350 BC
PRIOR ANALYTICS
by Aristotle
translated by A. J. Jenkinson
Book I
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.
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.
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.
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.
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.
6
But if one term belongs to all, and another to none, of a third,
or if both belong to all, or to none, of it, I call such a figure
the third; by middle term in it I mean that of which both the
predicates are predicated, by extremes I mean the predicates, by the
major extreme that which is further from the middle, by the
minor that
which is nearer to it. The middle term stands outside the extremes,
and is last in position. A syllogism cannot be perfect in this
figure either, but it may be valid whether the terms are related
universally or not to the middle term.
If they are universal, whenever both P and R belong to S,
it follows
that P will necessarily belong to some R. For, since the affirmative
statement is convertible, S will belong to some R: consequently
since P belongs to all S, and S to some R, P must belong to some R:
for a syllogism in the first figure is produced. It is possible to
demonstrate this also per impossibile and by exposition. For
if both P
and R belong to all S, should one of the Ss, e.g. N, be taken, both
P and R will belong to this, and thus P will belong to some R.
If R belongs to all S, and P to no S, there will be a syllogism to
prove that P will necessarily not belong to some R. This may be
demonstrated in the same way as before by converting the premiss RS.
It might be proved also per impossibile, as in the former cases. But
if R belongs to no S, P to all S, there will be no syllogism. Terms
for the positive relation are animal, horse, man: for the negative
relation animal, inanimate, man.
Nor can there be a syllogism when both terms are asserted of no S.
Terms for the positive relation are animal, horse, inanimate; for
the negative relation man, horse, inanimate-inanimate being
the middle
term.
It is clear then in this figure also when a syllogism will be
possible and when not, if the terms are related universally. For
whenever both the terms are affirmative, there will be a syllogism
to prove that one extreme belongs to some of the other; but when
they are negative, no syllogism will be possible. But when one is
negative, the other affirmative, if the major is negative, the minor
affirmative, there will be a syllogism to prove that the one extreme
does not belong to some of the other: but if the relation is
reversed,
no syllogism will be possible. If one term is related universally to
the middle, the other in part only, when both are affirmative there
must be a syllogism, no matter which of the premisses is universal.
For if R belongs to all S, P to some S, P must belong to some R. For
since the affirmative statement is convertible S will belong to some
P: consequently since R belongs to all S, and S to some P, R
must also
belong to some P: therefore P must belong to some R.
Again if R belongs to some S, and P to all S, P must belong to
some R. This may be demonstrated in the same way as the
preceding. And
it is possible to demonstrate it also per impossibile and by
exposition, as in the former cases. But if one term is affirmative,
the other negative, and if the affirmative is universal, a syllogism
will be possible whenever the minor term is affirmative. For if R
belongs to all S, but P does not belong to some S, it is necessary
that P does not belong to some R. For if P belongs to all R, and R
belongs to all S, then P will belong to all S: but we assumed that
it did not. Proof is possible also without reduction ad impossibile,
if one of the Ss be taken to which P does not belong.
But whenever the major is affirmative, no syllogism will be
possible, e.g. if P belongs to all S and R does not belong to some
S. Terms for the universal affirmative relation are animate, man,
animal. For the universal negative relation it is not possible to
get terms, if R belongs to some S, and does not belong to some S.
For if P belongs to all S, and R to some S, then P will
belong to some
R: but we assumed that it belongs to no R. We must put the matter as
before.' Since the expression 'it does not belong to some' is
indefinite, it may be used truly of that also which belongs to none.
But if R belongs to no S, no syllogism is possible, as has
been shown.
Clearly then no syllogism will be possible here.
But if the negative term is universal, whenever the major is
negative and the minor affirmative there will be a
syllogism. For if P
belongs to no S, and R belongs to some S, P will not belong
to some R:
for we shall have the first figure again, if the premiss RS is
converted.
But when the minor is negative, there will be no syllogism. Terms
for the positive relation are animal, man, wild: for the negative
relation, animal, science, wild-the middle in both being the term
wild.
