should: and the premiss converts like
other negative statements. For if it is possible for no man to be
a horse, it is also admissible for no horse to be a man; and if it
is admissible for no garment to be white, it is also admissible for
nothing white to be a garment. For if any white thing must be a garment,
then some garment will necessarily be white. This has been already
proved. The particular negative also must be treated like those dealt
with above. But if anything is said to be possible because it is the
general rule and natural (and it is in this way we define the possible),
the negative premisses can no longer be converted like the simple
negatives; the universal negative premiss does not convert, and the
particular does. This will be plain when we speak about the possible.
At present we may take this much as clear in addition to what has
been said: the statement that it is possible that no B is A or some
B is not A is affirmative in form: for the expression 'is possible'
ranks along with 'is', and 'is' makes an affirmation always and in
every case, whatever the terms to which it is added, in predication,
e.g. 'it is not-good' or 'it is not-white' or in a word 'it is not-this'.
But this also will be proved in the sequel. In conversion these premisses
will behave like the other affirmative propositions.
Part 4
After these distinctions we now state by what means, when, and how
every syllogism is produced; subsequently we must speak of demonstration.
Syllogism should be discussed before demonstration because syllogism
is the general: the demonstration is a sort of syllogism, but not
every syllogism is a demonstration.
Whenever three terms are so related to one another that the last is
contained in the middle as in a whole, and the middle is either contained
in, or excluded from, the first as in or from a whole, the extremes
must be related by a perfect syllogism. I call that term middle which
is itself contained in another and contains another in itself: in
position also this comes in the middle. By extremes I mean both that
term which is itself contained in another and that in which another
is contained. If A is predicated of all B, and B of all C, A must
be predicated of all C: we have already explained what we mean by
'predicated of all'. Similarly also, if A is predicated of no B, and
B of all C, it is necessary that no C will be A.
But if the first term belongs to all the middle, but the middle to
none of the last term, there will be no syllogism in respect of the
extremes; for nothing necessary follows from the terms being so related;
for it is possible that the first should belong either to all or to
none of the last, so that neither a particular nor a universal conclusion
is necessary. But if there is no necessary consequence, there cannot
be a syllogism by means of these premisses. As an example of a universal
affirmative relation between the extremes we may take the terms animal,
man, horse; of a universal negative relation, the terms animal, man,
stone. Nor again can syllogism be formed when neither the first term
belongs to any of the middle, nor the middle to any of the last. As
an example of a positive relation between the extremes take the terms
science, line, medicine: of a negative relation science, line, unit.
If then the terms are universally related, it is clear in this figure
when a syllogism will be possible and when not, and that if a syllogism
is possible the terms must be related as described, and if they are
so related there will be a syllogism.
But if one term is related universally, the other in part only, to
its subject, there must be a perfect syllogism whenever universality
is posited with reference to the major term either affirmatively or
negatively, and particularity with reference to the minor term affirmatively:
but whenever the universality is posited in relation to the minor
term, or the terms are related in any other way, a syllogism is impossible.
I call that term the major in which the middle is contained and that
term the minor which comes under the middle. Let all B be A and some
C be B. Then if 'predicated of all' means what was said above, it
is necessary that some C is A. And if no B is A but some C is B, it
is necessary that some C is not A. The meaning of 'predicated of none'
has also been defined. So there will be a perfect syllogism. This
holds good also if the premiss BC should be indefinite, provided that
it is affirmative: for we shall have the same syllogism whether the
premiss is indefinite or particular.
But if the universality is posited with respect to the minor term
either affirmatively or negatively, a syllogism will not be 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.
Part 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.
It is possible to prove these results also by reductio ad 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.
Part 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.
Part 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 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.
Part 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.
Part 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.
Part 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.
Part 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
terms. 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.
Part 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 or simple premiss is assumed.
Part 13
Perhaps enough has been said about the proof of necessity, how it
comes about and how it differs from the proof of a simple statement.
We proceed to discuss that which is possible, when and how and by
what means it can be proved. I use the terms 'to be possible' and
'the possible' of that which is not necessary but, being assumed,
results in nothing impossible. We say indeed ambiguously of the necessary
that it is possible. But that my definition of the possible is correct
is clear from the phrases by which we deny or on the contrary affirm
possibility. For the expressions 'it is not possible to belong', 'it
is impossible to belong', and 'it is necessary not to belong' are
either identical or follow from one another; consequently their opposites
also, 'it is possible to belong', 'it is not impossible to belong',
and 'it is not necessary not to belong', will either be identical
or follow from one another. For of everything the affirmation or the
denial holds good. That which is possible then will be not necessary
and that which is not necessary will be possible. It results that
all premisses in the mode of possibility are convertible into one
another. I mean not that the affirmative are convertible into the
negative, but that those which are affirmative in form admit of conversion
by opposition, e.g. 'it is possible to belong' may be converted into
'it is possible not to belong', and 'it is possible for A to belong
to all B' into 'it is possible for A to belong to no B' or 'not to
all B', and 'it is possible for A to belong to some B' into 'it is
possible for A not to belong to some B'. And similarly the other propositions
in this mode can be converted. For since that which is possible is
not necessary, and that which is not necessary may possibly not belong,
it is clear that if it is possible that A should belong to B, it is
possible also that it should not belong to B: and if it is possible
that it should belong to all, it is also possible that it should not
belong to all. The same holds good in the case of particular affirmations:
for the proof is identical. And such premisses are affirmative and
not negative; for 'to be possible' is in the same rank as 'to be',
as was said above.
Having made these distinctions we next point out that the expression
'to be possible' is used in two ways. In one it means to happen generally
and fall short of necessity, e.g. man's turning grey or growing or
decaying, or generally what naturally belongs to a thing (for this
has not its necessity unbroken, since man's existence is not continuous
for ever, although if a man does exist, it comes about either necessarily
or generally). In another sense the expression means the indefinite,
which can be both thus and not thus, e.g. an animal's walking or an
earthquake's taking place while it is walking, or generally what happens
by chance: for none of these inclines by nature in the one way more
than in the opposite.
That which is possible in each of its two senses is convertible into
its opposite, not however in the same way: but what is natural is
convertible because it does not necessarily belong (for in this sense
it is possible that a man should not grow grey) and what is indefinite
is convertible because it inclines this way no more than that. Science
and demonstrative syllogism are not concerned with things which are
indefinite, because the middle term is uncertain; but they are concerned
with things that are natural, and as a rule arguments and inquiries
are made about things which are possible in this sense. Syllogisms
indeed can be made about the former, but it is unusual at any rate
to inquire about them.
These matters will be treated more definitely in the sequel; our business
at present is to state the moods and nature of the syllogism made
from possible premisses. The expression 'it is possible for this to
belong to that' may be understood in two senses: 'that' may mean either
that to which 'that' belongs or that to which it may belong; for the
expression 'A is possible of the subject of B' means that it is possible
either of that of which B is stated or of that of which B may possibly
be stated. It makes no difference whether we say, A is possible of
the subject of B, or all B admits of A. It is clear then that the
expression 'A may possibly belong to all B' might be used in two senses.
First then we must state the nature and characteristics of the syllogism
which arises if B is possible of the subject of C, and A is possible
of the subject of B. For thus both premisses are assumed in the mode
of possibility; but whenever A is possible of that of which B is true,
one premiss is a simple assertion, the other a problematic. Consequently
we must start from premisses which are similar in form, as in the
other cases.
Part 14
Whenever A may possibly belong to all B, and B to all C, there will
be a perfect syllogism to prove that A may possibly belong to all
C. This is clear from the definition: for it was in this way that
we explained 'to be possible for one term to belong to all of another'.
Similarly if it is possible for A to belong no B, and for B to belong
to all C, then it is possible for A to belong to no C. For the statement
that it is possible for A not to belong to that of which B may be
true means (as we saw) that none of those things which can possibly
fall under the term B is left out of account. But whenever A may belong
to all B, and B may belong to no C, then indeed no syllogism results
from the premisses assumed, but if the premiss BC is converted after
the manner of problematic propositions, the same syllogism results
as before. For since it is possible that B should belong to no C,
it is possible also that it should belong to all C. This has been
stated above. Consequently if B is possible for all C, and A is possible
for all B, the same syllogism again results. Similarly if in both
the premisses the negative is joined with 'it is possible': e.g. if
A may belong to none of the Bs, and B to none of the Cs. No syllogism
results from the assumed premisses, but if they are converted we shall
have the same syllogism as before. It is clear then that if the minor
premiss is negative, or if both premisses are negative, either no
syllogism results, or if one it is not perfect. For the necessity
results from the conversion.
But if one of the premisses is universal, the other particular, when
the major premiss is universal there will be a perfect syllogism.
For if A is possible for all B, and B for some C, then A is possible
for some C. This is clear from the definition of being possible. Again
if A may belong to no B, and B may belong to some of the Cs, it is
necessary that A may possibly not belong to some of the Cs. The proof
is the same as above. But if the particular premiss is negative, and
the universal is affirmative, the major still being universal and
the minor particular, e.g. A is possible for all B, B may possibly
not belong to some C, then a clear syllogism does not result from
the assumed premisses, but if the particular premiss is converted
and it is laid down that B possibly may belong to some C, we shall
have the same conclusion as before, as in the cases given at the beginning.
But if the major premiss is the minor universal, whether both are
affirmative, or negative, or different in quality, or if both are
indefinite or particular, in no way will a syllogism be possible.
For nothing prevents B from reaching beyond A, so that as predicates
cover unequal areas. Let C be that by which B extends beyond A. To
C it is not possible that A should belong-either to all or to none
or to some or not to some, since premisses in the mode of possibility
are convertible and it is possible for B to belong to more things
than A can. Further, this is obvious if we take terms; for if the
premisses are as assumed, the major term is both possible for none
of the minor and must belong to all of it. Take as terms common to
all the cases under consideration 'animal'-'white'-'man', where the
major belongs necessarily to the minor; 'animal'-'white'-'garment',
where it is not possible that the major should belong to the minor.
It is clear then that if the terms are related in this manner, no
syllogism results. For every syllogism proves that something belongs
either simply or necessarily or possibly. It is clear that there is
no proof of the first or of the second. For the affirmative is destroyed
by the negative, and the negative by the affirmative. There remains
the proof of possibility. But this is impossible. For it has been
proved that if the terms are related in this manner it is both necessary
that the major should belong to all the minor and not possible that
it should belong to any. Consequently there cannot be a syllogism
to prove the possibility; for the necessary (as we stated) is not
possible.
It is clear that if the terms are universal in possible premisses
a syllogism always results in the first figure, whether they are affirmative
or negative, only a perfect syllogism results in the first case, an
imperfect in the second. But possibility must be understood according
to the definition laid down, not as covering necessity. This is sometimes
forgotten.
Part 15
If one premiss is a simple proposition, the other a problematic, whenever
the major premiss indicates possibility all the syllogisms will be
perfect and establish possibility in the sense defined; but whenever
the minor premiss indicates possibility all the syllogisms will be
imperfect, and those which are negative will establish not possibility
according to the definition, but that the major does not necessarily
belong to any, or to all, of the minor. For if this is so, we say
it is possible that it should belong to none or not to all. Let A
be possible for all B, and let B belong to all C. Since C falls under
B, and A is possible for all B, clearly it is possible for all C also.
So a perfect syllogism results. Likewise if the premiss AB is negative,
and the premiss BC is affirmative, the former stating possible, the
latter simple attribution, a perfect syllogism results proving that
A possibly belongs to no C.
It is clear that perfect syllogisms result if the minor premiss states
simple belonging: but that syllogisms will result if the modality
of the premisses is reversed, must be proved per impossibile. At the
same time it will be evident that they are imperfect: for the proof
proceeds not from the premisses assumed. First we must state that
if B's being follows necessarily from A's being, B's possibility will
follow necessarily from A's possibility. Suppose, the terms being
so related, that A is possible, and B is impossible. If then that
which is possible, when it is possible for it to be, might happen,
and if that which is impossible, when it is impossible, could not
happen, and if at the same time A is possible and B impossible, it
would be possible for A to happen without B, and if to happen, then
to be. For that which has happened, when it has happened, is. But
we must take the impossible and the possible not only in the sphere
of becoming, but also in the spheres of truth and predicability, and
the various other spheres in which we speak of the possible: for it
will be alike in all. Further we must understand the statement that
B's being depends on A's being, not as meaning that if some single
thing A is, B will be: for nothing follows of necessity from the being
of some one thing, but from two at least, i.e. when the premisses
are related in the manner stated to be that of the syllogism. For
if C is predicated of D, and D of F, then C is necessarily predicated
of F. And if each is possible, the conclusion also is possible. If
then, for example, one should indicate the premisses by A, and the
conclusion by B, it would not only result that if A is necessary B
is necessary, but also that if A is possible, B is possible.
Since this is proved it is evident that if a false and not impossible
assumption is made, the consequence of the assumption will also be
false and not impossible: e.g. if A is false, but not impossible,
and if B is the consequence of A, B also will be false but not impossible.
For since it has been proved that if B's being is the consequence
of A's being, then B's possibility will follow from A's possibility
(and A is assumed to be possible), consequently B will be possible:
for if it were impossible, the same thing would at the same time be
possible and impossible.
Since we have defined these points, let A belong to all B, and B be
possible for all C: it is necessary then that should be a possible
attribute for all C. Suppose that it is not possible, but assume that
B belongs to all C: this is false but not impossible. If then A is
not possible for C but B belongs to all C, then A is not possible
for all B: for a syllogism is formed in the third degree. But it was
assumed that A is a possible attribute for all B. It is necessary
then that A is possible for all C. For though the assumption we made
is false and not impossible, the conclusion is impossible. It is possible
also in the first figure to bring about the impossibility, by assuming
that B belongs to C. For if B belongs to all C, and A is possible
for all B, then A would be possible for all C. But the assumption
was made that A is not possible for all C.
We must understand 'that which belongs to all' with no limitation
in respect of time, e.g. to the present or to a particular period,
but simply without qualification. For it is by the help of such premisses
that we make syllogisms, since if the premiss is understood with reference
to the present moment, there cannot be a syllogism. For nothing perhaps
prevents 'man' belonging at a particular time to everything that is
moving, i.e. if nothing else were moving: but 'moving' is possible
for every horse; yet 'man' is possible for no horse. Further let the
major term be 'animal', the middle 'moving', the the minor 'man'.
The premisses then will be as before, but the conclusion necessary,
not possible. For man is necessarily animal. It is clear then that
the universal must be understood simply, without limitation in respect
of time.
Again let the premiss AB be universal and negative, and assume that
A belongs to no B, but B possibly belongs to all C. These propositions
being laid down, it is necessary that A possibly belongs to no C.
Suppose that it cannot belong, and that B belongs to C, as above.
It is necessary then that A belongs to some B: for we have a syllogism
in the third figure: but this is impossible. Thus it will be possible
for A to belong to no C; for if at is supposed false, the consequence
is an impossible one. This syllogism then does not establish that
which is possible according to the definition, but that which does
not necessarily belong to any part of the subject (for this is the
contradictory of the assumption which was made: for it was supposed
that A necessarily belongs to some C, but the syllogism per impossibile
establishes the contradictory which is opposed to this). Further,
it is clear also from an example that the conclusion will not establish
possibility. Let A be 'raven', B 'intelligent', and C 'man'. A then
belongs to no B: for no intelligent thing is a raven. But B is possible
for all C: for every man may possibly be intelligent. But A necessarily
belongs to no C: so the conclusion does not establish possibility.
But neither is it always necessary. Let A be 'moving', B 'science',
C 'man'. A then will belong to no B; but B is possible for all C.
And the conclusion will not be necessary. For it is not necessary
that no man should move; rather it is not necessary that any man should
move. Clearly then the conclusion establishes that one term does not
necessarily belong to any instance of another term. But we must take
our terms better.
If the minor premiss is negative and indicates possibility, from the
actual premisses taken there can be no syllogism, but if the problematic
premiss is converted, a syllogism will be possible, as before. Let
A belong to all B, and let B possibly belong to no C. If the terms
are arranged thus, nothing necessarily follows: but if the proposition
BC is converted and it is assumed that B is possible for all C, a
syllogism results as before: for the terms are in the same relative
positions. Likewise if both the relations are negative, if the major
premiss states that A does not belong to B, and the minor premiss
indicates that B may possibly belong to no C. Through the premisses
actually taken nothing necessary results in any way; but if the problematic
premiss is converted, we shall have a syllogism. Suppose that A belongs
to no B, and B may possibly belong to no C. Through these comes nothing
necessary. But if B is assumed to be possible for all C (and this
is true) and if the premiss AB remains as before, we shall again have
the same syllogism. But if it be assumed that B does not belong to
any C, instead of possibly not belonging, there cannot be a syllogism
anyhow, whether the premiss AB is negative or affirmative. As common
instances of a necessary and positive relation we may take the terms
white-animal-snow: of a necessary and negative relation, white-animal-pitch.
