Book I
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Part 1
We must first state the subject of our inquiry and the faculty to which
it belongs: its subject is demonstration and the faculty that carries it
out demonstrative science. We must next define a premiss, a term, and a
syllogism, and the nature of a perfect and of an imperfect syllogism; and
after that, the inclusion or noninclusion of one term in another as in
a whole, and what we mean by predicating one term of all, or none, of
another.
A premiss then is a sentence affirming or denying one thing of
another. This is either universal or particular or indefinite. By universal
I mean the statement that something belongs to all or none of something
else; by particular that it belongs to some or not to some or not to all;
by indefinite that it does or does not belong, without any mark to show
whether it is universal or particular, e.g. 'contraries are subjects of
the same science', or 'pleasure is not good'. The demonstrative premiss
differs from the dialectical, because the demonstrative premiss is the
assertion of one of two contradictory statements (the demonstrator does
not ask for his premiss, but lays it down), whereas the dialectical premiss
depends on the adversary's choice between two contradictories. But this
will make no difference to the production of a syllogism in either case;
for both the demonstrator and the dialectician argue syllogistically after
stating that something does or does not belong to something else. Therefore
a syllogistic premiss without qualification will be an affirmation or denial
of something concerning something else in the way we have described; it
will be demonstrative, if it is true and obtained through the first principles
of its science; while a dialectical premiss is the giving of a choice between
two contradictories, when a man is proceeding by question, but when he
is syllogizing it is the assertion of that which is apparent and generally
admitted, as has been said in the Topics. The nature then of a premiss
and the difference between syllogistic, demonstrative, and dialectical
premisses, may be taken as sufficiently defined by us in relation to our
present need, but will be stated accurately in the sequel.
I call that a term into which the premiss is resolved, i.e. both
the predicate and that of which it is predicated, 'being' being added and
'not being' removed, or vice versa.
A syllogism is discourse in which, certain things being stated,
something other than what is stated follows of necessity from their being
so. I mean by the last phrase that they produce the consequence, and by
this, that no further term is required from without in order to make the
consequence necessary.
I call that a perfect syllogism which needs nothing other than
what has been stated to make plain what necessarily follows; a syllogism
is imperfect, if it needs either one or more propositions, which are indeed
the necessary consequences of the terms set down, but have not been expressly
stated as premisses.
That one term should be included in another as in a whole is the
same as for the other to be predicated of all of the first. And we say
that one term is predicated of all of another, whenever no instance of
the subject can be found of which the other term cannot be asserted: 'to
be predicated of none' must be understood in the same
way.
Part 2
Every premiss states that something either is or must be or may
be the attribute of something else; of premisses of these three kinds some
are affirmative, others negative, in respect of each of the three modes
of attribution; again some affirmative and negative premisses are universal,
others particular, others indefinite. It is necessary then that in universal
attribution the terms of the negative premiss should be convertible, e.g.
if no pleasure is good, then no good will be pleasure; the terms of the
affirmative must be convertible, not however, universally, but in part,
e.g. if every pleasure,is good, some good must be pleasure; the particular
affirmative must convert in part (for if some pleasure is good, then some
good will be pleasure); but the particular negative need not convert, for
if some animal is not man, it does not follow that some man is not
animal.
First then take a universal negative with the terms A and B. If
no B is A, neither can any A be B. For if some A (say C) were B, it would
not be true that no B is A; for C is a B. But if every B is A then some
A is B. For if no A were B, then no B could be A. But we assumed that every
B is A. Similarly too, if the premiss is particular. For if some B is A,
then some of the As must be B. For if none were, then no B would be A.
But if some B is not A, there is no necessity that some of the As should
not be B; e.g. let B stand for animal and A for man. Not every animal is
a man; but every man is an animal.
Part 3
The same manner of conversion will hold good also in respect of
necessary premisses. The universal negative converts universally; each
of the affirmatives converts into a particular. If it is necessary that
no B is A, it is necessary also that no A is B. For if it is possible that
some A is B, it would be possible also that some B is A. If all or some
B is A of necessity, it is necessary also that some A is B: for if there
were no necessity, neither would some of the Bs be A necessarily. But the
particular negative does not convert, for the same reason which we have
already stated.