Nor is a syllogism possible when both are stated in the negative,
but one is universal, the other particular. When the minor is
related universally to the middle, take the terms animal, science,
wild; animal, man, wild. When the major is related universally to
the middle, take as terms for a negative relation raven, snow,
white. For a positive relation terms cannot be found, if R belongs
to some S, and does not belong to some S. For if P belongs to all R,
and R to some S, then P belongs to some S: but we assumed that it
belongs to no S. Our point, then, must be proved from the indefinite
nature of the particular statement.
Nor is a syllogism possible anyhow, if each of the extremes
belongs to some of the middle or does not belong, or one belongs and
the other does not to some of the middle, or one belongs to some of
the middle, the other not to all, or if the premisses are
indefinite. Common terms for all are animal, man, white: animal,
inanimate, white.
It is clear then in this figure also when a syllogism will be
possible, and when not; and that if the terms are as stated, a
syllogism results of necessity, and if there is a syllogism,
the terms
must be so related. It is clear also that all the syllogisms in this
figure are imperfect (for all are made perfect by certain
supplementary assumptions), and that it will not be possible to
reach a universal conclusion by means of this figure,
whether negative
or affirmative.
7
It is evident also that in all the figures, whenever a proper
syllogism does not result, if both the terms are affirmative or
negative nothing necessary follows at all, but if one is
affirmative, the other negative, and if the negative is stated
universally, a syllogism always results relating the minor to the
major term, e.g. if A belongs to all or some B, and B
belongs to no C:
for if the premisses are converted it is necessary that C does not
belong to some A. Similarly also in the other figures: a syllogism
always results by means of conversion. It is evident also that the
substitution of an indefinite for a particular affirmative
will effect
the same syllogism in all the figures.
It is clear too that all the imperfect syllogisms are made perfect
by means of the first figure. For all are brought to a conclusion
either ostensively or per impossibile. In both ways the first figure
is formed: if they are made perfect ostensively, because (as we saw)
all are brought to a conclusion by means of conversion, and
conversion
produces the first figure: if they are proved per
impossibile, because
on the assumption of the false statement the syllogism comes about
by means of the first figure, e.g. in the last figure, if A and B
belong to all C, it follows that A belongs to some B: for if A
belonged to no B, and B belongs to all C, A would belong to no C:
but (as we stated) it belongs to all C. Similarly also with the rest.
It is possible also to reduce all syllogisms to the universal
syllogisms in the first figure. Those in the second figure
are clearly
made perfect by these, though not all in the same way; the universal
syllogisms are made perfect by converting the negative premiss, each
of the particular syllogisms by reductio ad impossibile. In the
first figure particular syllogisms are indeed made perfect by
themselves, but it is possible also to prove them by means of the
second figure, reducing them ad impossibile, e.g. if A belongs to
all B, and B to some C, it follows that A belongs to some C.
For if it
belonged to no C, and belongs to all B, then B will belong to no C:
this we know by means of the second figure. Similarly also
demonstration will be possible in the case of the negative. For if A
belongs to no B, and B belongs to some C, A will not belong
to some C:
for if it belonged to all C, and belongs to no B, then B will belong
to no C: and this (as we saw) is the middle figure. Consequently,
since all syllogisms in the middle figure can be reduced to
universal syllogisms in the first figure, and since particular
syllogisms in the first figure can be reduced to syllogisms in the
middle figure, it is clear that particular syllogisms can be reduced
to universal syllogisms in the first figure. Syllogisms in the third
figure, if the terms are universal, are directly made
perfect by means
of those syllogisms; but, when one of the premisses is particular,
by means of the particular syllogisms in the first figure: and these
(we have seen) may be reduced to the universal syllogisms in
the first
figure: consequently also the particular syllogisms in the third
figure may be so reduced. It is clear then that all syllogisms may
be reduced to the universal syllogisms in the first figure.
We have stated then how syllogisms which prove that something
belongs or does not belong to something else are
constituted, both how
syllogisms of the same figure are constituted in themselves, and how
syllogisms of different figures are related to one another.
8
Since there is a difference according as something belongs,
necessarily belongs, or may belong to something else (for many
things belong indeed, but not necessarily, others neither
necessarily nor indeed at all, but it is possible for them
to belong),
it is clear that there will be different syllogisms to prove each of
these relations, and syllogisms with differently related terms, one
syllogism concluding from what is necessary, another from what is, a
third from what is possible.