Clearly then if the terms are universal, and one of the premisses
is assertoric, the other problematic, whenever the minor premiss is
problematic a syllogism always results, only sometimes it results
from the premisses that are taken, sometimes it requires the conversion
of one premiss. We have stated when each of these happens and the
reason why. But if one of the relations is universal, the other particular,
then whenever the major premiss is universal and problematic, whether
affirmative or negative, and the particular is affirmative and assertoric,
there will be a perfect syllogism, just as when the terms are universal.
The demonstration is the same as before. But whenever the major premiss
is universal, but assertoric, not problematic, and the minor is particular
and problematic, whether both premisses are negative or affirmative,
or one is negative, the other affirmative, in all cases there will
be an imperfect syllogism. Only some of them will be proved per impossibile,
others by the conversion of the problematic premiss, as has been shown
above. And a syllogism will be possible by means of conversion when
the major premiss is universal and assertoric, whether positive or
negative, and the minor particular, negative, and problematic, e.g.
if A belongs to all B or to no B, and B may possibly not belong to
some C. For if the premiss BC is converted in respect of possibility,
a syllogism results. But whenever the particular premiss is assertoric
and negative, there cannot be a syllogism. As instances of the positive
relation we may take the terms white-animal-snow; of the negative,
white-animal-pitch. For the demonstration must be made through the
indefinite nature of the particular premiss. But if the minor premiss
is universal, and the major particular, whether either premiss is
negative or affirmative, problematic or assertoric, nohow is a syllogism
possible. Nor is a syllogism possible when the premisses are particular
or indefinite, whether problematic or assertoric, or the one problematic,
the other assertoric. The demonstration is the same as above. As instances
of the necessary and positive relation we may take the terms animal-white-man;
of the necessary and negative relation, animal-white-garment. It is
evident then that if the major premiss is universal, a syllogism always
results, but if the minor is universal nothing at all can ever be
proved.
Part 16
Whenever one premiss is necessary, the other problematic, there will
be a syllogism when the terms are related as before; and a perfect
syllogism when the minor premiss is necessary. If the premisses are
affirmative the conclusion will be problematic, not assertoric, whether
the premisses are universal or not: but if one is affirmative, the
other negative, when the affirmative is necessary the conclusion will
be problematic, not negative assertoric; but when the negative is
necessary the conclusion will be problematic negative, and assertoric
negative, whether the premisses are universal or not. Possibility
in the conclusion must be understood in the same manner as before.
There cannot be an inference to the necessary negative proposition:
for 'not necessarily to belong' is different from 'necessarily not
to belong'.
If the premisses are affirmative, clearly the conclusion which follows
is not necessary. Suppose A necessarily belongs to all B, and let
B be possible for all C. We shall have an imperfect syllogism to prove
that A may belong to all C. That it is imperfect is clear from the
proof: for it will be proved in the same manner as above. Again, let
A be possible for all B, and let B necessarily belong to all C. We
shall then have a syllogism to prove that A may belong to all C, not
that A does belong to all C: and it is perfect, not imperfect: for
it is completed directly through the original premisses.
But if the premisses are not similar in quality, suppose first that
the negative premiss is necessary, and let necessarily A not be possible
for any B, but let B be possible for all C. It is necessary then that
A belongs to no C. For suppose A to belong to all C or to some C.
Now we assumed that A is not possible for any B. Since then the negative
proposition is convertible, B is not possible for any A. But A is
supposed to belong to all C or to some C. Consequently B will not
be possible for any C or for all C. But it was originally laid down
that B is possible for all C. And it is clear that the possibility
of belonging can be inferred, since the fact of not belonging is inferred.
Again, let the affirmative premiss be necessary, and let A possibly
not belong to any B, and let B necessarily belong to all C. The syllogism
will be perfect, but it will establish a problematic negative, not
an assertoric negative. For the major premiss was problematic, and
further it is not possible to prove the assertoric conclusion per
impossibile. For if it were supposed that A belongs to some C, and
it is laid down that A possibly does not belong to any B, no impossible
relation between B and C follows from these premisses. But if the
minor premiss is negative, when it is problematic a syllogism is possible
by conversion, as above; but when it is necessary no syllogism can
be formed. Nor again when both premisses are negative, and the minor
is necessary. The same terms as before serve both for the positive
relation-white-animal-snow, and for the negative relation-white-animal-pitch.
The same relation will obtain in particular syllogisms. Whenever the
negative proposition is necessary, the conclusion will be negative
assertoric: e.g. if it is not possible that A should belong to any
B, but B may belong to some of the Cs, it is necessary that A should
not belong to some of the Cs. For if A belongs to all C, but cannot
belong to any B, neither can B belong to any A. So if A belongs to
all C, to none of the Cs can B belong. But it was laid down that B
may belong to some C. But when the particular affirmative in the negative
syllogism, e.g. BC the minor premiss, or the universal proposition
in the affirmative syllogism, e.g. AB the major premiss, is necessary,
there will not be an assertoric conclusion. The demonstration is the
same as before. But if the minor premiss is universal, and problematic,
whether affirmative or negative, and the major premiss is particular
and necessary, there cannot be a syllogism. Premisses of this kind
are possible both where the relation is positive and necessary, e.g.
animal-white-man, and where it is necessary and negative, e.g. animal-white-garment.
But when the universal is necessary, the particular problematic, if
the universal is negative we may take the terms animal-white-raven
to illustrate the positive relation, or animal-white-pitch to illustrate
the negative; and if the universal is affirmative we may take the
terms animal-white-swan to illustrate the positive relation, and animal-white-snow
to illustrate the negative and necessary relation. Nor again is a
syllogism possible when the premisses are indefinite, or both particular.
Terms applicable in either case to illustrate the positive relation
are animal-white-man: to illustrate the negative, animal-white-inanimate.
For the relation of animal to some white, and of white to some inanimate,
is both necessary and positive and necessary and negative. Similarly
if the relation is problematic: so the terms may be used for all cases.
Clearly then from what has been said a syllogism results or not from
similar relations of the terms whether we are dealing with simple
existence or necessity, with this exception, that if the negative
premiss is assertoric the conclusion is problematic, but if the negative
premiss is necessary the conclusion is both problematic and negative
assertoric. [It is clear also that all the syllogisms are imperfect
and are perfected by means of the figures above mentioned.]
Part 17
In the second figure whenever both premisses are problematic, no syllogism
is possible, whether the premisses are affirmative or negative, universal
or particular. But when one premiss is assertoric, the other problematic,
if the affirmative is assertoric no syllogism is possible, but if
the universal negative is assertoric a conclusion can always be drawn.
Similarly when one premiss is necessary, the other problematic. Here
also we must understand the term 'possible' in the conclusion, in
the same sense as before.
First we must point out that the negative problematic proposition
is not convertible, e.g. if A may belong to no B, it does not follow
that B may belong to no A. For suppose it to follow and assume that
B may belong to no A. Since then problematic affirmations are convertible
with negations, whether they are contraries or contradictories, and
since B may belong to no A, it is clear that B may belong to all A.
But this is false: for if all this can be that, it does not follow
that all that can be this: consequently the negative proposition is
not convertible. Further, these propositions are not incompatible,
'A may belong to no B', 'B necessarily does not belong to some of
the As'; e.g. it is possible that no man should be white (for it is
also possible that every man should be white), but it is not true
to say that it is possible that no white thing should be a man: for
many white things are necessarily not men, and the necessary (as we
saw) other than the possible.
Moreover it is not possible to prove the convertibility of these propositions
by a reductio ad absurdum, i.e. by claiming assent to the following
argument: 'since it is false that B may belong to no A, it is true
that it cannot belong to no A, for the one statement is the contradictory
of the other. But if this is so, it is true that B necessarily belongs
to some of the As: consequently A necessarily belongs to some of the
Bs. But this is impossible.' The argument cannot be admitted, for
it does not follow that some A is necessarily B, if it is not possible
that no A should be B. For the latter expression is used in two senses,
one if A some is necessarily B, another if some A is necessarily not
B. For it is not true to say that that which necessarily does not
belong to some of the As may possibly not belong to any A, just as
it is not true to say that what necessarily belongs to some A may
possibly belong to all A. If any one then should claim that because
it is not possible for C to belong to all D, it necessarily does not
belong to some D, he would make a false assumption: for it does belong
to all D, but because in some cases it belongs necessarily, therefore
we say that it is not possible for it to belong to all. Hence both
the propositions 'A necessarily belongs to some B' and 'A necessarily
does not belong to some B' are opposed to the proposition 'A belongs
to all B'. Similarly also they are opposed to the proposition 'A may
belong to no B'. It is clear then that in relation to what is possible
and not possible, in the sense originally defined, we must assume,
not that A necessarily belongs to some B, but that A necessarily does
not belong to some B. But if this is assumed, no absurdity results:
consequently no syllogism. It is clear from what has been said that
the negative proposition is not convertible.
This being proved, suppose it possible that A may belong to no B and
to all C. By means of conversion no syllogism will result: for the
major premiss, as has been said, is not convertible. Nor can a proof
be obtained by a reductio ad absurdum: for if it is assumed that B
can belong to all C, no false consequence results: for A may belong
both to all C and to no C. In general, if there is a syllogism, it
is clear that its conclusion will be problematic because neither of
the premisses is assertoric; and this must be either affirmative or
negative. But neither is possible. Suppose the conclusion is affirmative:
it will be proved by an example that the predicate cannot belong to
the subject. Suppose the conclusion is negative: it will be proved
that it is not problematic but necessary. Let A be white, B man, C
horse. It is possible then for A to belong to all of the one and to
none of the other. But it is not possible for B to belong nor not
to belong to C. That it is not possible for it to belong, is clear.
For no horse is a man. Neither is it possible for it not to belong.
For it is necessary that no horse should be a man, but the necessary
we found to be different from the possible. No syllogism then results.
A similar proof can be given if the major premiss is negative, the
minor affirmative, or if both are affirmative or negative. The demonstration
can be made by means of the same terms. And whenever one premiss is
universal, the other particular, or both are particular or indefinite,
or in whatever other way the premisses can be altered, the proof will
always proceed through the same terms. Clearly then, if both the premisses
are problematic, no syllogism results.
Part 18
But if one premiss is assertoric, the other problematic, if the affirmative
is assertoric and the negative problematic no syllogism will be possible,
whether the premisses are universal or particular. The proof is the
same as above, and by means of the same terms. But when the affirmative
premiss is problematic, and the negative assertoric, we shall have
a syllogism. Suppose A belongs to no B, but can belong to all C. If
the negative proposition is converted, B will belong to no A. But
ex hypothesi can belong to all C: so a syllogism is made, proving
by means of the first figure that B may belong to no C. Similarly
also if the minor premiss is negative. But if both premisses are negative,
one being assertoric, the other problematic, nothing follows necessarily
from these premisses as they stand, but if the problematic premiss
is converted into its complementary affirmative a syllogism is formed
to prove that B may belong to no C, as before: for we shall again
have the first figure. But if both premisses are affirmative, no syllogism
will be possible. This arrangement of terms is possible both when
the relation is positive, e.g. health, animal, man, and when it is
negative, e.g. health, horse, man.
The same will hold good if the syllogisms are particular. Whenever
the affirmative proposition is assertoric, whether universal or particular,
no syllogism is possible (this is proved similarly and by the same
examples as above), but when the negative proposition is assertoric,
a conclusion can be drawn by means of conversion, as before. Again
if both the relations are negative, and the assertoric proposition
is universal, although no conclusion follows from the actual premisses,
a syllogism can be obtained by converting the problematic premiss
into its complementary affirmative as before. But if the negative
proposition is assertoric, but particular, no 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.
Part 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.
Part 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. Similarly 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.
Part 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.
Part 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 universally
to the middle, the other in part. If both premisses are 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.
Part 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
pointed 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
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.
Part 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.
Part 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.
Part 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.
Part 27
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, those things which the
inferior term follows, e.g. take as subjects of the predicate 'animal'
what are really subjects of the predicate 'man'. It is necessary indeed,
if animal follows man, that it should follow all these also. But these
belong more properly to the choice of what concerns man. One must
apprehend also normal consequents and normal antecedents-, for propositions
which obtain normally are established syllogistically from premisses
which obtain normally, some if not all of them having this character
of normality. For the conclusion of each syllogism resembles its principles.
We must not however choose attributes which are consequent upon all
the terms: for no syllogism can be made out of such premisses. The
reason why this is so will be clear in the sequel.
Part 28
If men wish to establish something about some whole, they must look
to the subjects of that which is being established (the subjects of
which it happens to be asserted), and the attributes which follow
that of which it is to be predicated. For if any of these subjects
is the same as any of these attributes, the attribute originally in
question must belong to the subject originally in question. But if
the purpose is to establish not a universal but a particular proposition,
they must look for the terms of which the terms in question are predicable:
for if any of these are identical, the attribute in question must
belong to some of the subject in question. Whenever the one term has
to belong to none of the other, one must look to the consequents of
the subject, and to those attributes which cannot possibly be present
in the predicate in question: or conversely to the attributes which
cannot possibly be present in the subject, and to the consequents
of the predicate. If any members of these groups are identical, one
of the terms in question cannot possibly belong to any of the other.
For sometimes a syllogism in the first figure results, sometimes a
syllogism in the second. But if the object is to establish a particular
negative proposition, we must find antecedents of the subject in question
and attributes which cannot possibly belong to the predicate in question.
If any members of these two groups are identical, it follows that
one of the terms in question does not belong to some of the other.
Perhaps each of these statements will become clearer in the following
way. Suppose the consequents of A are designated by B, the antecedents
of A by C, attributes which cannot possibly belong to A by D. Suppose
again that the attributes of E are designated by F, the antecedents
of E by G, and attributes which cannot belong to E by H. If then one
of the Cs should be identical with one of the Fs, A must belong to
all E: for F belongs to all E, and A to all C, consequently A belongs
to all E. If C and G are identical, A must belong to some of the Es:
for A follows C, and E follows all G. If F and D are identical, A
will belong to none of the Es by a prosyllogism: for since the negative
proposition is convertible, and F is identical with D, A will belong
to none of the Fs, but F belongs to all E. Again, if B and H are identical,
A will belong to none of the Es: for B will belong to all A, but to
no E: for it was assumed to be identical with H, and H belonged to
none of the Es. If D and G are identical, A will not belong to some
of the Es: for it will not belong to G, because it does not belong
to D: but G falls under E: consequently A will not belong to some
of the Es. If B is identical with G, there will be a converted syllogism:
for E will belong to all A since B belongs to A and E to B (for B
was found to be identical with G): but that A should belong to all
E is not necessary, but it must belong to some E because it is possible
to convert the universal statement into a particular.
It is clear then that in every proposition which requires proof we
must look to the aforesaid relations of the subject and predicate
in question: for all syllogisms proceed through these. But if we are
seeking consequents and antecedents we must look for those which are
primary and most universal, e.g. in reference to E we must look to
Kf rather than to F alone, and in reference to A we must look to KC
rather than to C alone. For if A belongs to KF, it belongs both to
F and to E: but if it does not follow KF, it may yet follow F. Similarly
we must consider the antecedents of A itself: for if a term follows
the primary antecedents, it will follow those also which are subordinate,
but if it does not follow the former, it may yet follow the latter.
It is clear too that the inquiry proceeds through the three terms
and the two premisses, and that all the syllogisms proceed through
the aforesaid figures. For it is proved that A belongs to all E, whenever
an identical term is found among the Cs and Fs. This will be the middle
term; A and E will be the extremes. So the first figure is formed.
And A will belong to some E, whenever C and G are apprehended to be
the same. This is the last figure: for G becomes the middle term.
And A will belong to no E, when D and F are identical. Thus we have
both the first figure and the middle figure; the first, because A
belongs to no F, since the negative statement is convertible, and
F belongs to all E: the middle figure because D belongs to no A, and
to all E. And A will not belong to some E, whenever D and G are identical.
This is the last figure: for A will belong to no G, and E will belong
to all G. Clearly then all syllogisms proceed through the aforesaid
figures, and we must not select consequents of all the terms, because
no syllogism is produced from them. For (as we saw) it is not possible
at all to establish a proposition from consequents, and it is not
possible to refute by means of a consequent of both the terms in question:
for the middle term must belong to the one, and not belong to the
other.
It is clear too that other methods of inquiry by selection of middle
terms are useless to produce a syllogism, e.g. if the consequents
of the terms in question are identical, or if the antecedents of A
are identical with those attributes which cannot possibly belong to
E, or if those attributes are identical which cannot belong to either
term: for no syllogism is produced by means of these. For if the consequents
are identical, e.g. B and F, we have the middle figure with both premisses
affirmative: if the antecedents of A are identical with attributes
which cannot belong to E, e.g. C with H, we have the first figure
with its minor premiss negative. If attributes which cannot belong
to either term are identical, e.g. C and H, both premisses are negative,
either in the first or in the middle figure. But no syllogism is possible
in this way.