In respect of possible premisses, since possibility is used in
several senses (for we say that what is necessary and what is not necessary
and what is potential is possible), affirmative statements will all convert
in a manner similar to those described. For if it is possible that all
or some B is A, it will be possible that some A is B. For if that were
not possible, then no B could possibly be A. This has been already proved.
But in negative statements the case is different. Whatever is said to be
possible, either because B necessarily is A, or because B is not necessarily
A, admits of conversion like other negative statements, e.g. if one should
say, it is possible that man is not horse, or that no garment is white.
For in the former case the one term necessarily does not belong to the
other; in the latter there is no necessity that it should: and the premiss
converts like other negative statements. For if it is possible for no man
to be a horse, it is also admissible for no horse to be a man; and if it
is admissible for no garment to be white, it is also admissible for nothing
white to be a garment. For if any white thing must be a garment, then some
garment will necessarily be white. This has been already proved. The particular
negative also must be treated like those dealt with above. But if anything
is said to be possible because it is the general rule and natural (and
it is in this way we define the possible), the negative premisses can no
longer be converted like the simple negatives; the universal negative premiss
does not convert, and the particular does. This will be plain when we speak
about the possible. At present we may take this much as clear in addition
to what has been said: the statement that it is possible that no B is A
or some B is not A is affirmative in form: for the expression 'is possible'
ranks along with 'is', and 'is' makes an affirmation always and in every
case, whatever the terms to which it is added, in predication, e.g. 'it
is not-good' or 'it is not-white' or in a word 'it is not-this'. But this
also will be proved in the sequel. In conversion these premisses will behave
like the other affirmative propositions.
Part 4
After these distinctions we now state by what means, when, and
how every syllogism is produced; subsequently we must speak of demonstration.
Syllogism should be discussed before demonstration because syllogism is
the general: the demonstration is a sort of syllogism, but not every syllogism
is a demonstration.
Whenever three terms are so related to one another that the last
is contained in the middle as in a whole, and the middle is either contained
in, or excluded from, the first as in or from a whole, the extremes must
be related by a perfect syllogism. I call that term middle which is itself
contained in another and contains another in itself: in position also this
comes in the middle. By extremes I mean both that term which is itself
contained in another and that in which another is contained. If A is predicated
of all B, and B of all C, A must be predicated of all C: we have already
explained what we mean by 'predicated of all'. Similarly also, if A is
predicated of no B, and B of all C, it is necessary that no C will be
A.
But if the first term belongs to all the middle, but the middle
to none of the last term, there will be no syllogism in respect of the
extremes; for nothing necessary follows from the terms being so related;
for it is possible that the first should belong either to all or to none
of the last, so that neither a particular nor a universal conclusion is
necessary. But if there is no necessary consequence, there cannot be a
syllogism by means of these premisses. As an example of a universal affirmative
relation between the extremes we may take the terms animal, man, horse;
of a universal negative relation, the terms animal, man, stone. Nor again
can syllogism be formed when neither the first term belongs to any of the
middle, nor the middle to any of the last. As an example of a positive
relation between the extremes take the terms science, line, medicine: of
a negative relation science, line, unit.
If then the terms are universally related, it is clear in this
figure when a syllogism will be possible and when not, and that if a syllogism
is possible the terms must be related as described, and if they are so
related there will be a syllogism.
But if one term is related universally, the other in part only,
to its subject, there must be a perfect syllogism whenever universality
is posited with reference to the major term either affirmatively or negatively,
and particularity with reference to the minor term affirmatively: but whenever
the universality is posited in relation to the minor term, or the terms
are related in any other way, a syllogism is impossible. I call that term
the major in which the middle is contained and that term the minor which
comes under the middle. Let all B be A and some C be B. Then if 'predicated
of all' means what was said above, it is necessary that some C is A. And
if no B is A but some C is B, it is necessary that some C is not A. The
meaning of 'predicated of none' has also been defined. So there will be
a perfect syllogism. This holds good also if the premiss BC should be indefinite,
provided that it is affirmative: for we shall have the same syllogism whether
the premiss is indefinite or particular.