There is hardly any difference between syllogisms from necessary
premisses and syllogisms from premisses which merely assert. When
the terms are put in the same way, then, whether something belongs
or necessarily belongs (or does not belong) to something else, a
syllogism will or will not result alike in both cases, the only
difference being the addition of the expression 'necessarily' to the
terms. For the negative statement is convertible alike in both
cases, and we should give the same account of the expressions 'to be
contained in something as in a whole' and 'to be predicated of all
of something'. With the exceptions to be made below, the conclusion
will be proved to be necessary by means of conversion, in the same
manner as in the case of simple predication. But in the middle
figure when the universal statement is affirmative, and the
particular
negative, and again in the third figure when the universal is
affirmative and the particular negative, the demonstration will not
take the same form, but it is necessary by the 'exposition' of a
part of the subject of the particular negative proposition, to which
the predicate does not belong, to make the syllogism in reference to
this: with terms so chosen the conclusion will necessarily
follow. But
if the relation is necessary in respect of the part taken, it must
hold of some of that term in which this part is included:
for the part
taken is just some of that. And each of the resulting
syllogisms is in
the appropriate figure.
9
It happens sometimes also that when one premiss is necessary the
conclusion is necessary, not however when either premiss is
necessary,
but only when the major is, e.g. if A is taken as necessarily
belonging or not belonging to B, but B is taken as simply
belonging to
C: for if the premisses are taken in this way, A will necessarily
belong or not belong to C. For since necessarily belongs, or does
not belong, to every B, and since C is one of the Bs, it is
clear that
for C also the positive or the negative relation to A will hold
necessarily. But if the major premiss is not necessary, but the
minor is necessary, the conclusion will not be necessary. For if it
were, it would result both through the first figure and through the
third that A belongs necessarily to some B. But this is false; for B
may be such that it is possible that A should belong to none of it.
Further, an example also makes it clear that the conclusion not be
necessary, e.g. if A were movement, B animal, C man: man is an
animal necessarily, but an animal does not move necessarily, nor
does man. Similarly also if the major premiss is negative; for the
proof is the same.
In particular syllogisms, if the universal premiss is necessary,
then the conclusion will be necessary; but if the particular, the
conclusion will not be necessary, whether the universal premiss is
negative or affirmative. First let the universal be
necessary, and let
A belong to all B necessarily, but let B simply belong to some C: it
is necessary then that A belongs to some C necessarily: for C falls
under B, and A was assumed to belong necessarily to all B. Similarly
also if the syllogism should be negative: for the proof will be the
same. But if the particular premiss is necessary, the conclusion
will not be necessary: for from the denial of such a conclusion
nothing impossible results, just as it does not in the universal
syllogisms. The same is true of negative syllogisms. Try the terms
movement, animal, white.
10
In the second figure, if the negative premiss is
necessary, then the
conclusion will be necessary, but if the affirmative, not necessary.
First let the negative be necessary; let A be possible of no B, and
simply belong to C. Since then the negative statement is
convertible, B is possible of no A. But A belongs to all C;
consequently B is possible of no C. For C falls under A. The same
result would be obtained if the minor premiss were negative: for if
A is possible be of no C, C is possible of no A: but A belongs to
all B, consequently C is possible of none of the Bs: for
again we have
obtained the first figure. Neither then is B possible of C: for
conversion is possible without modifying the relation.
But if the affirmative premiss is necessary, the
conclusion will not
be necessary. Let A belong to all B necessarily, but to no C simply.
If then the negative premiss is converted, the first figure results.
But it has been proved in the case of the first figure that if the
negative major premiss is not necessary the conclusion will not be
necessary either. Therefore the same result will obtain
here. Further,
if the conclusion is necessary, it follows that C
necessarily does not
belong to some A. For if B necessarily belongs to no C, C will
necessarily belong to no B. But B at any rate must belong to some A,
if it is true (as was assumed) that A necessarily belongs to all B.