It is evident too that we must find out which terms in this inquiry
are identical, not which are different or contrary, first because
the object of our investigation is the middle term, and the middle
term must be not diverse but identical. Secondly, wherever it happens
that a syllogism results from taking contraries or terms which cannot
belong to the same thing, all arguments can be reduced to the aforesaid
moods, e.g. if B and F are contraries or cannot belong to the same
thing. For if these are taken, a syllogism will be formed to prove
that A belongs to none of the Es, not however from the premisses taken
but in the aforesaid mood. For B will belong to all A and to no E.
Consequently B must be identical with one of the Hs. Again, if B and
G cannot belong to the same thing, it follows that A will not belong
to some of the Es: for then too we shall have the middle figure: for
B will belong to all A and to no G. Consequently B must be identical
with some of the Hs. For the fact that B and G cannot belong to the
same thing differs in no way from the fact that B is identical with
some of the Hs: for that includes everything which cannot belong to
E.
It is clear then that from the inquiries taken by themselves no syllogism
results; but if B and F are contraries B must be identical with one
of the Hs, and the syllogism results through these terms. It turns
out then that those who inquire in this manner are looking gratuitously
for some other way than the necessary way because they have failed
to observe the identity of the Bs with the Hs.
Part 29
Syllogisms which lead to impossible conclusions are similar to ostensive
syllogisms; they also are formed by means of the consequents and antecedents
of the terms in question. In both cases the same inquiry is involved.
For what is proved ostensively may also be concluded syllogistically
per impossibile by means of the same terms; and what is proved per
impossibile may also be proved ostensively, e.g. that A belongs to
none of the Es. For suppose A to belong to some E: then since B belongs
to all A and A to some of the Es, B will belong to some of the Es:
but it was assumed that it belongs to none. Again we may prove that
A belongs to some E: for if A belonged to none of the Es, and E belongs
to all G, A will belong to none of the Gs: but it was assumed to belong
to all. Similarly with the other propositions requiring proof. The
proof per impossibile will always and in all cases be from the consequents
and antecedents of the terms in question. Whatever the problem the
same inquiry is necessary whether one wishes to use an ostensive syllogism
or a reduction to impossibility. For both the demonstrations start
from the same terms, e.g. suppose it has been proved that A belongs
to no E, because it turns out that otherwise B belongs to some of
the Es and this is impossible-if now it is assumed that B belongs
to no E and to all A, it is clear that A will belong to no E. Again
if it has been proved by an ostensive syllogism that A belongs to
no E, assume that A belongs to some E and it will be proved per impossibile
to belong to no E. Similarly with the rest. In all cases it is necessary
to find some common term other than the subjects of inquiry, to which
the syllogism establishing the false conclusion may relate, so that
if this premiss is converted, and the other remains as it is, the
syllogism will be ostensive by means of the same terms. For the ostensive
syllogism differs from the reductio ad impossibile in this: in the
ostensive syllogism both remisses are laid down in accordance with
the truth, in the reductio ad impossibile one of the premisses is
assumed falsely.
These points will be made clearer by the sequel, when we discuss the
reduction to impossibility: at present this much must be clear, that
we must look to terms of the kinds mentioned whether we wish to use
an ostensive syllogism or a reduction to impossibility. In the other
hypothetical syllogisms, I mean those which proceed by substitution,
or by positing a certain quality, the inquiry will be directed to
the terms of the problem to be proved-not the terms of the original
problem, but the new terms introduced; and the method of the inquiry
will be the same as before. But we must consider and determine in
how many ways hypothetical syllogisms are possible.
Each of the problems then can be proved in the manner described; but
it is possible to establish some of them syllogistically in another
way, e.g. universal problems by the inquiry which leads up to a particular
conclusion, with the addition of an hypothesis. For if the Cs and
the Gs should be identical, but E should be assumed to belong to the
Gs only, then A would belong to every E: and again if the Ds and the
Gs should be identical, but E should be predicated of the Gs only,
it follows that A will belong to none of the Es. Clearly then we must
consider the matter in this way also. The method is the same whether
the relation is necessary or possible. For the inquiry will be the
same, and the syllogism will proceed through terms arranged in the
same order whether a possible or a pure proposition is proved. We
must find in the case of possible relations, as well as terms that
belong, terms which can belong though they actually do not: for we
have proved that the syllogism which establishes a possible relation
proceeds through these terms as well. Similarly also with the other
modes of predication.
It is clear then from what has been said not only that all syllogisms
can be formed in this way, but also that they cannot be formed in
any other. For every syllogism has been proved to be formed through
one of the aforementioned figures, and these cannot be composed through
other terms than the consequents and antecedents of the terms in question:
for from these we obtain the premisses and find the middle term. Consequently
a syllogism cannot be formed by means of other terms.
Part 30
The method is the same in all cases, in philosophy, in any art or
study. We must look for the attributes and the subjects of both our
terms, and we must supply ourselves with as many of these as possible,
and consider them by means of the three terms, refuting statements
in one way, confirming them in another, in the pursuit of truth starting
from premisses in which the arrangement of the terms is in accordance
with truth, while if we look for dialectical syllogisms we must start
from probable premisses. The principles of syllogisms have been stated
in general terms, both how they are characterized and how we must
hunt for them, so as not to look to everything that is said about
the terms of the problem or to the same points whether we are confirming
or refuting, or again whether we are confirming of all or of some,
and whether we are refuting of all or some. we must look to fewer
points and they must be definite. We have also stated how we must
select with reference to everything that is, e.g. about good or knowledge.
But in each science the principles which are peculiar are the most
numerous. Consequently it is the business of experience to give the
principles which belong to each subject. I mean for example that astronomical
experience supplies the principles of astronomical science: for once
the phenomena were adequately apprehended, the demonstrations of astronomy
were discovered. Similarly with any other art or science. Consequently,
if the attributes of the thing are apprehended, our business will
then be to exhibit readily the demonstrations. For if none of the
true attributes of things had been omitted in the historical survey,
we should be able to discover the proof and demonstrate everything
which admitted of proof, and to make that clear, whose nature does
not admit of proof.
In general then we have explained fairly well how we must select premisses:
we have discussed the matter accurately in the treatise concerning
dialectic.
Part 31
It is easy to see that division into classes is a small part of the
method we have described: for division is, so to speak, a weak syllogism;
for what it ought to prove, it begs, and it always establishes something
more general than the attribute in question. First, this very point
had escaped all those who used the method of division; and they attempted
to persuade men that it was possible to make a demonstration of substance
and essence. Consequently they did not understand what it is possible
to prove syllogistically by division, nor did they understand that
it was possible to prove syllogistically in the manner we have described.
In demonstrations, when there is a need to prove a positive statement,
the middle term through which the syllogism is formed must always
be inferior to and not comprehend the first of the extremes. But division
has a contrary intention: for it takes the universal as middle. Let
animal be the term signified by A, mortal by B, and immortal by C,
and let man, whose definition is to be got, be signified by D. The
man who divides assumes that every animal is either mortal or immortal:
i.e. whatever is A is all either B or C. Again, always dividing, he
lays it down that man is an animal, so he assumes A of D as belonging
to it. Now the true conclusion is that every D is either B or C, consequently
man must be either mortal or immortal, but it is not necessary that
man should be a mortal animal-this is begged: and this is what ought
to have been proved syllogistically. And again, taking A as mortal
animal, B as footed, C as footless, and D as man, he assumes in the
same way that A inheres either in B or in C (for every mortal animal
is either footed or footless), and he assumes A of D (for he assumed
man, as we saw, to be a mortal animal); consequently it is necessary
that man should be either a footed or a footless animal; but it is
not necessary that man should be footed: this he assumes: and it is
just this again which he ought to have demonstrated. Always dividing
then in this way it turns out that these logicians assume as middle
the universal term, and as extremes that which ought to have been
the subject of demonstration and the differentiae. In conclusion,
they do not make it clear, and show it to be necessary, that this
is man or whatever the subject of inquiry may be: for they pursue
the other method altogether, never even suspecting the presence of
the rich supply of evidence which might be used. It is clear that
it is neither possible to refute a statement by this method of division,
nor to draw a conclusion about an accident or property of a thing,
nor about its genus, nor in cases in which it is unknown whether it
is thus or thus, e.g. whether the diagonal is incommensurate. For
if he assumes that every length is either commensurate or incommensurate,
and the diagonal is a length, he has proved that the diagonal is either
incommensurate or commensurate. But if he should assume that it is
incommensurate, he will have assumed what he ought to have proved.
He cannot then prove it: for this is his method, but proof is not
possible by this method. Let A stand for 'incommensurate or commensurate',
B for 'length', C for 'diagonal'. It is clear then that this method
of investigation is not suitable for every inquiry, nor is it useful
in those cases in which it is thought to be most suitable.
From what has been said it is clear from what elements demonstrations
are formed and in what manner, and to what points we must look in
each problem.
Part 32
Our next business is to state how we can reduce syllogisms to the
aforementioned figures: for this part of the inquiry still remains.
If we should investigate the production of the syllogisms and had
the power of discovering them, and further if we could resolve the
syllogisms produced into the aforementioned figures, our original
problem would be brought to a conclusion. It will happen at the same
time that what has been already said will be confirmed and its truth
made clearer by what we are about to say. For everything that is true
must in every respect agree with itself First then we must attempt
to select the two premisses of the syllogism (for it is easier to
divide into large parts than into small, and the composite parts are
larger than the elements out of which they are made); next we must
inquire which are universal and which particular, and if both premisses
have not been stated, we must ourselves assume the one which is missing.
For sometimes men put forward the universal premiss, but do not posit
the premiss which is contained in it, either in writing or in discussion:
or men put forward the premisses of the principal syllogism, but omit
those through which they are inferred, and invite the concession of
others to no purpose. We must inquire then whether anything unnecessary
has been assumed, or anything necessary has been omitted, and we must
posit the one and take away the other, until we have reached the two
premisses: for unless we have these, we cannot reduce arguments put
forward in the way described. In some arguments it is easy to see
what is wanting, but some escape us, and appear to be syllogisms,
because something necessary results from what has been laid down,
e.g. if the assumptions were made that substance is not annihilated
by the annihilation of what is not substance, and that if the elements
out of which a thing is made are annihilated, then that which is made
out of them is destroyed: these propositions being laid down, it is
necessary that any part of substance is substance; this has not however
been drawn by syllogism from the propositions assumed, but premisses
are wanting. Again if it is necessary that animal should exist, if
man does, and that substance should exist, if animal does, it is necessary
that substance should exist if man does: but as yet the conclusion
has not been drawn syllogistically: for the premisses are not in the
shape we required. We are deceived in such cases because something
necessary results from what is assumed, since the syllogism also is
necessary. But that which is necessary is wider than the syllogism:
for every syllogism is necessary, but not everything which is necessary
is a syllogism. Consequently, though something results when certain
propositions are assumed, we must not try to reduce it directly, but
must first state the two premisses, then divide them into their terms.
We must take that term as middle which is stated in both the remisses:
for it is necessary that the middle should be found in both premisses
in all the figures.
If then the middle term is a predicate and a subject of predication,
or if it is a predicate, and something else is denied of it, we shall
have the first figure: if it both is a predicate and is denied of
something, the middle figure: if other things are predicated of it,
or one is denied, the other predicated, the last figure. For it was
thus that we found the middle term placed in each figure. It is placed
similarly too if the premisses are not universal: for the middle term
is determined in the same way. Clearly then, if the same term is not
stated more than once in the course of an argument, a syllogism cannot
be made: for a middle term has not been taken. Since we know what
sort of thesis is established in each figure, and in which the universal,
in what sort the particular is described, clearly we must not look
for all the figures, but for that which is appropriate to the thesis
in hand. If the thesis is established in more figures than one, we
shall recognize the figure by the position of the middle term.
Part 33
Men are frequently deceived about syllogisms because the inference
is necessary, as has been said above; sometimes they are deceived
by the similarity in the positing of the terms; and this ought not
to escape our notice. E.g. if A is stated of B, and B of C: it would
seem that a syllogism is possible since the terms stand thus: but
nothing necessary results, nor does a syllogism. Let A represent the
term 'being eternal', B 'Aristomenes as an object of thought', C 'Aristomenes'.
It is true then that A belongs to B. For Aristomenes as an object
of thought is eternal. But B also belongs to C: for Aristomenes is
Aristomenes as an object of thought. But A does not belong to C: for
Aristomenes is perishable. For no syllogism was made although the
terms stood thus: that required that the premiss Ab should be stated
universally. But this is false, that every Aristomenes who is an object
of thought is eternal, since Aristomenes is perishable. Again let
C stand for 'Miccalus', B for 'musical Miccalus', A for 'perishing
to-morrow'. It is true to predicate B of C: for Miccalus is musical
Miccalus. Also A can be predicated of B: for musical Miccalus might
perish to-morrow. But to state A of C is false at any rate. This argument
then is identical with the former; for it is not true universally
that musical Miccalus perishes to-morrow: but unless this is assumed,
no syllogism (as we have shown) is possible.
This deception then arises through ignoring a small distinction. For
if we accept the conclusion as though it made no difference whether
we said 'This belong to that' or 'This belongs to all of that'.
Part 34
Men will frequently fall into fallacies through not setting out the
terms of the premiss well, e.g. suppose A to be health, B disease,
C man. It is true to say that A cannot belong to any B (for health
belongs to no disease) and again that B belongs to every C (for every
man is capable of disease). It would seem to follow that health cannot
belong to any man. The reason for this is that the terms are not set
out well in the statement, since if the things which are in the conditions
are substituted, no syllogism can be made, e.g. if 'healthy' is substituted
for 'health' and 'diseased' for 'disease'. For it is not true to say
that being healthy cannot belong to one who is diseased. But unless
this is assumed no conclusion results, save in respect of possibility:
but such a conclusion is not impossible: for it is possible that health
should belong to no man. Again the fallacy may occur in a similar
way in the middle figure: 'it is not possible that health should belong
to any disease, but it is possible that health should belong to every
man, consequently it is not possible that disease should belong to
any man'. In the third figure the fallacy results in reference to
possibility. For health and diseae and knowledge and ignorance, and
in general contraries, may possibly belong to the same thing, but
cannot belong to one another. This is not in agreement with what was
said before: for we stated that when several things could belong to
the same thing, they could belong to one another.
It is evident then that in all these cases the fallacy arises from
the setting out of the terms: for if the things that are in the conditions
are substituted, no fallacy arises. It is clear then that in such
premisses what possesses the condition ought always to be substituted
for the condition and taken as the term.
Part 35
We must not always seek to set out the terms a single word: for we
shall often have complexes of words to which a single name is not
given. Hence it is difficult to reduce syllogisms with such terms.
Sometimes too fallacies will result from such a search, e.g. the belief
that syllogism can establish that which has no mean. Let A stand for
two right angles, B for triangle, C for isosceles triangle. A then
belongs to C because of B: but A belongs to B without the mediation
of another term: for the triangle in virtue of its own nature contains
two right angles, consequently there will be no middle term for the
proposition AB, although it is demonstrable. For it is clear that
the middle must not always be assumed to be an individual thing, but
sometimes a complex of words, as happens in the case mentioned.
Part 36
That the first term belongs to the middle, and the middle to the extreme,
must not be understood in the sense that they can always be predicated
of one another or that the first term will be predicated of the middle
in the same way as the middle is predicated of the last term. The
same holds if the premisses are negative. But we must suppose the
verb 'to belong' to have as many meanings as the senses in which the
verb 'to be' is used, and in which the assertion that a thing 'is'
may be said to be true. Take for example the statement that there
is a single science of contraries. Let A stand for 'there being a
single science', and B for things which are contrary to one another.
Then A belongs to B, not in the sense that contraries are the fact
of there being a single science of them, but in the sense that it
is true to say of the contraries that there is a single science of
them.
It happens sometimes that the first term is stated of the middle,
but the middle is not stated of the third term, e.g. if wisdom is
knowledge, and wisdom is of the good, the conclusion is that there
is knowledge of the good. The good then is not knowledge, though wisdom
is knowledge. Sometimes the middle term is stated of the third, but
the first is not stated of the middle, e.g. if there is a science
of everything that has a quality, or is a contrary, and the good both
is a contrary and has a quality, the conclusion is that there is a
science of the good, but the good is not science, nor is that which
has a quality or is a contrary, though the good is both of these.
Sometimes neither the first term is stated of the middle, nor the
middle of the third, while the first is sometimes stated of the third,
and sometimes not: e.g. if there is a genus of that of which there
is a science, and if there is a science of the good, we conclude that
there is a genus of the good. But nothing is predicated of anything.
And if that of which there is a science is a genus, and if there is
a science of the good, we conclude that the good is a genus. The first
term then is predicated of the extreme, but in the premisses one thing
is not stated of another.