But if the universality is posited with respect to the minor term
either affirmatively or negatively, a syllogism will not be possible, whether
the major premiss is positive or negative, indefinite or particular: e.g.
if some B is or is not A, and all C is B. As an example of a positive relation
between the extremes take the terms good, state, wisdom: of a negative
relation, good, state, ignorance. Again if no C is B, but some B is or
is not A or not every B is A, there cannot be a syllogism. Take the terms
white, horse, swan: white, horse, raven. The same terms may be taken also
if the premiss BA is indefinite.
Nor when the major premiss is universal, whether affirmative or
negative, and the minor premiss is negative and particular, can there be
a syllogism, whether the minor premiss be indefinite or particular: e.g.
if all B is A and some C is not B, or if not all C is B. For the major
term may be predicable both of all and of none of the minor, to some of
which the middle term cannot be attributed. Suppose the terms are animal,
man, white: next take some of the white things of which man is not predicated-swan
and snow: animal is predicated of all of the one, but of none of the other.
Consequently there cannot be a syllogism. Again let no B be A, but let
some C not be B. Take the terms inanimate, man, white: then take some white
things of which man is not predicated-swan and snow: the term inanimate
is predicated of all of the one, of none of the other.
Further since it is indefinite to say some C is not B, and it is
true that some C is not B, whether no C is B, or not all C is B, and since
if terms are assumed such that no C is B, no syllogism follows (this has
already been stated) it is clear that this arrangement of terms will not
afford a syllogism: otherwise one would have been possible with a universal
negative minor premiss. A similar proof may also be given if the universal
premiss is negative.
Nor can there in any way be a syllogism if both the relations of
subject and predicate are particular, either positively or negatively,
or the one negative and the other affirmative, or one indefinite and the
other definite, or both indefinite. Terms common to all the above are animal,
white, horse: animal, white, stone.
It is clear then from what has been said that if there is a syllogism
in this figure with a particular conclusion, the terms must be related
as we have stated: if they are related otherwise, no syllogism is possible
anyhow. It is evident also that all the syllogisms in this figure are perfect
(for they are all completed by means of the premisses originally taken)
and that all conclusions are proved by this figure, viz. universal and
particular, affirmative and negative. Such a figure I call the
first.
Part 5
Whenever the same thing belongs to all of one subject, and to none
of another, or to all of each subject or to none of either, I call such
a figure the second; by middle term in it I mean that which is predicated
of both subjects, by extremes the terms of which this is said, by major
extreme that which lies near the middle, by minor that which is further
away from the middle. The middle term stands outside the extremes, and
is first in position. A syllogism cannot be perfect anyhow in this figure,
but it may be valid whether the terms are related universally or
not.
If then the terms are related universally a syllogism will be possible,
whenever the middle belongs to all of one subject and to none of another
(it does not matter which has the negative relation), but in no other way.
Let M be predicated of no N, but of all O. Since, then, the negative relation
is convertible, N will belong to no M: but M was assumed to belong to all
O: consequently N will belong to no O. This has already been proved. Again
if M belongs to all N, but to no O, then N will belong to no O. For if
M belongs to no O, O belongs to no M: but M (as was said) belongs to all
N: O then will belong to no N: for the first figure has again been formed.
But since the negative relation is convertible, N will belong to no O.
Thus it will be the same syllogism that proves both
conclusions.
It is possible to prove these results also by reductio ad
impossibile.
It is clear then that a syllogism is formed when the terms are
so related, but not a perfect syllogism; for necessity is not perfectly
established merely from the original premisses; others also are
needed.
But if M is predicated of every N and O, there cannot be a syllogism.
Terms to illustrate a positive relation between the extremes are substance,
animal, man; a negative relation, substance, animal, number-substance being
the middle term.
Nor is a syllogism possible when M is predicated neither of any
N nor of any O. Terms to illustrate a positive relation are line, animal,
man: a negative relation, line, animal, stone.
It is clear then that if a syllogism is formed when the terms are
universally related, the terms must be related as we stated at the outset:
for if they are otherwise related no necessary consequence
follows.