Consequently it is necessary that C does not belong to some A. But
nothing prevents such an A being taken that it is possible for C to
belong to all of it. Further one might show by an exposition of
terms that the conclusion is not necessary without qualification,
though it is a necessary conclusion from the premisses. For example
let A be animal, B man, C white, and let the premisses be assumed to
correspond to what we had before: it is possible that animal should
belong to nothing white. Man then will not belong to anything white,
but not necessarily: for it is possible for man to be born white,
not however so long as animal belongs to nothing white. Consequently
under these conditions the conclusion will be necessary, but
it is not
necessary without qualification.
Similar results will obtain also in particular syllogisms. For
whenever the negative premiss is both universal and necessary, then
the conclusion will be necessary: but whenever the
affirmative premiss
is universal, the negative particular, the conclusion will not be
necessary. First then let the negative premiss be both universal and
necessary: let it be possible for no B that A should belong
to it, and
let A simply belong to some C. Since the negative statement is
convertible, it will be possible for no A that B should belong to
it: but A belongs to some C; consequently B necessarily does not
belong to some of the Cs. Again let the affirmative premiss be both
universal and necessary, and let the major premiss be affirmative.
If then A necessarily belongs to all B, but does not belong
to some C,
it is clear that B will not belong to some C, but not
necessarily. For
the same terms can be used to demonstrate the point, which were used
in the universal syllogisms. Nor again, if the negative statement is
necessary but particular, will the conclusion be necessary. The
point can be demonstrated by means of the same terms.
11
In the last figure when the terms are related universally to the
middle, and both premisses are affirmative, if one of the two is
necessary, then the conclusion will be necessary. But if one is
negative, the other affirmative, whenever the negative is necessary
the conclusion also will be necessary, but whenever the
affirmative is
necessary the conclusion will not be necessary. First let both the
premisses be affirmative, and let A and B belong to all C, and let
AC be necessary. Since then B belongs to all C, C also will belong
to some B, because the universal is convertible into the particular:
consequently if A belongs necessarily to all C, and C belongs to
some B, it is necessary that A should belong to some B also. For B
is under C. The first figure then is formed. A similar proof will be
given also if BC is necessary. For C is convertible with some A:
consequently if B belongs necessarily to all C, it will belong
necessarily also to some A.
Again let AC be negative, BC affirmative, and let the negative
premiss be necessary. Since then C is convertible with some B, but A
necessarily belongs to no C, A will necessarily not belong to some B
either: for B is under C. But if the affirmative is necessary, the
conclusion will not be necessary. For suppose BC is affirmative and
necessary, while AC is negative and not necessary. Since then the
affirmative is convertible, C also will belong to some B
necessarily: consequently if A belongs to none of the Cs, while C
belongs to some of the Bs, A will not belong to some of the
Bs-but not
of necessity; for it has been proved, in the case of the
first figure,
that if the negative premiss is not necessary, neither will the
conclusion be necessary. Further, the point may be made clear by
considering the terms. Let the term A be 'good', let that which B
signifies be 'animal', let the term C be 'horse'. It is possible
then that the term good should belong to no horse, and it is
necessary
that the term animal should belong to every horse: but it is not
necessary that some animal should not be good, since it is possible
for every animal to be good. Or if that is not possible, take as the
term 'awake' or 'asleep': for every animal can accept these.
If, then, the premisses are universal, we have stated when the
conclusion will be necessary. But if one premiss is universal, the
other particular, and if both are affirmative, whenever the
universal is necessary the conclusion also must be necessary. The
demonstration is the same as before; for the particular affirmative
also is convertible. If then it is necessary that B should belong to
all C, and A falls under C, it is necessary that B should belong to
some A. But if B must belong to some A, then A must belong to some
B: for conversion is possible. Similarly also if AC should be
necessary and universal: for B falls under C. But if the particular
premiss is necessary, the conclusion will not be necessary. Let the
premiss BC be both particular and necessary, and let A belong to all
C, not however necessarily. If the proposition BC is converted the
first figure is formed, and the universal premiss is not necessary,
but the particular is necessary. But when the premisses were
thus, the
conclusion (as we proved was not necessary: consequently it is not
here either. Further, the point is clear if we look at the 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.
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.
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.
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.
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.
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.]
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.
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.
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.
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.
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.
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.
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.