The same holds good where the relation is negative. For 'that does
not belong to this' does not always mean that 'this is not that',
but sometimes that 'this is not of that' or 'for that', e.g. 'there
is not a motion of a motion or a becoming of a becoming, but there
is a becoming of pleasure: so pleasure is not a becoming.' Or again
it may be said that there is a sign of laughter, but there is not
a sign of a sign, consequently laughter is not a sign. This holds
in the other cases too, in which the thesis is refuted because the
genus is asserted in a particular way, in relation to the terms of
the thesis. Again take the inference 'opportunity is not the right
time: for opportunity belongs to God, but the right time does not,
since nothing is useful to God'. We must take as terms opportunity-right
time-God: but the premiss must be understood according to the case
of the noun. For we state this universally without qualification,
that the terms ought always to be stated in the nominative, e.g. man,
good, contraries, not in oblique cases, e.g. of man, of a good, of
contraries, but the premisses ought to be understood with reference
to the cases of each term-either the dative, e.g. 'equal to this',
or the genitive, e.g. 'double of this', or the accusative, e.g. 'that
which strikes or sees this', or the nominative, e.g. 'man is an animal',
or in whatever other way the word falls in the premiss.
Part 37
The expressions 'this belongs to that' and 'this holds true of that'
must be understood in as many ways as there are different categories,
and these categories must be taken either with or without qualification,
and further as simple or compound: the same holds good of the corresponding
negative expressions. We must consider these points and define them
better.
Part 38
A term which is repeated in the premisses ought to be joined to the
first extreme, not to the middle. I mean for example that if a syllogism
should be made proving that there is knowledge of justice, that it
is good, the expression 'that it is good' (or 'qua good') should be
joined to the first term. Let A stand for 'knowledge that it is good',
B for good, C for justice. It is true to predicate A of B. For of
the good there is knowledge that it is good. Also it is true to predicate
B of C. For justice is identical with a good. In this way an analysis
of the argument can be made. But if the expression 'that it is good'
were added to B, the conclusion will not follow: for A will be true
of B, but B will not be true of C. For to predicate of justice the
term 'good that it is good' is false and not intelligible. Similarly
if it should be proved that the healthy is an object of knowledge
qua good, of goat-stag an object of knowledge qua not existing, or
man perishable qua an object of sense: in every case in which an addition
is made to the predicate, the addition must be joined to the extreme.
The position of the terms is not the same when something is established
without qualification and when it is qualified by some attribute or
condition, e.g. when the good is proved to be an object of knowledge
and when it is proved to be an object of knowledge that it is good.
If it has been proved to be an object of knowledge without 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.
Part 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.
Part 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.
Part 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.
Part 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.
Part 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.
Part 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.
Part 45
Whatever problems are proved in more than one figure, if they have
been established in one figure by syllogism, can be reduced 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.
Part 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 has a corresponding negation.
The negation then of 'it is not-good' is '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
is not-white or all are not-white is false. Similarly also 'every
animal is not-white' is not the negation of 'every animal is white'
(for both are false): the proper negation is 'every animal is not
white'. Since it is clear that 'it is not-white' and 'it is not white'
mean different things, and one is an affirmation, the other a denial,
it is evident that the method of proving each cannot be the same,
e.g. that whatever is an animal is not white or may not be white,
and that it is true to call it not-white; for this means that it is
not-white. But we may prove that it is true to call it white or not-white
in the same way for both are proved constructively by means of the
first figure. For the expression 'it is true' stands on a similar
footing to 'it is'. For the negation of 'it is true to call it white'
is not 'it is true to call it not-white' but 'it is not true to call
it white'. If then it is to be true to say that whatever is a man
is musical or is not-musical, we must assume that whatever is an animal
either is musical or is not-musical; and the proof has been made.
That whatever is a man is not musical is proved destructively in the
three ways mentioned.
In general whenever A and B are such that they cannot belong at the
same time to the same thing, and one of the two necessarily belongs
to everything, and again C and D are related in the same way, and
A follows C but the relation cannot be reversed, then D must follow
B and the relation cannot be reversed. And A and D may belong to the
same thing, but B and C cannot. First it is clear from the following
consideration that D follows B. For since either C or D necessarily
belongs to everything; and since C cannot belong to that to which
B belongs, because it carries A along with it and A and B cannot belong
to the same thing; it is clear that D must follow B. Again since C
does not reciprocate with but A, but C or D belongs to everything,
it is possible that A and D should belong to the same thing. But B
and C cannot belong to the same thing, because A follows C; and so
something impossible results. It is clear then that B does not reciprocate
with D either, since it is possible that D and A should belong at
the same time to the same thing.
It results sometimes even in such an arrangement of terms that one
is deceived through not apprehending the opposites rightly, one of
which must belong to everything, e.g. we may reason that 'if A and
B cannot belong at the same time to the same thing, but it is necessary
that one of them should belong to whatever the other does not belong
to: and again C and D are related in the same way, and follows everything
which C follows: it will result that B belongs necessarily to everything
to which D belongs': but this is false. 'Assume that F stands for
the negation of A and B, and again that H stands for the negation
of C and D. It is necessary then that either A or F should belong
to everything: for either the affirmation or the denial must belong.
And again either C or H must belong to everything: for they are related
as affirmation and denial. And ex hypothesi A belongs to everything
ever thing to which C belongs. Therefore H belongs to everything to
which F belongs. Again since either F or B belongs to everything,
and similarly either H or D, and since H follows F, B must follow
D: for we know this. If then A follows C, B must follow D'. But this
is false: for as we proved the sequence is reversed in terms so constituted.
The fallacy arises because perhaps it is not necessary that A or F
should belong to everything, or that F or B should belong to everything:
for F is not the denial of A. For not good is the negation of good:
and not-good is not identical with 'neither good nor not-good'. Similarly
also with C and D. For two negations have been assumed in respect
to one term.
----------------------------------------------------------------------
BOOK II
Part 1
We have already explained the number of the figures, the character
and number of the premisses, when and how a syllogism is formed; further
what we must look for when a refuting and establishing propositions,
and how we should investigate a given problem in any branch of inquiry,
also by what means we shall obtain principles appropriate to each
subject. Since some syllogisms are universal, others particular, all
the universal syllogisms give more than one result, and of particular
syllogisms the affirmative yield more than one, the negative yield
only the stated conclusion. For all propositions are convertible save
only the particular negative: and the conclusion states one definite
thing about another definite thing. Consequently all syllogisms save
the particular negative yield more than one conclusion, e.g. if A
has been proved to to all or to some B, then B must belong to some
A: and if A has been proved to belong to no B, then B belongs to no
A. This is a different conclusion from the former. But if A does not
belong to some B, it is not necessary that B should not belong to
some A: for it may possibly belong to all A.
This then is the reason common to all syllogisms whether universal
or particular. But it is possible to give another reason concerning
those which are universal. For all the things that are subordinate
to the middle term or to the conclusion may be proved by the same
syllogism, if the former are placed in the middle, the latter in the
conclusion; e.g. if the conclusion AB is proved through C, whatever
is subordinate to B or C must accept the predicate A: for if D is
included in B as in a whole, and B is included in A, then D will be
included in A. Again if E is included in C as in a whole, and C is
included in A, then E will be included in A. Similarly if the syllogism
is negative. In the second figure it will be possible to infer only
that which is subordinate to the conclusion, e.g. if A belongs to
no B and to all C; we conclude that B belongs to no C. If then D is
subordinate to C, clearly B does not belong to it. But that B does
not belong to what is subordinate to A is not clear by means of the
syllogism. And yet B does not belong to E, if E is subordinate to
A. But while it has been proved through the syllogism that B belongs
to no C, it has been assumed without proof that B does not belong
to A, consequently it does not result through the syllogism that B
does not belong to E.
But in particular syllogisms there will be no necessity of inferring
what is subordinate to the conclusion (for a syllogism does not result
when this premiss is particular), but whatever is subordinate to the
middle term may be inferred, not however through the syllogism, e.g.
if A belongs to all B and B to some C. Nothing can be inferred about
that which is subordinate to C; something can be inferred about that
which is subordinate to B, but not through the preceding syllogism.
Similarly in the other figures. That which is subordinate to the conclusion
cannot be proved; the other subordinate can be proved, only not through
the syllogism, just as in the universal syllogisms what is subordinate
to the middle term is proved (as we saw) from a premiss which is not
demonstrated: consequently either a conclusion is not possible in
the case of universal syllogisms or else it is possible also in the
case of particular syllogisms.
Part 2
It is possible for the premisses of the syllogism to be true, or to
be false, or to be the one true, the other false. The conclusion is
either true or false necessarily. From true premisses it is not possible
to draw a false conclusion, but a true conclusion may be drawn from
false premisses, true however only in respect to the fact, not to
the reason. The reason cannot be established from false premisses:
why this is so will be explained in the sequel.
First then that it is not possible to draw a false conclusion from
true premisses, is made clear by this consideration. If it is necessary
that B should be when A is, it is necessary that A should not be when
B is not. If then A is true, B must be true: otherwise it will turn
out that the same thing both is and is not at the same time. But this
is impossible. Let it not, because A is laid down as a single term,
be supposed that it is possible, when a single fact is given, that
something should necessarily result. For that is not possible. For
what results necessarily is the conclusion, and the means by which
this comes about are at the least three terms, and two relations of
subject and predicate or premisses. If then it is true that A belongs
to all that to which B belongs, and that B belongs to all that to
which C belongs, it is necessary that A should belong to all that
to which C belongs, and this cannot be false: for then the same thing
will belong and not belong at the same time. So A is posited as one
thing, being two premisses taken together. The same holds good of
negative syllogisms: it is not possible to prove a false conclusion
from true premisses.
But from what is false a true conclusion may be drawn, whether both
the premisses are false or only one, provided that this is not either
of the premisses indifferently, if it is taken as wholly false: but
if the premiss is not taken as wholly false, it does not matter which
of the two is false. (1) Let A belong to the whole of C, but to none
of the Bs, neither let B belong to C. This is possible, e.g. animal
belongs to no stone, nor stone to any man. If then A is taken to belong
to all B and B to all C, A will belong to all C; consequently though
both the premisses are false the conclusion is true: for every man
is an animal. Similarly with the negative. For it is possible that
neither A nor B should belong to any C, although A belongs to all
B, e.g. if the same terms are taken and man is put as middle: for
neither animal nor man belongs to any stone, but animal belongs to
every man. Consequently if one term is taken to belong to none of
that to which it does belong, and the other term is taken to belong
to all of that to which it does not belong, though both the premisses
are false the conclusion will be true. (2) A similar proof may be
given if each premiss is partially false.
(3) But if one only of the premisses is false, when the first premiss
is wholly false, e.g. AB, the conclusion will not be true, but if
the premiss BC is wholly false, a true conclusion will be possible.
I mean by 'wholly false' the contrary of the truth, e.g. if what belongs
to none is assumed to belong to all, or if what belongs to all is
assumed to belong to none. Let A belong to no B, and B to all C. If
then the premiss BC which I take is true, and the premiss AB is wholly
false, viz. that A belongs to all B, it is impossible that the conclusion
should be true: for A belonged to none of the Cs, since A belonged
to nothing to which B belonged, and B belonged to all C. Similarly
there cannot be a true conclusion if A belongs to all B, and B to
all C, but while the true premiss BC is assumed, the wholly false
premiss AB is also assumed, viz. that A belongs to nothing to which
B belongs: here the conclusion must be false. For A will belong to
all C, since A belongs to everything to which B belongs, and B to
all C. It is clear then that when the first premiss is wholly false,
whether affirmative or negative, and the other premiss is true, the
conclusion cannot be true.
(4) But if the premiss is not wholly false, a true conclusion is possible.
For if A belongs to all C and to some B, and if B belongs to all C,
e.g. animal to every swan and to some white thing, and white to every
swan, then if we take as premisses that A belongs to all B, and B
to all C, A will belong to all C truly: for every swan is an animal.
Similarly if the statement AB is negative. For it is possible that
A should belong to some B and to no C, and that B should belong to
all C, e.g. animal to some white thing, but to no snow, and white
to all snow. If then one should assume that A belongs to no B, and
B to all C, then will belong to no C.
(5) But if the premiss AB, which is assumed, is wholly true, and the
premiss BC is wholly false, a true syllogism will be possible: for
nothing prevents A belonging to all B and to all C, though B belongs
to no C, e.g. these being species of the same genus which are not
subordinate one to the other: for animal belongs both to horse and
to man, but horse to no man. If then it is assumed that A belongs
to all B and B to all C, the conclusion will be true, although the
premiss BC is wholly false. Similarly if the premiss AB is negative.
For it is possible that A should belong neither to any B nor to any
C, and that B should not belong to any C, e.g. a genus to species
of another genus: for animal belongs neither to music nor to the art
of healing, nor does music belong to the art of healing. If then it
is assumed that A belongs to no B, and B to all C, the conclusion
will be true.
(6) And if the premiss BC is not wholly false but in part only, even
so the conclusion may be true. For nothing prevents A belonging to
the whole of B and of C, while B belongs to some C, e.g. a genus to
its species and difference: for animal belongs to every man and to
every footed thing, and man to some footed things though not to all.
If then it is assumed that A belongs to all B, and B to all C, A will
belong to all C: and this ex hypothesi is true. Similarly if the premiss
AB is negative. For it is possible that A should neither belong to
any B nor to any C, though B belongs to some C, e.g. a genus to the
species of another genus and its difference: for animal neither belongs
to any wisdom nor to any instance of 'speculative', but wisdom belongs
to some instance of 'speculative'. If then it should be assumed that
A belongs to no B, and B to all C, will belong to no C: and this ex
hypothesi is true.
In particular syllogisms it is possible when the first premiss is
wholly false, and the other true, that the conclusion should be true;
also when the first premiss is false in part, and the other true;
and when the first is true, and the particular is false; and when
both are false. (7) For nothing prevents A belonging to no B, but
to some C, and B to some C, e.g. animal belongs to no snow, but to
some white thing, and snow to some white thing. If then snow is taken
as middle, and animal as first term, and it is assumed that A belongs
to the whole of B, and B to some C, then the premiss BC is wholly
false, the premiss BC true, and the conclusion true. Similarly if
the premiss AB is negative: for it is possible that A should belong
to the whole of B, but not to some C, although B belongs to some C,
e.g. animal belongs to every man, but does not follow some white,
but man belongs to some white; consequently if man be taken as middle
term and it is assumed that A belongs to no B but B belongs to some
C, the conclusion will be true although the premiss AB is wholly false.
(If the premiss AB is false in part, the conclusion may be true. For
nothing prevents A belonging both to B and to some C, and B belonging
to some C, e.g. animal to something beautiful and to something great,
and beautiful belonging to something great. If then A is assumed to
belong to all B, and B to some C, the a premiss AB will be partially
false, the premiss BC will be true, and the conclusion true. Similarly
if the premiss AB is negative. For the same terms will serve, and
in the same positions, to prove the point.
(9) Again if the premiss AB is true, and the premiss BC is false,
the conclusion may be true. For nothing prevents A belonging to the
whole of B and to some C, while B belongs to no C, e.g. animal to
every swan and to some black things, though swan belongs to no black
thing. Consequently if it should be assumed that A belongs to all
B, and B to some C, the conclusion will be true, although the statement
Bc is false. Similarly if the premiss AB is negative. For it is possible
that A should belong to no B, and not to some C, while B belongs to
no C, e.g. a genus to the species of another genus and to the accident
of its own species: for animal belongs to no number and not to some
white things, and number belongs to nothing white. If then number
is taken as middle, and it is assumed that A belongs to no B, and
B to some C, then A will not belong to some C, which ex hypothesi
is true. And the premiss AB is true, the premiss BC false.
(10) Also if the premiss AB is partially false, and the premiss BC
is false too, the conclusion may be true. For nothing prevents A belonging
to some B and to some C, though B belongs to no C, e.g. if B is the
contrary of C, and both are accidents of the same genus: for animal
belongs to some white things and to some black things, but white belongs
to no black thing. If then it is assumed that A belongs to all B,
and B to some C, the conclusion will be true. Similarly if the premiss
AB is negative: for the same terms arranged in the same way will serve
for the proof.
(11) Also though both premisses are false the conclusion may be true.
For it is possible that A may belong to no B and to some C, while
B belongs to no C, e.g. a genus in relation to the species of another
genus, and to the accident of its own species: for animal belongs
to no number, but to some white things, and number to nothing white.
If then it is assumed that A belongs to all B and B to some C, the
conclusion will be true, though both premisses are false. Similarly
also if the premiss AB is negative. For nothing prevents A belonging
to the whole of B, and not to some C, while B belongs to no C, e.g.
animal belongs to every swan, and not to some black things, and swan
belongs to nothing black. Consequently if it is assumed that A belongs
to no B, and B to some C, then A does not belong to some C. The conclusion
then is true, but the premisses arc false.
Part 3
In the middle figure it is possible in every way to reach a true conclusion
through false premisses, whether the syllogisms are universal or particular,
viz. when both premisses are wholly false; when each is partially
false; when one is true, the other wholly false (it does not matter
which of the two premisses is false); if both premisses are partially
false; if one is quite true, the other partially false; if one is
wholly false, the other partially true. For (1) if A belongs to no
B and to all C, e.g. animal to no stone and to every horse, then if
the premisses are stated contrariwise and it is assumed that A belongs
to all B and to no C, though the premisses are wholly false they will
yield a true conclusion. Similarly if A belongs to all B and to no
C: for we shall have the same syllogism.