If the middle term is related universally to one of the extremes,
a particular negative syllogism must result whenever the middle term is
related universally to the major whether positively or negatively, and
particularly to the minor and in a manner opposite to that of the universal
statement: by 'an opposite manner' I mean, if the universal statement is
negative, the particular is affirmative: if the universal is affirmative,
the particular is negative. For if M belongs to no N, but to some O, it
is necessary that N does not belong to some O. For since the negative statement
is convertible, N will belong to no M: but M was admitted to belong to
some O: therefore N will not belong to some O: for the result is reached
by means of the first figure. Again if M belongs to all N, but not to some
O, it is necessary that N does not belong to some O: for if N belongs to
all O, and M is predicated also of all N, M must belong to all O: but we
assumed that M does not belong to some O. And if M belongs to all N but
not to all O, we shall conclude that N does not belong to all O: the proof
is the same as the above. But if M is predicated of all O, but not of all
N, there will be no syllogism. Take the terms animal, substance, raven;
animal, white, raven. Nor will there be a conclusion when M is predicated
of no O, but of some N. Terms to illustrate a positive relation between
the extremes are animal, substance, unit: a negative relation, animal,
substance, science.
If then the universal statement is opposed to the particular, we
have stated when a syllogism will be possible and when not: but if the
premisses are similar in form, I mean both negative or both affirmative,
a syllogism will not be possible anyhow. First let them be negative, and
let the major premiss be universal, e.g. let M belong to no N, and not
to some O. It is possible then for N to belong either to all O or to no
O. Terms to illustrate the negative relation are black, snow, animal. But
it is not possible to find terms of which the extremes are related positively
and universally, if M belongs to some O, and does not belong to some O.
For if N belonged to all O, but M to no N, then M would belong to no O:
but we assumed that it belongs to some O. In this way then it is not admissible
to take terms: our point must be proved from the indefinite nature of the
particular statement. For since it is true that M does not belong to some
O, even if it belongs to no O, and since if it belongs to no O a syllogism
is (as we have seen) not possible, clearly it will not be possible now
either.
Again let the premisses be affirmative, and let the major premiss
as before be universal, e.g. let M belong to all N and to some O. It is
possible then for N to belong to all O or to no O. Terms to illustrate
the negative relation are white, swan, stone. But it is not possible to
take terms to illustrate the universal affirmative relation, for the reason
already stated: the point must be proved from the indefinite nature of
the particular statement. But if the minor premiss is universal, and M
belongs to no O, and not to some N, it is possible for N to belong either
to all O or to no O. Terms for the positive relation are white, animal,
raven: for the negative relation, white, stone, raven. If the premisses
are affirmative, terms for the negative relation are white, animal, snow;
for the positive relation, white, animal, swan. Evidently then, whenever
the premisses are similar in form, and one is universal, the other particular,
a syllogism can, not be formed anyhow. Nor is one possible if the middle
term belongs to some of each of the extremes, or does not belong to some
of either, or belongs to some of the one, not to some of the other, or
belongs to neither universally, or is related to them indefinitely. Common
terms for all the above are white, animal, man: white, animal, inanimate.
It is clear then from what has been said that if the terms are related
to one another in the way stated, a syllogism results of necessity; and
if there is a syllogism, the terms must be so related. But it is evident
also that all the syllogisms in this figure are imperfect: for all are
made perfect by certain supplementary statements, which either are contained
in the terms of necessity or are assumed as hypotheses, i.e. when we prove
per impossibile. And it is evident that an affirmative conclusion is not
attained by means of this figure, but all are negative, whether universal
or particular.
Part 6
But if one term belongs to all, and another to none, of a third,
or if both belong to all, or to none, of it, I call such a figure the third;
by middle term in it I mean that of which both the predicates are predicated,
by extremes I mean the predicates, by the major extreme that which is further
from the middle, by the minor that which is nearer to it. The middle term
stands outside the extremes, and is last in position. A syllogism cannot
be perfect in this figure either, but it may be valid whether the terms
are related universally or not to the middle term.
If they are universal, whenever both P and R belong to S, it follows
that P will necessarily belong to some R. For, since the affirmative statement
is convertible, S will belong to some R: consequently since P belongs to
all S, and S to some R, P must belong to some R: for a syllogism in the
first figure is produced. It is possible to demonstrate this also per impossibile
and by exposition. For if both P and R belong to all S, should one of the
Ss, e.g. N, be taken, both P and R will belong to this, and thus P will
belong to some R.