24
Further in every syllogism one of the premisses must be
affirmative,
and universality must be present: unless one of the premisses is
universal either a syllogism will not be possible, or it will not
refer to the subject proposed, or the original position will be
begged. Suppose we have to prove that pleasure in music is good. If
one should claim as a premiss that pleasure is good without adding
'all', no syllogism will be possible; if one should claim that some
pleasure is good, then if it is different from pleasure in music, it
is not relevant to the subject proposed; if it is this very
pleasure, one is assuming that which was proposed at the outset to
be proved. This is more obvious in geometrical proofs, e.g. that the
angles at the base of an isosceles triangle are equal. Suppose the
lines A and B have been drawn to the centre. If then one
should assume
that the angle AC is equal to the angle BD, without claiming
generally
that angles of semicircles are equal; and again if one should assume
that the angle C is equal to the angle D, without the additional
assumption that every angle of a segment is equal to every
other angle
of the same segment; and further if one should assume that when
equal angles are taken from the whole angles, which are themselves
equal, the remainders E and F are equal, he will beg the thing to be
proved, unless he also states that when equals are taken from equals
the remainders are equal.
It is clear then that in every syllogism there must be a universal
premiss, and that a universal statement is proved only when all the
premisses are universal, while a particular statement is proved both
from two universal premisses and from one only: consequently if the
conclusion is universal, the premisses also must be universal, but
if the premisses are universal it is possible that the conclusion
may not be universal. And it is clear also that in every syllogism
either both or one of the premisses must be like the conclusion. I
mean not only in being affirmative or negative, but also in being
necessary, pure, problematic. We must consider also the
other forms of
predication.
It is clear also when a syllogism in general can be made
and when it
cannot; and when a valid, when a perfect syllogism can be formed;
and that if a syllogism is formed the terms must be arranged
in one of
the ways that have been mentioned.
25
It is clear too that every demonstration will proceed through
three terms and no more, unless the same conclusion is established
by different pairs of propositions; e.g. the conclusion E may be
established through the propositions A and B, and through the
propositions C and D, or through the propositions A and B,
or A and C,
or B and C. For nothing prevents there being several middles for the
same terms. But in that case there is not one but several
syllogisms. Or again when each of the propositions A and B
is obtained
by syllogistic inference, e.g. by means of D and E, and again B by
means of F and G. Or one may be obtained by syllogistic, the other
by inductive inference. But thus also the syllogisms are
many; for the
conclusions are many, e.g. A and B and C. But if this can be called
one syllogism, not many, the same conclusion may be reached by more
than three terms in this way, but it cannot be reached as C is
established by means of A and B. Suppose that the proposition E is
inferred from the premisses A, B, C, and D. It is necessary then
that of these one should be related to another as whole to part: for
it has already been proved that if a syllogism is formed some of its
terms must be related in this way. Suppose then that A stands in
this relation to B. Some conclusion then follows from them. It must
either be E or one or other of C and D, or something other
than these.
(1) If it is E the syllogism will have A and B for its sole
premisses. But if C and D are so related that one is whole, the
other part, some conclusion will follow from them also; and
it must be
either E, or one or other of the propositions A and B, or something
other than these. And if it is (i) E, or (ii) A or B, either (i) the
syllogisms will be more than one, or (ii) the same thing
happens to be
inferred by means of several terms only in the sense which we saw to
be possible. But if (iii) the conclusion is other than E or A or B,
the syllogisms will be many, and unconnected with one another. But
if C is not so related to D as to make a syllogism, the propositions
will have been assumed to no purpose, unless for the sake of
induction
or of obscuring the argument or something of the sort.
(2) But if from the propositions A and B there follows not E but
some other conclusion, and if from C and D either A or B follows or
something else, then there are several syllogisms, and they do not
establish the conclusion proposed: for we assumed that the syllogism
proved E. And if no conclusion follows from C and D, it
turns out that
these propositions have been assumed to no purpose, and the
syllogism does not prove the original proposition.
So it is clear that every demonstration and every syllogism will
proceed through three terms only.