(2) Again if one premiss is wholly false, the other wholly true: for
nothing prevents A belonging to all B and to all C, though B belongs
to no C, e.g. a genus to its co-ordinate species. For animal belongs
to every horse and man, and no man is a horse. If then it is assumed
that animal belongs to all of the one, and none of the other, the
one premiss will be wholly false, the other wholly true, and the conclusion
will be true whichever term the negative statement concerns.
(3) Also if one premiss is partially false, the other wholly true.
For it is possible that A should belong to some B and to all C, though
B belongs to no C, e.g. animal to some white things and to every raven,
though white belongs to no raven. If then it is assumed that A belongs
to no B, but to the whole of C, the premiss AB is partially false,
the premiss AC wholly true, and the conclusion true. Similarly if
the negative statement is transposed: the proof can be made by means
of the same terms. Also if the affirmative premiss is partially false,
the negative wholly true, a true conclusion is possible. For nothing
prevents A belonging to some B, but not to C as a whole, while B belongs
to no C, e.g. animal belongs to some white things, but to no pitch,
and white belongs to no pitch. Consequently if it is assumed that
A belongs to the whole of B, but to no C, the premiss AB is partially
false, the premiss AC is wholly true, and the conclusion is true.
(4) And if both the premisses are partially false, the conclusion
may be true. For it is possible that A should belong to some B and
to some C, and B to no C, e.g. animal to some white things and to
some black things, though white belongs to nothing black. If then
it is assumed that A belongs to all B and to no C, both premisses
are partially false, but the conclusion is true. Similarly, if the
negative premiss is transposed, the proof can be made by means of
the same terms.
It is clear also that our thesis holds in particular syllogisms. For
(5) nothing prevents A belonging to all B and to some C, though B
does not belong to some C, e.g. animal to every man and to some white
things, though man will not belong to some white things. If then it
is stated that A belongs to no B and to some C, the universal premiss
is wholly false, the particular premiss is true, and the conclusion
is true. Similarly if the premiss AB is affirmative: for it is possible
that A should belong to no B, and not to some C, though B does not
belong to some C, e.g. animal belongs to nothing lifeless, and does
not belong to some white things, and lifeless will not belong to some
white things. If then it is stated that A belongs to all B and not
to some C, the premiss AB which is universal is wholly false, the
premiss AC is true, and the conclusion is true. Also a true conclusion
is possible when the universal premiss is true, and the particular
is false. For nothing prevents A following neither B nor C at all,
while B does not belong to some C, e.g. animal belongs to no number
nor to anything lifeless, and number does not follow some lifeless
things. If then it is stated that A belongs to no B and to some C,
the conclusion will be true, and the universal premiss true, but the
particular false. Similarly if the premiss which is stated universally
is affirmative. For it is possible that should A belong both to B
and to C as wholes, though B does not follow some C, e.g. a genus
in relation to its species and difference: for animal follows every
man and footed things as a whole, but man does not follow every footed
thing. Consequently if it is assumed that A belongs to the whole of
B, but does not belong to some C, the universal premiss is true, the
particular false, and the conclusion true.
(6) It is clear too that though both premisses are false they may
yield a true conclusion, since it is possible that A should belong
both to B and to C as wholes, though B does not follow some C. For
if it is assumed that A belongs to no B and to some C, the premisses
are both false, but the conclusion is true. Similarly if the universal
premiss is affirmative and the particular negative. For it is possible
that A should follow no B and all C, though B does not belong to some
C, e.g. animal follows no science but every man, though science does
not follow every man. If then A is assumed to belong to the whole
of B, and not to follow some C, the premisses are false but the conclusion
is true.
Part 4
In the last figure a true conclusion may come through what is false,
alike when both premisses are wholly false, when each is partly false,
when one premiss is wholly true, the other false, when one premiss
is partly false, the other wholly true, and vice versa, and in every
other way in which it is possible to alter the premisses. For (1)
nothing prevents neither A nor B from belonging to any C, while A
belongs to some B, e.g. neither man nor footed follows anything lifeless,
though man belongs to some footed things. If then it is assumed that
A and B belong to all C, the premisses will be wholly false, but the
conclusion true. Similarly if one premiss is negative, the other affirmative.
For it is possible that B should belong to no C, but A to all C, and
that should not belong to some B, e.g. black belongs to no swan, animal
to every swan, and animal not to everything black. Consequently if
it is assumed that B belongs to all C, and A to no C, A will not belong
to some B: and the conclusion is true, though the premisses are false.
(2) Also if each premiss is partly false, the conclusion may be true.
For nothing prevents both A and B from belonging to some C while A
belongs to some B, e.g. white and beautiful belong to some animals,
and white to some beautiful things. If then it is stated that A and
B belong to all C, the premisses are partially false, but the conclusion
is true. Similarly if the premiss AC is stated as negative. For nothing
prevents A from not belonging, and B from belonging, to some C, while
A does not belong to all B, e.g. white does not belong to some animals,
beautiful belongs to some animals, and white does not belong to everything
beautiful. Consequently if it is assumed that A belongs to no C, and
B to all C, both premisses are partly false, but the conclusion is
true.
(3) Similarly if one of the premisses assumed is wholly false, the
other wholly true. For it is possible that both A and B should follow
all C, though A does not belong to some B, e.g. animal and white follow
every swan, though animal does not belong to everything white. Taking
these then as terms, if one assumes that B belongs to the whole of
C, but A does not belong to C at all, the premiss BC will be wholly
true, the premiss AC wholly false, and the conclusion true. Similarly
if the statement BC is false, the statement AC true, the conclusion
may be true. The same terms will serve for the proof. Also if both
the premisses assumed are affirmative, the conclusion may be true.
For nothing prevents B from following all C, and A from not belonging
to C at all, though A belongs to some B, e.g. animal belongs to every
swan, black to no swan, and black to some animals. Consequently if
it is assumed that A and B belong to every C, the premiss BC is wholly
true, the premiss AC is wholly false, and the conclusion is true.
Similarly if the premiss AC which is assumed is true: the proof can
be made through the same terms.
(4) Again if one premiss is wholly true, the other partly false, the
conclusion may be true. For it is possible that B should belong to
all C, and A to some C, while A belongs to some B, e.g. biped belongs
to every man, beautiful not to every man, and beautiful to some bipeds.
If then it is assumed that both A and B belong to the whole of C,
the premiss BC is wholly true, the premiss AC partly false, the conclusion
true. Similarly if of the premisses assumed AC is true and BC partly
false, a true conclusion is possible: this can be proved, if the same
terms as before are transposed. Also the conclusion may be true if
one premiss is negative, the other affirmative. For since it is possible
that B should belong to the whole of C, and A to some C, and, when
they are so, that A should not belong to all B, therefore it is assumed
that B belongs to the whole of C, and A to no C, the negative premiss
is partly false, the other premiss wholly true, and the conclusion
is true. Again since it has been proved that if A belongs to no C
and B to some C, it is possible that A should not belong to some C,
it is clear that if the premiss AC is wholly true, and the premiss
BC partly false, it is possible that the conclusion should be true.
For if it is assumed that A belongs to no C, and B to all C, the premiss
AC is wholly true, and the premiss BC is partly false.
(5) It is clear also in the case of particular syllogisms that a true
conclusion may come through what is false, in every possible way.
For the same terms must be taken as have been taken when the premisses
are universal, positive terms in positive syllogisms, negative terms
in negative. For it makes no difference to the setting out of the
terms, whether one assumes that what belongs to none belongs to all
or that what belongs to some belongs to all. The same applies to negative
statements.
It is clear then that if the conclusion is false, the premisses of
the argument must be false, either all or some of them; but when the
conclusion is true, it is not necessary that the premisses should
be true, either one or all, yet it is possible, though no part of
the syllogism is true, that the conclusion may none the less be true;
but it is not necessitated. The reason is that when two things are
so related to one another, that if the one is, the other necessarily
is, then if the latter is not, the former will not be either, but
if the latter is, it is not necessary that the former should be. But
it is impossible that the same thing should be necessitated by the
being and by the not-being of the same thing. I mean, for example,
that it is impossible that B should necessarily be great since A is
white and that B should necessarily be great since A is not white.
For whenever since this, A, is white it is necessary that that, B,
should be great, and since B is great that C should not be white,
then it is necessary if is white that C should not be white. And whenever
it is necessary, since one of two things is, that the other should
be, it is necessary, if the latter is not, that the former (viz. A)
should not be. If then B is not great A cannot be white. But if, when
A is not white, it is necessary that B should be great, it necessarily
results that if B is not great, B itself is great. (But this is impossible.)
For if B is not great, A will necessarily not be white. If then when
this is not white B must be great, it results that if B is not great,
it is great, just as if it were proved through three terms.
Part 5
Circular and reciprocal proof means proof by means of the conclusion,
i.e. by converting one of the premisses simply and inferring the premiss
which was assumed in the original syllogism: e.g. suppose it has been
necessary to prove that A belongs to all C, and it has been proved
through B; suppose that A should now be proved to belong to B by assuming
that A belongs to C, and C to B-so A belongs to B: but in the first
syllogism the converse was assumed, viz. that B belongs to C. Or suppose
it is necessary to prove that B belongs to C, and A is assumed to
belong to C, which was the conclusion of the first syllogism, and
B to belong to A but the converse was assumed in the earlier syllogism,
viz. that A belongs to B. In no other way is reciprocal proof possible.
If another term is taken as middle, the proof is not circular: for
neither of the propositions assumed is the same as before: if one
of the accepted terms is taken as middle, only one of the premisses
of the first syllogism can be assumed in the second: for if both of
them are taken the same conclusion as before will result: but it must
be different. If the terms are not convertible, one of the premisses
from which the syllogism results must be undemonstrated: for it is
not possible to demonstrate through these terms that the third belongs
to the middle or the middle to the first. If the terms are convertible,
it is possible to demonstrate everything reciprocally, e.g. if A and
B and C are convertible with one another. Suppose the proposition
AC has been demonstrated through B as middle term, and again the proposition
AB through the conclusion and the premiss BC converted, and similarly
the proposition BC through the conclusion and the premiss AB converted.
But it is necessary to prove both the premiss CB, and the premiss
BA: for we have used these alone without demonstrating them. If then
it is assumed that B belongs to all C, and C to all A, we shall have
a syllogism relating B to A. Again if it is assumed that C belongs
to all A, and A to all B, C must belong to all B. In both these syllogisms
the premiss CA has been assumed without being demonstrated: the other
premisses had ex hypothesi been proved. Consequently if we succeed
in demonstrating this premiss, all the premisses will have been proved
reciprocally. If then it is assumed that C belongs to all B, and B
to all A, both the premisses assumed have been proved, and C must
belong to A. It is clear then that only if the terms are convertible
is circular and reciprocal demonstration possible (if the terms are
not convertible, the matter stands as we said above). But it turns
out in these also that we use for the demonstration the very thing
that is being proved: for C is proved of B, and B of by assuming that
C is said of and C is proved of A through these premisses, so that
we use the conclusion for the demonstration.
In negative syllogisms reciprocal proof is as follows. Let B belong
to all C, and A to none of the Bs: we conclude that A belongs to none
of the Cs. If again it is necessary to prove that A belongs to none
of the Bs (which was previously assumed) A must belong to no C, and
C to all B: thus the previous premiss is reversed. If it is necessary
to prove that B belongs to C, the proposition AB must no longer be
converted as before: for the premiss 'B belongs to no A' is identical
with the premiss 'A belongs to no B'. But we must assume that B belongs
to all of that to none of which longs. Let A belong to none of the
Cs (which was the previous conclusion) and assume that B belongs to
all of that to none of which A belongs. It is necessary then that
B should belong to all C. Consequently each of the three propositions
has been made a conclusion, and this is circular demonstration, to
assume the conclusion and the converse of one of the premisses, and
deduce the remaining premiss.
In particular syllogisms it is not possible to demonstrate the universal
premiss through the other propositions, but the particular premiss
can be demonstrated. Clearly it is impossible to demonstrate the universal
premiss: for what is universal is proved through propositions which
are universal, but the conclusion is not universal, and the proof
must start from the conclusion and the other premiss. Further a syllogism
cannot be made at all if the other premiss is converted: for the result
is that both premisses are particular. But the particular premiss
may be proved. Suppose that A has been proved of some C through B.
If then it is assumed that B belongs to all A and the conclusion is
retained, B will belong to some C: for we obtain the first figure
and A is middle. But if the syllogism is negative, it is not possible
to prove the universal premiss, for the reason given above. But it
is possible to prove the particular premiss, if the proposition AB
is converted as in the universal syllogism, i.e 'B belongs to some
of that to some of which A does not belong': otherwise no syllogism
results because the particular premiss is negative.
Part 6
In the second figure it is not possible to prove an affirmative proposition
in this way, but a negative proposition may be proved. An affirmative
proposition is not proved because both premisses of the new syllogism
are not affirmative (for the conclusion is negative) but an affirmative
proposition is (as we saw) proved from premisses which are both affirmative.
The negative is proved as follows. Let A belong to all B, and to no
C: we conclude that B belongs to no C. If then it is assumed that
B belongs to all A, it is necessary that A should belong to no C:
for we get the second figure, with B as middle. But if the premiss
AB was negative, and the other affirmative, we shall have the first
figure. For C belongs to all A and B to no C, consequently B belongs
to no A: neither then does A belong to B. Through the conclusion,
therefore, and one premiss, we get no syllogism, but if another premiss
is assumed in addition, a syllogism will be possible. But if the syllogism
not universal, the universal premiss cannot be proved, for the same
reason as we gave above, but the particular premiss can be proved
whenever the universal statement is affirmative. Let A belong to all
B, and not to all C: the conclusion is BC. If then it is assumed that
B belongs to all A, but not to all C, A will not belong to some C,
B being middle. But if the universal premiss is negative, the premiss
AC will not be demonstrated by the conversion of AB: for it turns
out that either both or one of the premisses is negative; consequently
a syllogism will not be possible. But the proof will proceed as in
the universal syllogisms, if it is assumed that A belongs to some
of that to some of which B does not belong.
Part 7
In the third figure, when both premisses are taken universally, it
is not possible to prove them reciprocally: for that which is universal
is proved through statements which are universal, but the conclusion
in this figure is always particular, so that it is clear that it is
not possible at all to prove through this figure the universal premiss.
But if one premiss is universal, the other particular, proof of the
latter will sometimes be possible, sometimes not. When both the premisses
assumed are affirmative, and the universal concerns the minor extreme,
proof will be possible, but when it concerns the other extreme, impossible.
Let A belong to all C and B to some C: the conclusion is the statement
AB. If then it is assumed that C belongs to all A, it has been proved
that C belongs to some B, but that B belongs to some C has not been
proved. And yet it is necessary, if C belongs to some B, that B should
belong to some C. But it is not the same that this should belong to
that, and that to this: but we must assume besides that if this belongs
to some of that, that belongs to some of this. But if this is assumed
the syllogism no longer results from the conclusion and the other
premiss. But if B belongs to all C, and A to some C, it will be possible
to prove the proposition AC, when it is assumed that C belongs to
all B, and A to some B. For if C belongs to all B and A to some B,
it is necessary that A should belong to some C, B being middle. And
whenever one premiss is affirmative the other negative, and the affirmative
is universal, the other premiss can be proved. Let B belong to all
C, and A not to some C: the conclusion is that A does not belong to
some B. If then it is assumed further that C belongs to all B, it
is necessary that A should not belong to some C, B being middle. But
when the negative premiss is universal, the other premiss is not except
as before, viz. if it is assumed that that belongs to some of that,
to some of which this does not belong, e.g. if A belongs to no C,
and B to some C: the conclusion is that A does not belong to some
B. If then it is assumed that C belongs to some of that to some of
which does not belong, it is necessary that C should belong to some
of the Bs. In no other way is it possible by converting the universal
premiss to prove the other: for in no other way can a syllogism be
formed.
It is clear then that in the first figure reciprocal proof is made
both through the third and through the first figure-if the conclusion
is affirmative through the first; if the conclusion is negative through
the last. For it is assumed that that belongs to all of that to none
of which this belongs. In the middle figure, when the syllogism is
universal, proof is possible through the second figure and through
the first, but when particular through the second and the last. In
the third figure all proofs are made through itself. It is clear also
that in the third figure and in the middle figure those syllogisms
which are not made through those figures themselves either are not
of the nature of circular proof or are imperfect.
Part 8
To convert a syllogism means to alter the conclusion and make another
syllogism to prove that either the extreme cannot belong to the middle
or the middle to the last term. For it is necessary, if the conclusion
has been changed into its opposite and one of the premisses stands,
that the other premiss should be destroyed. For if it should stand,
the conclusion also must stand. It makes a difference whether the
conclusion is converted into its contradictory or into its contrary.