This being evident, it is clear that a syllogistic conclusion
follows from two premisses and not from more than two. For the three
terms make two premisses, unless a new premiss is assumed,
as was said
at the beginning, to perfect the syllogisms. It is clear therefore
that in whatever syllogistic argument the premisses through which
the main conclusion follows (for some of the preceding conclusions
must be premisses) are not even in number, this argument either has
not been drawn syllogistically or it has assumed more than was
necessary to establish its thesis.
If then syllogisms are taken with respect to their main premisses,
every syllogism will consist of an even number of premisses
and an odd
number of terms (for the terms exceed the premisses by one), and the
conclusions will be half the number of the premisses. But whenever a
conclusion is reached by means of prosyllogisms or by means
of several
continuous middle terms, e.g. the proposition AB by means of the
middle terms C and D, the number of the terms will similarly exceed
that of the premisses by one (for the extra term must either be
added outside or inserted: but in either case it follows that the
relations of predication are one fewer than the terms related), and
the premisses will be equal in number to the relations of
predication.
The premisses however will not always be even, the terms
odd; but they
will alternate-when the premisses are even, the terms must be odd;
when the terms are even, the premisses must be odd: for
along with one
term one premiss is added, if a term is added from any quarter.
Consequently since the premisses were (as we saw) even, and the
terms odd, we must make them alternately even and odd at each
addition. But the conclusions will not follow the same arrangement
either in respect to the terms or to the premisses. For if
one term is
added, conclusions will be added less by one than the pre-existing
terms: for the conclusion is drawn not in relation to the single
term last added, but in relation to all the rest, e.g. if to ABC the
term D is added, two conclusions are thereby added, one in
relation to
A, the other in relation to B. Similarly with any further additions.
And similarly too if the term is inserted in the middle: for in
relation to one term only, a syllogism will not be constructed.
Consequently the conclusions will be much more numerous than
the terms
or the premisses.
26
Since we understand the subjects with which syllogisms are
concerned, what sort of conclusion is established in each figure,
and in how many moods this is done, it is evident to us both
what sort
of problem is difficult and what sort is easy to prove. For
that which
is concluded in many figures and through many moods is easier; that
which is concluded in few figures and through few moods is more
difficult to attempt. The universal affirmative is proved by means
of the first figure only and by this in only one mood; the universal
negative is proved both through the first figure and through the
second, through the first in one mood, through the second in two.
The particular affirmative is proved through the first and
through the
last figure, in one mood through the first, in three moods
through the
last. The particular negative is proved in all the figures, but once
in the first, in two moods in the second, in three moods in
the third.
It is clear then that the universal affirmative is most difficult to
establish, most easy to overthrow. In general, universals are easier
game for the destroyer than particulars: for whether the predicate
belongs to none or not to some, they are destroyed: and the
particular
negative is proved in all the figures, the universal negative in
two. Similarly with universal negatives: the original statement is
destroyed, whether the predicate belongs to all or to some: and this
we found possible in two figures. But particular statements can be
refuted in one way only-by proving that the predicate belongs either
to all or to none. But particular statements are easier to
establish: for proof is possible in more figures and through more
moods. And in general we must not forget that it is possible
to refute
statements by means of one another, I mean, universal statements by
means of particular, and particular statements by means of
universal: but it is not possible to establish universal
statements by
means of particular, though it is possible to establish particular
statements by means of universal. At the same time it is evident
that it is easier to refute than to establish.
The manner in which every syllogism is produced, the number of the
terms and premisses through which it proceeds, the relation of the
premisses to one another, the character of the problem proved in
each figure, and the number of the figures appropriate to each
problem, all these matters are clear from what has been said.
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.
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.
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.
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.
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.
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.
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'.
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.
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.
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.
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.
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.
39
We ought also to exchange terms which have the same value, word
for word, and phrase for phrase, and word and phrase, and always
take a word in preference to a phrase: for thus the setting
out of the
terms will be easier. For example if it makes no difference
whether we
say that the supposable is not the genus of the opinable or that the
opinable is not identical with a particular kind of supposable (for
what is meant is the same in both statements), it is better
to take as
the terms the supposable and the opinable in preference to the
phrase suggested.