For the same syllogism does not result whichever form the conversion
takes. This will be made clear by the sequel. By contradictory opposition
I mean the opposition of 'to all' to 'not to all', and of 'to some'
to 'to none'; by contrary opposition I mean the opposition of 'to
all' to 'to none', and of 'to some' to 'not to some'. Suppose that
A been proved of C, through B as middle term. If then it should be
assumed that A belongs to no C, but to all B, B will belong to no
C. And if A belongs to no C, and B to all C, A will belong, not to
no B at all, but not to all B. For (as we saw) the universal is not
proved through the last figure. In a word it is not possible to refute
universally by conversion the premiss which concerns the major extreme:
for the refutation always proceeds through the third since it is necessary
to take both premisses in reference to the minor extreme. Similarly
if the syllogism is negative. Suppose it has been proved that A belongs
to no C through B. Then if it is assumed that A belongs to all C,
and to no B, B will belong to none of the Cs. And if A and B belong
to all C, A will belong to some B: but in the original premiss it
belonged to no B.
If the conclusion is converted into its contradictory, the syllogisms
will be contradictory and not universal. For one premiss is particular,
so that the conclusion also will be particular. Let the syllogism
be affirmative, and let it be converted as stated. Then if A belongs
not to all C, but to all B, B will belong not to all C. And if A belongs
not to all C, but B belongs to all C, A will belong not to all B.
Similarly if the syllogism is negative. For if A belongs to some C,
and to no B, B will belong, not to no C at all, but-not to some C.
And if A belongs to some C, and B to all C, as was originally assumed,
A will belong to some B.
In particular syllogisms when the conclusion is converted into its
contradictory, both premisses may be refuted, but when it is converted
into its contrary, neither. For the result is no longer, as in the
universal syllogisms, refutation in which the conclusion reached by
O, conversion lacks universality, but no refutation at all. Suppose
that A has been proved of some C. If then it is assumed that A belongs
to no C, and B to some C, A will not belong to some B: and if A belongs
to no C, but to all B, B will belong to no C. Thus both premisses
are refuted. But neither can be refuted if the conclusion is converted
into its contrary. For if A does not belong to some C, but to all
B, then B will not belong to some C. But the original premiss is not
yet refuted: for it is possible that B should belong to some C, and
should not belong to some C. The universal premiss AB cannot be affected
by a syllogism at all: for if A does not belong to some of the Cs,
but B belongs to some of the Cs, neither of the premisses is universal.
Similarly if the syllogism is negative: for if it should be assumed
that A belongs to all C, both premisses are refuted: but if the assumption
is that A belongs to some C, neither premiss is refuted. The proof
is the same as before.
Part 9
In the second figure it is not possible to refute the premiss which
concerns the major extreme by establishing something contrary to it,
whichever form the conversion of the conclusion may take. For the
conclusion of the refutation will always be in the third figure, and
in this figure (as we saw) there is no universal syllogism. The other
premiss can be refuted in a manner similar to the conversion: I mean,
if the conclusion of the first syllogism is converted into its contrary,
the conclusion of the refutation will be the contrary of the minor
premiss of the first, if into its contradictory, the contradictory.
Let A belong to all B and to no C: conclusion BC. If then it is assumed
that B belongs to all C, and the proposition AB stands, A will belong
to all C, since the first figure is produced. If B belongs to all
C, and A to no C, then A belongs not to all B: the figure is the last.
But if the conclusion BC is converted into its contradictory, the
premiss AB will be refuted as before, the premiss, AC by its contradictory.
For if B belongs to some C, and A to no C, then A will not belong
to some B. Again if B belongs to some C, and A to all B, A will belong
to some C, so that the syllogism results in the contradictory of the
minor premiss. A similar proof can be given if the premisses are transposed
in respect of their quality.
If the syllogism is particular, when the conclusion is converted into
its contrary neither premiss can be refuted, as also happened in the
first figure,' if the conclusion is converted into its contradictory,
both premisses can be refuted. Suppose that A belongs to no B, and
to some C: the conclusion is BC. If then it is assumed that B belongs
to some C, and the statement AB stands, the conclusion will be that
A does not belong to some C. But the original statement has not been
refuted: for it is possible that A should belong to some C and also
not to some C. Again if B belongs to some C and A to some C, no syllogism
will be possible: for neither of the premisses taken is universal.
Consequently the proposition AB is not refuted. But if the conclusion
is converted into its contradictory, both premisses can be refuted.
For if B belongs to all C, and A to no B, A will belong to no C: but
it was assumed to belong to some C. Again if B belongs to all C and
A to some C, A will belong to some B. The same proof can be given
if the universal statement is affirmative.
Part 10
In the third figure when the conclusion is converted into its contrary,
neither of the premisses can be refuted in any of the syllogisms,
but when the conclusion is converted into its contradictory, both
premisses may be refuted and in all the moods. Suppose it has been
proved that A belongs to some B, C being taken as middle, and the
premisses being universal. If then it is assumed that A does not belong
to some B, but B belongs to all C, no syllogism is formed about A
and C. Nor if A does not belong to some B, but belongs to all C, will
a syllogism be possible about B and C. A similar proof can be given
if the premisses are not universal. For either both premisses arrived
at by the conversion must be particular, or the universal premiss
must refer to the minor extreme. But we found that no syllogism is
possible thus either in the first or in the middle figure. But if
the conclusion is converted into its contradictory, both the premisses
can be refuted. For if A belongs to no B, and B to all C, then A belongs
to no C: again if A belongs to no B, and to all C, B belongs to no
C. And similarly if one of the premisses is not universal. For if
A belongs to no B, and B to some C, A will not belong to some C: if
A belongs to no B, and to C, B will belong to no C.
Similarly if the original syllogism is negative. Suppose it has been
proved that A does not belong to some B, BC being affirmative, AC
being negative: for it was thus that, as we saw, a syllogism could
be made. Whenever then the contrary of the conclusion is assumed a
syllogism will not be possible. For if A belongs to some B, and B
to all C, no syllogism is possible (as we saw) about A and C. Nor,
if A belongs to some B, and to no C, was a syllogism possible concerning
B and C. Therefore the premisses are not refuted. But when the contradictory
of the conclusion is assumed, they are refuted. For if A belongs to
all B, and B to C, A belongs to all C: but A was supposed originally
to belong to no C. Again if A belongs to all B, and to no C, then
B belongs to no C: but it was supposed to belong to all C. A similar
proof is possible if the premisses are not universal. For AC becomes
universal and negative, the other premiss particular and affirmative.
If then A belongs to all B, and B to some C, it results that A belongs
to some C: but it was supposed to belong to no C. Again if A belongs
to all B, and to no C, then B belongs to no C: but it was assumed
to belong to some C. If A belongs to some B and B to some C, no syllogism
results: nor yet if A belongs to some B, and to no C. Thus in one
way the premisses are refuted, in the other way they are not.
From what has been said it is clear how a syllogism results in each
figure when the conclusion is converted; when a result contrary to
the premiss, and when a result contradictory to the premiss, is obtained.
It is clear that in the first figure the syllogisms are formed through
the middle and the last figures, and the premiss which concerns the
minor extreme is alway refuted through the middle figure, the premiss
which concerns the major through the last figure. In the second figure
syllogisms proceed through the first and the last figures, and the
premiss which concerns the minor extreme is always refuted through
the first figure, the premiss which concerns the major extreme through
the last. In the third figure the refutation proceeds through the
first and the middle figures; the premiss which concerns the major
is always refuted through the first figure, the premiss which concerns
the minor through the middle figure.
Part 11
It is clear then what conversion is, how it is effected in each figure,
and what syllogism results. The syllogism per impossibile is proved
when the contradictory of the conclusion stated and another premiss
is assumed; it can be made in all the figures. For it resembles conversion,
differing only in this: conversion takes place after a syllogism has
been formed and both the premisses have been taken, but a reduction
to the impossible takes place not because the contradictory has been
agreed to already, but because it is clear that it is true. The terms
are alike in both, and the premisses of both are taken in the same
way. For example if A belongs to all B, C being middle, then if it
is supposed that A does not belong to all B or belongs to no B, but
to all C (which was admitted to be true), it follows that C belongs
to no B or not to all B. But this is impossible: consequently the
supposition is false: its contradictory then is true. Similarly in
the other figures: for whatever moods admit of conversion admit also
of the reduction per impossibile.
All the problems can be proved per impossibile in all the figures,
excepting the universal affirmative, which is proved in the middle
and third figures, but not in the first. Suppose that A belongs not
to all B, or to no B, and take besides another premiss concerning
either of the terms, viz. that C belongs to all A, or that B belongs
to all D; thus we get the first figure. If then it is supposed that
A does not belong to all B, no syllogism results whichever term the
assumed premiss concerns; but if it is supposed that A belongs to
no B, when the premiss BD is assumed as well we shall prove syllogistically
what is false, but not the problem proposed. For if A belongs to no
B, and B belongs to all D, A belongs to no D. Let this be impossible:
it is false then A belongs to no B. But the universal affirmative
is not necessarily true if the universal negative is false. But if
the premiss CA is assumed as well, no syllogism results, nor does
it do so when it is supposed that A does not belong to all B. Consequently
it is clear that the universal affirmative cannot be proved in the
first figure per impossibile.
But the particular affirmative and the universal and particular negatives
can all be proved. Suppose that A belongs to no B, and let it have
been assumed that B belongs to all or to some C. Then it is necessary
that A should belong to no C or not to all C. But this is impossible
(for let it be true and clear that A belongs to all C): consequently
if this is false, it is necessary that A should belong to some B.
But if the other premiss assumed relates to A, no syllogism will be
possible. Nor can a conclusion be drawn when the contrary of the conclusion
is supposed, e.g. that A does not belong to some B. Clearly then we
must suppose the contradictory.
Again suppose that A belongs to some B, and let it have been assumed
that C belongs to all A. It is necessary then that C should belong
to some B. But let this be impossible, so that the supposition is
false: in that case it is true that A belongs to no B. We may proceed
in the same way if the proposition CA has been taken as negative.
But if the premiss assumed concerns B, no syllogism will be possible.
If the contrary is supposed, we shall have a syllogism and an impossible
conclusion, but the problem in hand is not proved. Suppose that A
belongs to all B, and let it have been assumed that C belongs to all
A. It is necessary then that C should belong to all B. But this is
impossible, so that it is false that A belongs to all B. But we have
not yet shown it to be necessary that A belongs to no B, if it does
not belong to all B. Similarly if the other premiss taken concerns
B; we shall have a syllogism and a conclusion which is impossible,
but the hypothesis is not refuted. Therefore it is the contradictory
that we must suppose.
To prove that A does not belong to all B, we must suppose that it
belongs to all B: for if A belongs to all B, and C to all A, then
C belongs to all B; so that if this is impossible, the hypothesis
is false. Similarly if the other premiss assumed concerns B. The same
results if the original proposition CA was negative: for thus also
we get a syllogism. But if the negative proposition concerns B, nothing
is proved. If the hypothesis is that A belongs not to all but to some
B, it is not proved that A belongs not to all B, but that it belongs
to no B. For if A belongs to some B, and C to all A, then C will belong
to some B. If then this is impossible, it is false that A belongs
to some B; consequently it is true that A belongs to no B. But if
this is proved, the truth is refuted as well; for the original conclusion
was that A belongs to some B, and does not belong to some B. Further
the impossible does not result from the hypothesis: for then the hypothesis
would be false, since it is impossible to draw a false conclusion
from true premisses: but in fact it is true: for A belongs to some
B. Consequently we must not suppose that A belongs to some B, but
that it belongs to all B. Similarly if we should be proving that A
does not belong to some B: for if 'not to belong to some' and 'to
belong not to all' have the same meaning, the demonstration of both
will be identical.
It is clear then that not the contrary but the contradictory ought
to be supposed in all the syllogisms. For thus we shall have necessity
of inference, and the claim we make is one that will be generally
accepted. For if of everything one or other of two contradictory statements
holds good, then if it is proved that the negation does not hold,
the affirmation must be true. Again if it is not admitted that the
affirmation is true, the claim that the negation is true will be generally
accepted. But in neither way does it suit to maintain the contrary:
for it is not necessary that if the universal negative is false, the
universal affirmative should be true, nor is it generally accepted
that if the one is false the other is true.
Part 12
It is clear then that in the first figure all problems except the
universal affirmative are proved per impossibile. But in the middle
and the last figures this also is proved. Suppose that A does not
belong to all B, and let it have been assumed that A belongs to all
C. If then A belongs not to all B, but to all C, C will not belong
to all B. But this is impossible (for suppose it to be clear that
C belongs to all B): consequently the hypothesis is false. It is true
then that A belongs to all B. But if the contrary is supposed, we
shall have a syllogism and a result which is impossible: but the problem
in hand is not proved. For if A belongs to no B, and to all C, C will
belong to no B. This is impossible; so that it is false that A belongs
to no B. But though this is false, it does not follow that it is true
that A belongs to all B.
When A belongs to some B, suppose that A belongs to no B, and let
A belong to all C. It is necessary then that C should belong to no
B. Consequently, if this is impossible, A must belong to some B. But
if it is supposed that A does not belong to some B, we shall have
the same results as in the first figure.
Again suppose that A belongs to some B, and let A belong to no C.
It is necessary then that C should not belong to some B. But originally
it belonged to all B, consequently the hypothesis is false: A then
will belong to no B.
When A does not belong to an B, suppose it does belong to all B, and
to no C. It is necessary then that C should belong to no B. But this
is impossible: so that it is true that A does not belong to all B.
It is clear then that all the syllogisms can be formed in the middle
figure.
Part 13
Similarly they can all be formed in the last figure. Suppose that
A does not belong to some B, but C belongs to all B: then A does not
belong to some C. If then this is impossible, it is false that A does
not belong to some B; so that it is true that A belongs to all B.
But if it is supposed that A belongs to no B, we shall have a syllogism
and a conclusion which is impossible: but the problem in hand is not
proved: for if the contrary is supposed, we shall have the same results
as before.
But to prove that A belongs to some B, this hypothesis must be made.
If A belongs to no B, and C to some B, A will belong not to all C.
If then this is false, it is true that A belongs to some B.
When A belongs to no B, suppose A belongs to some B, and let it have
been assumed that C belongs to all B. Then it is necessary that A
should belong to some C. But ex hypothesi it belongs to no C, so that
it is false that A belongs to some B. But if it is supposed that A
belongs to all B, the problem is not proved.
But this hypothesis must be made if we are prove that A belongs not
to all B. For if A belongs to all B and C to some B, then A belongs
to some C. But this we assumed not to be so, so it is false that A
belongs to all B. But in that case it is true that A belongs not to
all B. If however it is assumed that A belongs to some B, we shall
have the same result as before.
It is clear then that in all the syllogisms which proceed per impossibile
the contradictory must be assumed. And it is plain that in the middle
figure an affirmative conclusion, and in the last figure a universal
conclusion, are proved in a way.
Part 14
Demonstration per impossibile differs from ostensive proof in that
it posits what it wishes to refute by reduction to a statement admitted
to be false; whereas ostensive proof starts from admitted positions.
Both, indeed, take two premisses that are admitted, but the latter
takes the premisses from which the syllogism starts, the former takes
one of these, along with the contradictory of the original conclusion.
Also in the ostensive proof it is not necessary that the conclusion
should be known, nor that one should suppose beforehand that it is
true or not: in the other it is necessary to suppose beforehand that
it is not true. It makes no difference whether the conclusion is affirmative
or negative; the method is the same in both cases. Everything which
is concluded ostensively can be proved per impossibile, and that which
is proved per impossibile can be proved ostensively, through the same
terms. Whenever the syllogism is formed in the first figure, the truth
will be found in the middle or the last figure, if negative in the
middle, if affirmative in the last. Whenever the syllogism is formed
in the middle figure, the truth will be found in the first, whatever
the problem may be. Whenever the syllogism is formed in the last figure,
the truth will be found in the first and middle figures, if affirmative
in first, if negative in the middle. Suppose that A has been proved
to belong to no B, or not to all B, through the first figure. Then
the hypothesis must have been that A belongs to some B, and the original
premisses that C belongs to all A and to no B. For thus the syllogism
was made and the impossible conclusion reached. But this is the middle
figure, if C belongs to all A and to no B. And it is clear from these
premisses that A belongs to no B. Similarly if has been proved not
to belong to all B. For the hypothesis is that A belongs to all B;
and the original premisses are that C belongs to all A but not to
all B. Similarly too, if the premiss CA should be negative: for thus
also we have the middle figure. Again suppose it has been proved that
A belongs to some B. The hypothesis here is that is that A belongs
to no B; and the original premisses that B belongs to all C, and A
either to all or to some C: for in this way we shall get what is impossible.
But if A and B belong to all C, we have the last figure. And it is
clear from these premisses that A must belong to some B. Similarly
if B or A should be assumed to belong to some C.
Again suppose it has been proved in the middle figure that A belongs
to all B. Then the hypothesis must have been that A belongs not to
all B, and the original premisses that A belongs to all C, and C to
all B: for thus we shall get what is impossible. But if A belongs
to all C, and C to all B, we have the first figure. Similarly if it
has been proved that A belongs to some B: for the hypothesis then
must have been that A belongs to no B, and the original premisses
that A belongs to all C, and C to some B. If the syllogism is negative,
the hypothesis must have been that A belongs to some B, and the original
premisses that A belongs to no C, and C to all B, so that the first
figure results. If the syllogism is not universal, but proof has been
given that A does not belong to some B, we may infer in the same way.
The hypothesis is that A belongs to all B, the original premisses
that A belongs to no C, and C belongs to some B: for thus we get the
first figure.
Again suppose it has been proved in the third figure that A belongs
to all B. Then the hypothesis must have been that A belongs not to
all B, and the original premisses that C belongs to all B, and A belongs
to all C; for thus we shall get what is impossible. And the original
premisses form the first figure. Similarly if the demonstration establishes
a particular proposition: the hypothesis then must have been that
A belongs to no B, and the original premisses that C belongs to some
B, and A to all C. If the syllogism is negative, the hypothesis must
have been that A belongs to some B, and the original premisses that
C belongs to no A and to all B, and this is the middle figure. Similarly
if the demonstration is not universal. The hypothesis will then be
that A belongs to all B, the premisses that C belongs to no A and
to some B: and this is the middle figure.
It is clear then that it is possible through the same terms to prove
each of the problems ostensively as well. Similarly it will be possible
if the syllogisms are ostensive to reduce them ad impossibile in the
terms which have been taken, whenever the contradictory of the conclusion
of the ostensive syllogism is taken as a premiss. For the syllogisms
become identical with those which are obtained by means of conversion,
so that we obtain immediately the figures through which each problem
will be solved. It is clear then that every thesis can be proved in
both ways, i.e. per impossibile and ostensively, and it is not possible
to separate one method from the other.
Part 15
In what figure it is possible to draw a conclusion from premisses
which are opposed, and in what figure this is not possible, will be
made clear in this way. Verbally four kinds of opposition are possible,
viz. universal affirmative to universal negative, universal affirmative
to particular negative, particular affirmative to universal negative,
and particular affirmative to particular negative: but really there
are only three: for the particular affirmative is only verbally opposed
to the particular negative. Of the genuine opposites I call those
which are universal contraries, the universal affirmative and the
universal negative, e.g. 'every science is good', 'no science is good';
the others I call contradictories.
In the first figure no syllogism whether affirmative or negative can
be made out of opposed premisses: no affirmative syllogism is possible
because both premisses must be affirmative, but opposites are, the
one affirmative, the other negative: no negative syllogism is possible
because opposites affirm and deny the same predicate of the same subject,
and the middle term in the first figure is not predicated of both
extremes, but one thing is denied of it, and it is affirmed of something
else: but such premisses are not opposed.
In the middle figure a syllogism can be made both oLcontradictories
and of contraries. Let A stand for good, let B and C stand for science.
If then one assumes that every science is good, and no science is
good, A belongs to all B and to no C, so that B belongs to no C: no
science then is a science. Similarly if after taking 'every science
is good' one took 'the science of medicine is not good'; for A belongs
to all B but to no C, so that a particular science will not be a science.
Again, a particular science will not be a science if A belongs to
all C but to no B, and B is science, C medicine, and A supposition:
for after taking 'no science is supposition', one has assumed that
a particular science is supposition. This syllogism differs from the
preceding because the relations between the terms are reversed: before,
the affirmative statement concerned B, now it concerns C. Similarly
if one premiss is not universal: for the middle term is always that
which is stated negatively of one extreme, and affirmatively of the
other. Consequently it is possible that contradictories may lead to
a conclusion, though not always or in every mood, but only if the
terms subordinate to the middle are such that they are either identical
or related as whole to part. Otherwise it is impossible: for the premisses
cannot anyhow be either contraries or contradictories.
In the third figure an affirmative syllogism can never be made out
of opposite premisses, for the reason given in reference to the first
figure; but a negative syllogism is possible whether the terms are
universal or not. Let B and C stand for science, A for medicine. If
then one should assume that all medicine is science and that no medicine
is science, he has assumed that B belongs to all A and C to no A,
so that a particular science will not be a science. Similarly if the
premiss BA is not assumed universally. For if some medicine is science
and again no medicine is science, it results that some science is
not science, The premisses are contrary if the terms are taken universally;
if one is particular, they are contradictory.
We must recognize that it is possible to take opposites in the way
we said, viz. 'all science is good' and 'no science is good' or 'some
science is not good'. This does not usually escape notice. But it
is possible to establish one part of a contradiction through other
premisses, or to assume it in the way suggested in the Topics. Since
there are three oppositions to affirmative statements, it follows
that opposite statements may be assumed as premisses in six ways;
we may have either universal affirmative and negative, or universal
affirmative and particular negative, or particular affirmative and
universal negative, and the relations between the terms may be reversed;
e.g. A may belong to all B and to no C, or to all C and to no B, or
to all of the one, not to all of the other; here too the relation
between the terms may be reversed. Similarly in the third figure.
So it is clear in how many ways and in what figures a syllogism can
be made by means of premisses which are opposed.
It is clear too that from false premisses it is possible to draw a
true conclusion, as has been said before, but it is not possible if
the premisses are opposed. For the syllogism is always contrary to
the fact, e.g. if a thing is good, it is proved that it is not good,
if an animal, that it is not an animal because the syllogism springs
out of a contradiction and the terms presupposed are either identical
or related as whole and part. It is evident also that in fallacious
reasonings nothing prevents a contradiction to the hypothesis from
resulting, e.g. if something is odd, it is not odd. For the syllogism
owed its contrariety to its contradictory premisses; if we assume
such premisses we shall get a result that contradicts our hypothesis.
But we must recognize that contraries cannot be inferred from a single
syllogism in such a way that we conclude that what is not good is
good, or anything of that sort unless a self-contradictory premiss
is at once assumed, e.g. 'every animal is white and not white', and
we proceed 'man is an animal'. Either we must introduce the contradiction
by an additional assumption, assuming, e.g., that every science is
supposition, and then assuming 'Medicine is a science, but none of
it is supposition' (which is the mode in which refutations are made),
or we must argue from two syllogisms. In no other way than this, as
was said before, is it possible that the premisses should be really
contrary.
Part 16
To beg and assume the original question is a species of failure to
demonstrate the problem proposed; but this happens in many ways. A
man may not reason syllogistically at all, or he may argue from premisses
which are less known or equally unknown, or he may establish the antecedent
by means of its consequents; for demonstration proceeds from what
is more certain and is prior. Now begging the question is none of
these: but since we get to know some things naturally through themselves,
and other things by means of something else (the first principles
through themselves, what is subordinate to them through something
else), whenever a man tries to prove what is not self-evident by means
of itself, then he begs the original question. This may be done by
assuming what is in question at once; it is also possible to make
a transition to other things which would naturally be proved through
the thesis proposed, and demonstrate it through them, e.g. if A should
be proved through B, and B through C, though it was natural that C
should be proved through A: for it turns out that those who reason
thus are proving A by means of itself. This is what those persons
do who suppose that they are constructing parallel straight lines:
for they fail to see that they are assuming facts which it is impossible
to demonstrate unless the parallels exist. So it turns out that those
who reason thus merely say a particular thing is, if it is: in this
way everything will be self-evident. But that is impossible.
If then it is uncertain whether A belongs to C, and also whether A
belongs to B, and if one should assume that A does belong to B, it
is not yet clear whether he begs the original question, but it is
evident that he is not demonstrating: for what is as uncertain as
the question to be answered cannot be a principle of a demonstration.
If however B is so related to C that they are identical, or if they
are plainly convertible, or the one belongs to the other, the original
question is begged. For one might equally well prove that A belongs
to B through those terms if they are convertible. But if they are
not convertible, it is the fact that they are not that prevents such
a demonstration, not the method of demonstrating. But if one were
to make the conversion, then he would be doing what we have described
and effecting a reciprocal proof with three propositions.
Similarly if he should assume that B belongs to C, this being as uncertain
as the question whether A belongs to C, the question is not yet begged,
but no demonstration is made. If however A and B are identical either
because they are convertible or because A follows B, then the question
is begged for the same reason as before. For we have explained the
meaning of begging the question, viz. proving that which is not self-evident
by means of itself.
If then begging the question is proving what is not self-evident by
means of itself, in other words failing to prove when the failure
is due to the thesis to be proved and the premiss through which it
is proved being equally uncertain, either because predicates which
are identical belong to the same subject, or because the same predicate
belongs to subjects which are identical, the question may be begged
in the middle and third figures in both ways, though, if the syllogism
is affirmative, only in the third and first figures. If the syllogism
is negative, the question is begged when identical predicates are
denied of the same subject; and both premisses do not beg the question
indifferently (in a similar way the question may be begged in the
middle figure), because the terms in negative syllogisms are not convertible.
In scientific demonstrations the question is begged when the terms
are really related in the manner described, in dialectical arguments
when they are according to common opinion so related.
Part 17
The objection that 'this is not the reason why the result is false',
which we frequently make in argument, is made primarily in the case
of a reductio ad impossibile, to rebut the proposition which was being
proved by the reduction. For unless a man has contradicted this proposition
he will not say, 'False cause', but urge that something false has
been assumed in the earlier parts of the argument; nor will he use
the formula in the case of an ostensive proof; for here what one denies
is not assumed as a premiss. Further when anything is refuted ostensively
by the terms ABC, it cannot be objected that the syllogism does not
depend on the assumption laid down. For we use the expression 'false
cause', when the syllogism is concluded in spite of the refutation
of this position; but that is not possible in ostensive proofs: since
if an assumption is refuted, a syllogism can no longer be drawn in
reference to it. It is clear then that the expression 'false cause'
can only be used in the case of a reductio ad impossibile, and when
the original hypothesis is so related to the impossible conclusion,
that the conclusion results indifferently whether the hypothesis is
made or not. The most obvious case of the irrelevance of an assumption
to a conclusion which is false is when a syllogism drawn from middle
terms to an impossible conclusion is independent of the hypothesis,
as we have explained in the Topics. For to put that which is not the
cause as the cause, is just this: e.g. if a man, wishing to prove
that the diagonal of the square is incommensurate with the side, should
try to prove Zeno's theorem that motion is impossible, and so establish
a reductio ad impossibile: for Zeno's false theorem has no connexion
at all with the original assumption. Another case is where the impossible
conclusion is connected with the hypothesis, but does not result from
it. This may happen whether one traces the connexion upwards or downwards,
e.g. if it is laid down that A belongs to B, B to C, and C to D, and
it should be false that B belongs to D: for if we eliminated A and
assumed all the same that B belongs to C and C to D, the false conclusion
would not depend on the original hypothesis. Or again trace the connexion
upwards; e.g. suppose that A belongs to B, E to A and F to E, it being
false that F belongs to A. In this way too the impossible conclusion
would result, though the original hypothesis were eliminated. But
the impossible conclusion ought to be connected with the original
terms: in this way it will depend on the hypothesis, e.g. when one
traces the connexion downwards, the impossible conclusion must be
connected with that term which is predicate in the hypothesis: for
if it is impossible that A should belong to D, the false conclusion
will no longer result after A has been eliminated. If one traces the
connexion upwards, the impossible conclusion must be connected with
that term which is subject in the hypothesis: for if it is impossible
that F should belong to B, the impossible conclusion will disappear
if B is eliminated. Similarly when the syllogisms are negative.
It is clear then that when the impossibility is not related to the
original terms, the false conclusion does not result on account of
the assumption. Or perhaps even so it may sometimes be independent.
For if it were laid down that A belongs not to B but to K, and that
K belongs to C and C to D, the impossible conclusion would still stand.
Similarly if one takes the terms in an ascending series. Consequently
since the impossibility results whether the first assumption is suppressed
or not, it would appear to be independent of that assumption. Or perhaps
we ought not to understand the statement that the false conclusion
results independently of the assumption, in the sense that if something
else were supposed the impossibility would result; but rather we mean
that when the first assumption is eliminated, the same impossibility
results through the remaining premisses; since it is not perhaps absurd
that the same false result should follow from several hypotheses,
e.g. that parallels meet, both on the assumption that the interior
angle is greater than the exterior and on the assumption that a triangle
contains more than two right angles.
Part 18
A false argument depends on the first false statement in it. Every
syllogism is made out of two or more premisses. If then the false
conclusion is drawn from two premisses, one or both of them must be
false: for (as we proved) a false syllogism cannot be drawn from two
premisses. But if the premisses are more than two, e.g. if C is established
through A and B, and these through D, E, F, and G, one of these higher
propositions must be false, and on this the argument depends: for
A and B are inferred by means of D, E, F, and G. Therefore the conclusion
and the error results from one of them.
Part 19
In order to avoid having a syllogism drawn against us we must take
care, whenever an opponent asks us to admit the reason without the
conclusions, not to grant him the same term twice over in his premisses,
since we know that a syllogism cannot be drawn without a middle term,
and that term which is stated more than once is the middle. How we
ought to watch the middle in reference to each conclusion, is evident
from our knowing what kind of thesis is proved in each figure. This
will not escape us since we know how we are maintaining the argument.
That which we urge men to beware of in their admissions, they ought
in attack to try to conceal. This will be possible first, if, instead
of drawing the conclusions of preliminary syllogisms, they take the
necessary premisses and leave the conclusions in the dark; secondly
if instead of inviting assent to propositions which are closely connected
they take as far as possible those that are not connected by middle
terms. For example suppose that A is to be inferred to be true of
F, B, C, D, and E being middle terms. One ought then to ask whether
A belongs to B, and next whether D belongs to E, instead of asking
whether B belongs to C; after that he may ask whether B belongs to
C, and so on. If the syllogism is drawn through one middle term, he
ought to begin with that: in this way he will most likely deceive
his opponent.
Part 20
Since we know when a syllogism can be formed and how its terms must
be related, it is clear when refutation will be possible and when
impossible. A refutation is possible whether everything is conceded,
or the answers alternate (one, I mean, being affirmative, the other
negative). For as has been shown a syllogism is possible whether the
terms are related in affirmative propositions or one proposition is
affirmative, the other negative: consequently, if what is laid down
is contrary to the conclusion, a refutation must take place: for a
refutation is a syllogism which establishes the contradictory. But
if nothing is conceded, a refutation is impossible: for no syllogism
is possible (as we saw) when all the terms are negative: therefore
no refutation is possible. For if a refutation were possible, a syllogism
must be possible; although if a syllogism is possible it does not
follow that a refutation is possible. Similarly refutation is not
possible if nothing is conceded universally: since the fields of refutation
and syllogism are defined in the same way.
Part 21
It sometimes happens that just as we are deceived in the arrangement
of the terms, so error may arise in our thought about them, e.g. if
it is possible that the same predicate should belong to more than
one subject immediately, but although knowing the one, a man may forget
the other and think the opposite true. Suppose that A belongs to B
and to C in virtue of their nature, and that B and C belong to all
D in the same way. If then a man thinks that A belongs to all B, and
B to D, but A to no C, and C to all D, he will both know and not know
the same thing in respect of the same thing. Again if a man were to
make a mistake about the members of a single series; e.g. suppose
A belongs to B, B to C, and C to D, but some one thinks that A belongs
to all B, but to no C: he will both know that A belongs to D, and
think that it does not. Does he then maintain after this simply that
what he knows, he does not think? For he knows in a way that A belongs
to C through B, since the part is included in the whole; so that what
he knows in a way, this he maintains he does not think at all: but
that is impossible.
In the former case, where the middle term does not belong to the same
series, it is not possible to think both the premisses with reference
to each of the two middle terms: e.g. that A belongs to all B, but
to no C, and both B and C belong to all D. For it turns out that the
first premiss of the one syllogism is either wholly or partially contrary
to the first premiss of the other. For if he thinks that A belongs
to everything to which B belongs, and he knows that B belongs to D,
then he knows that A belongs to D. Consequently if again he thinks
that A belongs to nothing to which C belongs, he thinks that A does
not belong to some of that to which B belongs; but if he thinks that
A belongs to everything to which B belongs, and again thinks that
A does not belong to some of that to which B belongs, these beliefs
are wholly or partially contrary. In this way then it is not possible
to think; but nothing prevents a man thinking one premiss of each
syllogism of both premisses of one of the two syllogisms: e.g. A belongs
to all B, and B to D, and again A belongs to no C. An error of this
kind is similar to the error into which we fall concerning particulars:
e.g. if A belongs to all B, and B to all C, A will belong to all C.