40
Since the expressions 'pleasure is good' and 'pleasure is the
good' are not identical, we must not set out the terms in the same
way; but if the syllogism is to prove that pleasure is the good, the
term must be 'the good', but if the object is to prove that pleasure
is good, the term will be 'good'. Similarly in all other cases.
41
It is not the same, either in fact or in speech, that A belongs to
all of that to which B belongs, and that A belongs to all of that to
all of which B belongs: for nothing prevents B from belonging to C,
though not to all C: e.g. let B stand for beautiful, and C for
white. If beauty belongs to something white, it is true to say that
beauty belongs to that which is white; but not perhaps to everything
that is white. If then A belongs to B, but not to everything of
which B is predicated, then whether B belongs to all C or merely
belongs to C, it is not necessary that A should belong, I do not say
to all C, but even to C at all. But if A belongs to everything of
which B is truly stated, it will follow that A can be said of all of
that of all of which B is said. If however A is said of that
of all of
which B may be said, nothing prevents B belonging to C, and yet A
not belonging to all C or to any C at all. If then we take
three terms
it is clear that the expression 'A is said of all of which B is
said' means this, 'A is said of all the things of which B is said'.
And if B is said of all of a third term, so also is A: but
if B is not
said of all of the third term, there is no necessity that A should
be said of all of it.
We must not suppose that something absurd results through setting
out the terms: for we do not use the existence of this particular
thing, but imitate the geometrician who says that 'this line a foot
long' or 'this straight line' or 'this line without breadth' exists
although it does not, but does not use the diagrams in the sense
that he reasons from them. For in general, if two things are not
related as whole to part and part to whole, the prover does not
prove from them, and so no syllogism a is formed. We (I mean the
learner) use the process of setting out terms like perception by
sense, not as though it were impossible to demonstrate without these
illustrative terms, as it is to demonstrate without the premisses of
the syllogism.
42
We should not forget that in the same syllogism not all
conclusions are reached through one figure, but one through one
figure, another through another. Clearly then we must analyse
arguments in accordance with this. Since not every problem is proved
in every figure, but certain problems in each figure, it is
clear from
the conclusion in what figure the premisses should be sought.
43
In reference to those arguments aiming at a definition which have
been directed to prove some part of the definition, we must take as
a term the point to which the argument has been directed, not the
whole definition: for so we shall be less likely to be disturbed by
the length of the term: e.g. if a man proves that water is a
drinkable
liquid, we must take as terms drinkable and water.
44
Further we must not try to reduce hypothetical syllogisms; for
with the given premisses it is not possible to reduce them. For they
have not been proved by syllogism, but assented to by agreement. For
instance if a man should suppose that unless there is one faculty of
contraries, there cannot be one science, and should then argue that
not every faculty is of contraries, e.g. of what is healthy and what
is sickly: for the same thing will then be at the same time healthy
and sickly. He has shown that there is not one faculty of all
contraries, but he has not proved that there is not a
science. And yet
one must agree. But the agreement does not come from a syllogism,
but from an hypothesis. This argument cannot be reduced: but
the proof
that there is not a single faculty can. The latter argument perhaps
was a syllogism: but the former was an hypothesis.
The same holds good of arguments which are brought to a conclusion
per impossibile. These cannot be analysed either; but the
reduction to
what is impossible can be analysed since it is proved by syllogism,
though the rest of the argument cannot, because the conclusion is
reached from an hypothesis. But these differ from the previous
arguments: for in the former a preliminary agreement must be reached
if one is to accept the conclusion; e.g. an agreement that
if there is
proved to be one faculty of contraries, then contraries fall
under the
same science; whereas in the latter, even if no preliminary
agreement has been made, men still accept the reasoning, because the
falsity is patent, e.g. the falsity of what follows from the
assumption that the diagonal is commensurate, viz. that then odd
numbers are equal to evens.
Many other arguments are brought to a conclusion by the help of an
hypothesis; these we ought to consider and mark out clearly. We
shall describe in the sequel their differences, and the various ways
in which hypothetical arguments are formed: but at present this much
must be clear, that it is not possible to resolve such arguments
into the figures. And we have explained the reason.
45
Whatever problems are proved in more than one figure, if they have
been established in one figure by syllogism, can be 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.
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
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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-
.