If then a man knows that A belongs to everything to which B belongs,
he knows that A belongs to C. But nothing prevents his being ignorant
that C exists; e.g. let A stand for two right angles, B for triangle,
C for a particular diagram of a triangle. A man might think that C
did not exist, though he knew that every triangle contains two right
angles; consequently he will know and not know the same thing at the
same time. For the expression 'to know that every triangle has its
angles equal to two right angles' is ambiguous, meaning to have the
knowledge either of the universal or of the particulars. Thus then
he knows that C contains two right angles with a knowledge of the
universal, but not with a knowledge of the particulars; consequently
his knowledge will not be contrary to his ignorance. The argument
in the Meno that learning is recollection may be criticized in a similar
way. For it never happens that a man starts with a foreknowledge of
the particular, but along with the process of being led to see the
general principle he receives a knowledge of the particulars, by an
act (as it were) of recognition. For we know some things directly;
e.g. that the angles are equal to two right angles, if we know that
the figure is a triangle. Similarly in all other cases.
By a knowledge of the universal then we see the particulars, but we
do not know them by the kind of knowledge which is proper to them;
consequently it is possible that we may make mistakes about them,
but not that we should have the knowledge and error that are contrary
to one another: rather we have the knowledge of the universal but
make a mistake in apprehending the particular. Similarly in the cases
stated above. The error in respect of the middle term is not contrary
to the knowledge obtained through the syllogism, nor is the thought
in respect of one middle term contrary to that in respect of the other.
Nothing prevents a man who knows both that A belongs to the whole
of B, and that B again belongs to C, thinking that A does not belong
to C, e.g. knowing that every mule is sterile and that this is a mule,
and thinking that this animal is with foal: for he does not know that
A belongs to C, unless he considers the two propositions together.
So it is evident that if he knows the one and does not know the other,
he will fall into error. And this is the relation of knowledge of
the universal to knowledge of the particular. For we know no sensible
thing, once it has passed beyond the range of our senses, even if
we happen to have perceived it, except by means of the universal and
the possession of the knowledge which is proper to the particular,
but without the actual exercise of that knowledge. For to know is
used in three senses: it may mean either to have knowledge of the
universal or to have knowledge proper to the matter in hand or to
exercise such knowledge: consequently three kinds of error also are
possible. Nothing then prevents a man both knowing and being mistaken
about the same thing, provided that his knowledge and his error are
not contrary. And this happens also to the man whose knowledge is
limited to each of the premisses and who has not previously considered
the particular question. For when he thinks that the mule is with
foal he has not the knowledge in the sense of its actual exercise,
nor on the other hand has his thought caused an error contrary to
his knowledge: for the error contrary to the knowledge of the universal
would be a syllogism.
But he who thinks the essence of good is the essence of bad will think
the same thing to be the essence of good and the essence of bad. Let
A stand for the essence of good and B for the essence of bad, and
again C for the essence of good. Since then he thinks B and C identical,
he will think that C is B, and similarly that B is A, consequently
that C is A. For just as we saw that if B is true of all of which
C is true, and A is true of all of which B is true, A is true of C,
similarly with the word 'think'. Similarly also with the word 'is';
for we saw that if C is the same as B, and B as A, C is the same as
A. Similarly therefore with 'opine'. Perhaps then this is necessary
if a man will grant the first point. But presumably that is false,
that any one could suppose the essence of good to be the essence of
bad, save incidentally. For it is possible to think this in many different
ways. But we must consider this matter better.
Part 22
Whenever the extremes are convertible it is necessary that the middle
should be convertible with both. For if A belongs to C through B,
then if A and C are convertible and C belongs everything to which
A belongs, B is convertible with A, and B belongs to everything to
which A belongs, through C as middle, and C is convertible with B
through A as middle. Similarly if the conclusion is negative, e.g.
if B belongs to C, but A does not belong to B, neither will A belong
to C. If then B is convertible with A, C will be convertible with
A. Suppose B does not belong to A; neither then will C: for ex hypothesi
B belonged to all C. And if C is convertible with B, B is convertible
also with A, for C is said of that of all of which B is said. And
if C is convertible in relation to A and to B, B also is convertible
in relation to A. For C belongs to that to which B belongs: but C
does not belong to that to which A belongs. And this alone starts
from the conclusion; the preceding moods do not do so as in the affirmative
syllogism. Again if A and B are convertible, and similarly C and D,
and if A or C must belong to anything whatever, then B and D will
be such that one or other belongs to anything whatever. For since
B belongs to that to which A belongs, and D belongs to that to which
C belongs, and since A or C belongs to everything, but not together,
it is clear that B or D belongs to everything, but not together. For
example if that which is uncreated is incorruptible and that which
is incorruptible is uncreated, it is necessary that what is created
should be corruptible and what is corruptible should have been created.
For two syllogisms have been put together. Again if A or B belongs
to everything and if C or D belongs to everything, but they cannot
belong together, then when A and C are convertible B and D are convertible.
For if B does not belong to something to which D belongs, it is clear
that A belongs to it. But if A then C: for they are convertible. Therefore
C and D belong together. But this is impossible. When A belongs to
the whole of B and to C and is affirmed of nothing else, and B also
belongs to all C, it is necessary that A and B should be convertible:
for since A is said of B and C only, and B is affirmed both of itself
and of C, it is clear that B will be said of everything of which A
is said, except A itself. Again when A and B belong to the whole of
C, and C is convertible with B, it is necessary that A should belong
to all B: for since A belongs to all C, and C to B by conversion,
A will belong to all B.
When, of two opposites A and B, A is preferable to B, and similarly
D is preferable to C, then if A and C together are preferable to B
and D together, A must be preferable to D. For A is an object of desire
to the same extent as B is an object of aversion, since they are opposites:
and C is similarly related to D, since they also are opposites. If
then A is an object of desire to the same extent as D, B is an object
of aversion to the same extent as C (since each is to the same extent
as each-the one an object of aversion, the other an object of desire).
Therefore both A and C together, and B and D together, will be equally
objects of desire or aversion. But since A and C are preferable to
B and D, A cannot be equally desirable with D; for then B along with
D would be equally desirable with A along with C. But if D is preferable
to A, then B must be less an object of aversion than C: for the less
is opposed to the less. But the greater good and lesser evil are preferable
to the lesser good and greater evil: the whole BD then is preferable
to the whole AC. But ex hypothesi this is not so. A then is preferable
to D, and C consequently is less an object of aversion than B. If
then every lover in virtue of his love would prefer A, viz. that the
beloved should be such as to grant a favour, and yet should not grant
it (for which C stands), to the beloved's granting the favour (represented
by D) without being such as to grant it (represented by B), it is
clear that A (being of such a nature) is preferable to granting the
favour. To receive affection then is preferable in love to sexual
intercourse. Love then is more dependent on friendship than on intercourse.
And if it is most dependent on receiving affection, then this is its
end. Intercourse then either is not an end at all or is an end relative
to the further end, the receiving of affection. And indeed the same
is true of the other desires and arts.
Part 23
It is clear then how the terms are related in conversion, and in respect
of being in a higher degree objects of aversion or of desire. We must
now state that not only dialectical and demonstrative syllogisms are
formed by means of the aforesaid figures, but also rhetorical syllogisms
and in general any form of persuasion, however it may be presented.
For every belief comes either through syllogism or from induction.
Now induction, or rather the syllogism which springs out of induction,
consists in establishing syllogistically a relation between one extreme
and the middle by means of the other extreme, e.g. if B is the middle
term between A and C, it consists in proving through C that A belongs
to B. For this is the manner in which we make inductions. For example
let A stand for long-lived, B for bileless, and C for the particular
long-lived animals, e.g. man, horse, mule. A then belongs to the whole
of C: for whatever is bileless is long-lived. But B also ('not possessing
bile') belongs to all C. If then C is convertible with B, and the
middle term is not wider in extension, it is necessary that A should
belong to B. For it has already been proved that if two things belong
to the same thing, and the extreme is convertible with one of them,
then the other predicate will belong to the predicate that is converted.
But we must apprehend C as made up of all the particulars. For induction
proceeds through an enumeration of all the cases.
Such is the syllogism which establishes the first and immediate premiss:
for where there is a middle term the syllogism proceeds through the
middle term; when there is no middle term, through induction. And
in a way induction is opposed to syllogism: for the latter proves
the major term to belong to the third term by means of the middle,
the former proves the major to belong to the middle by means of the
third. In the order of nature, syllogism through the middle term is
prior and better known, but syllogism through induction is clearer
to us.
Part 24
We have an 'example' when the major term is proved to belong to the
middle by means of a term which resembles the third. It ought to be
known both that the middle belongs to the third term, and that the
first belongs to that which resembles the third. For example let A
be evil, B making war against neighbours, C Athenians against Thebans,
D Thebans against Phocians. If then we wish to prove that to fight
with the Thebans is an evil, we must assume that to fight against
neighbours is an evil. Evidence of this is obtained from similar cases,
e.g. that the war against the Phocians was an evil to the Thebans.
Since then to fight against neighbours is an evil, and to fight against
the Thebans is to fight against neighbours, it is clear that to fight
against the Thebans is an evil. Now it is clear that B belongs to
C and to D (for both are cases of making war upon one's neighbours)
and that A belongs to D (for the war against the Phocians did not
turn out well for the Thebans): but that A belongs to B will be proved
through D. Similarly if the belief in the relation of the middle term
to the extreme should be produced by several similar cases. Clearly
then to argue by example is neither like reasoning from part to whole,
nor like reasoning from whole to part, but rather reasoning from part
to part, when both particulars are subordinate to the same term, and
one of them is known. It differs from induction, because induction
starting from all the particular cases proves (as we saw) that the
major term belongs to the middle, and does not apply the syllogistic
conclusion to the minor term, whereas argument by example does make
this application and does not draw its proof from all the particular
cases.
Part 25
By reduction we mean an argument in which the first term clearly belongs
to the middle, but the relation of the middle to the last term is
uncertain though equally or more probable than the conclusion; or
again an argument in which the terms intermediate between the last
term and the middle are few. For in any of these cases it turns out
that we approach more nearly to knowledge. For example let A stand
for what can be taught, B for knowledge, C for justice. Now it is
clear that knowledge can be taught: but it is uncertain whether virtue
is knowledge. If now the statement BC is equally or more probable
than Ac, we have a reduction: for we are nearer to knowledge, since
we have taken a new term, being so far without knowledge that A belongs
to C. Or again suppose that the terms intermediate between B and C
are few: for thus too we are nearer knowledge. For example let D stand
for squaring, E for rectilinear figure, F for circle. If there were
only one term intermediate between E and F (viz. that the circle is
made equal to a rectilinear figure by the help of lunules), we should
be near to knowledge. But when BC is not more probable than AC, and
the intermediate terms are not few, I do not call this reduction:
nor again when the statement BC is immediate: for such a statement
is knowledge.
Part 26
An objection is a premiss contrary to a premiss. It differs from a
premiss, because it may be particular, but a premiss either cannot
be particular at all or not in universal syllogisms. An objection
is brought in two ways and through two figures; in two ways because
every objection is either universal or particular, by two figures
because objections are brought in opposition to the premiss, and opposites
can be proved only in the first and third figures. If a man maintains
a universal affirmative, we reply with a universal or a particular
negative; the former is proved from the first figure, the latter from
the third. For example let stand for there being a single science,
B for contraries. If a man premises that contraries are subjects of
a single science, the objection may be either that opposites are never
subjects of a single science, and contraries are opposites, so that
we get the first figure, or that the knowable and the unknowable are
not subjects of a single science: this proof is in the third figure:
for it is true of C (the knowable and the unknowable) that they are
contraries, and it is false that they are the subjects of a single
science.
Similarly if the premiss objected to is negative. For if a man maintains
that contraries are not subjects of a single science, we reply either
that all opposites or that certain contraries, e.g. what is healthy
and what is sickly, are subjects of the same science: the former argument
issues from the first, the latter from the third figure.
In general if a man urges a universal objection he must frame his
contradiction with reference to the universal of the terms taken by
his opponent, e.g. if a man maintains that contraries are not subjects
of the same science, his opponent must reply that there is a single
science of all opposites. Thus we must have the first figure: for
the term which embraces the original subject becomes the middle term.
If the objection is particular, the objector must frame his contradiction
with reference to a term relatively to which the subject of his opponent's
premiss is universal, e.g. he will point out that the knowable and
the unknowable are not subjects of the same science: 'contraries'
is universal relatively to these. And we have the third figure: for
the particular term assumed is middle, e.g. the knowable and the unknowable.
Premisses from which it is possible to draw the contrary conclusion
are what we start from when we try to make objections. Consequently
we bring objections in these figures only: for in them only are opposite
syllogisms possible, since the second figure cannot produce an affirmative
conclusion.
Besides, an objection in the middle figure would require a fuller
argument, e.g. if it should not be granted that A belongs to B, because
C does not follow B. This can be made clear only by other premisses.
But an objection ought not to turn off into other things, but have
its new premiss quite clear immediately. For this reason also this
is the only figure from which proof by signs cannot be obtained.
We must consider later the other kinds of objection, namely the objection
from contraries, from similars, and from common opinion, and inquire
whether a particular objection cannot be elicited from the first figure
or a negative objection from the second.
Part 27
A probability and a sign are not identical, but a probability is a
generally approved proposition: what men know to happen or not to
happen, to be or not to be, for the most part thus and thus, is a
probability, e.g. 'the envious hate', 'the beloved show affection'.
A sign means a demonstrative proposition necessary or generally approved:
for anything such that when it is another thing is, or when it has
come into being the other has come into being before or after, is
a sign of the other's being or having come into being. Now an enthymeme
is a syllogism starting from probabilities or signs, and a sign may
be taken in three ways, corresponding to the position of the middle
term in the figures. For it may be taken as in the first figure or
the second or the third. For example the proof that a woman is with
child because she has milk is in the first figure: for to have milk
is the middle term. Let A represent to be with child, B to have milk,
C woman. The proof that wise men are good, since Pittacus is good,
comes through the last figure. Let A stand for good, B for wise men,
C for Pittacus. It is true then to affirm both A and B of C: only
men do not say the latter, because they know it, though they state
the former. The proof that a woman is with child because she is pale
is meant to come through the middle figure: for since paleness follows
women with child and is a concomitant of this woman, people suppose
it has been proved that she is with child. Let A stand for paleness,
B for being with child, C for woman. Now if the one proposition is
stated, we have only a sign, but if the other is stated as well, a
syllogism, e.g. 'Pittacus is generous, since ambitious men are generous
and Pittacus is ambitious.' Or again 'Wise men are good, since Pittacus
is not only good but wise.' In this way then syllogisms are formed,
only that which proceeds through the first figure is irrefutable if
it is true (for it is universal), that which proceeds through the
last figure is refutable even if the conclusion is true, since the
syllogism is not universal nor correlative to the matter in question:
for though Pittacus is good, it is not therefore necessary that all
other wise men should be good. But the syllogism which proceeds through
the middle figure is always refutable in any case: for a syllogism
can never be formed when the terms are related in this way: for though
a woman with child is pale, and this woman also is pale, it is not
necessary that she should be with child. Truth then may be found in
signs whatever their kind, but they have the differences we have stated.
We must either divide signs in the way stated, and among them designate
the middle term as the index (for people call that the index which
makes us know, and the middle term above all has this character),
or else we must call the arguments derived from the extremes signs,
that derived from the middle term the index: for that which is proved
through the first figure is most generally accepted and most true.
It is possible to infer character from features, if it is granted
that the body and the soul are changed together by the natural affections:
I say 'natural', for though perhaps by learning music a man has made
some change in his soul, this is not one of those affections which
are natural to us; rather I refer to passions and desires when I speak
of natural emotions. If then this were granted and also that for each
change there is a corresponding sign, and we could state the affection
and sign proper to each kind of animal, we shall be able to infer
character from features. For if there is an affection which belongs
properly to an individual kind, e.g. courage to lions, it is necessary
that there should be a sign of it: for ex hypothesi body and soul
are affected together. Suppose this sign is the possession of large
extremities: this may belong to other kinds also though not universally.
For the sign is proper in the sense stated, because the affection
is proper to the whole kind, though not proper to it alone, according
to our usual manner of speaking. The same thing then will be found
in another kind, and man may be brave, and some other kinds of animal
as well. They will then have the sign: for ex hypothesi there is one
sign corresponding to each affection. If then this is so, and we can
collect signs of this sort in these animals which have only one affection
proper to them-but each affection has its sign, since it is necessary
that it should have a single sign-we shall then be able to infer character
from features. But if the kind as a whole has two properties, e.g.
if the lion is both brave and generous, how shall we know which of
the signs which are its proper concomitants is the sign of a particular
affection? Perhaps if both belong to some other kind though not to
the whole of it, and if, in those kinds in which each is found though
not in the whole of their members, some members possess one of the
affections and not the other: e.g. if a man is brave but not generous,
but possesses, of the two signs, large extremities, it is clear that
this is the sign of courage in the lion also. To judge character from
features, then, is possible in the first figure if the middle term
is convertible with the first extreme, but is wider than the third
term and not convertible with it: e.g. let A stand for courage, B
for large extremities, and C for lion. B then belongs to everything
to which C belongs, but also to others. But A belongs to everything
to which B belongs, and to nothing besides, but is convertible with
B: otherwise, there would not be a single sign correlative with each
affection.
THE END
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