If R belongs to all S, and P to no S, there will be a syllogism
to prove that P will necessarily not belong to some R. This may be demonstrated
in the same way as before by converting the premiss RS. It might be proved
also per impossibile, as in the former cases. But if R belongs to no S,
P to all S, there will be no syllogism. Terms for the positive relation
are animal, horse, man: for the negative relation animal, inanimate,
man.
Nor can there be a syllogism when both terms are asserted of no
S. Terms for the positive relation are animal, horse, inanimate; for the
negative relation man, horse, inanimate-inanimate being the middle
term.
It is clear then in this figure also when a syllogism will be possible
and when not, if the terms are related universally. For whenever both the
terms are affirmative, there will be a syllogism to prove that one extreme
belongs to some of the other; but when they are negative, no syllogism
will be possible. But when one is negative, the other affirmative, if the
major is negative, the minor affirmative, there will be a syllogism to
prove that the one extreme does not belong to some of the other: but if
the relation is reversed, no syllogism will be possible. If one term is
related universally to the middle, the other in part only, when both are
affirmative there must be a syllogism, no matter which of the premisses
is universal. For if R belongs to all S, P to some S, P must belong to
some R. For since the affirmative statement is convertible S will belong
to some P: consequently since R belongs to all S, and S to some P, R must
also belong to some P: therefore P must belong to some
R.
Again if R belongs to some S, and P to all S, P must belong to
some R. This may be demonstrated in the same way as the preceding. And
it is possible to demonstrate it also per impossibile and by exposition,
as in the former cases. But if one term is affirmative, the other negative,
and if the affirmative is universal, a syllogism will be possible whenever
the minor term is affirmative. For if R belongs to all S, but P does not
belong to some S, it is necessary that P does not belong to some R. For
if P belongs to all R, and R belongs to all S, then P will belong to all
S: but we assumed that it did not. Proof is possible also without reduction
ad impossibile, if one of the Ss be taken to which P does not
belong.
But whenever the major is affirmative, no syllogism will be possible,
e.g. if P belongs to all S and R does not belong to some S. Terms for the
universal affirmative relation are animate, man, animal. For the universal
negative relation it is not possible to get terms, if R belongs to some
S, and does not belong to some S. For if P belongs to all S, and R to some
S, then P will belong to some R: but we assumed that it belongs to no R.
We must put the matter as before.' Since the expression 'it does not belong
to some' is indefinite, it may be used truly of that also which belongs
to none. But if R belongs to no S, no syllogism is possible, as has been
shown. Clearly then no syllogism will be possible here.
But if the negative term is universal, whenever the major is negative
and the minor affirmative there will be a syllogism. For if P belongs to
no S, and R belongs to some S, P will not belong to some R: for we shall
have the first figure again, if the premiss RS is converted.
But when the minor is negative, there will be no syllogism. Terms
for the positive relation are animal, man, wild: for the negative relation,
animal, science, wild-the middle in both being the term
wild.
Nor is a syllogism possible when both are stated in the negative,
but one is universal, the other particular. When the minor is related universally
to the middle, take the terms animal, science, wild; animal, man, wild.
When the major is related universally to the middle, take as terms for
a negative relation raven, snow, white. For a positive relation terms cannot
be found, if R belongs to some S, and does not belong to some S. For if
P belongs to all R, and R to some S, then P belongs to some S: but we assumed
that it belongs to no S. Our point, then, must be proved from the indefinite
nature of the particular statement.
Nor is a syllogism possible anyhow, if each of the extremes belongs
to some of the middle or does not belong, or one belongs and the other
does not to some of the middle, or one belongs to some of the middle, the
other not to all, or if the premisses are indefinite. Common terms for
all are animal, man, white: animal, inanimate, white.
It is clear then in this figure also when a syllogism will be possible,
and when not; and that if the terms are as stated, a syllogism results
of necessity, and if there is a syllogism, the terms must be so related.
It is clear also that all the syllogisms in this figure are imperfect (for
all are made perfect by certain supplementary assumptions), and that it
will not be possible to reach a universal conclusion by means of this figure,
whether negative or affirmative.
Part 7
It is evident also that in all the figures, whenever a proper syllogism
does not result, if both the terms are affirmative or negative nothing
necessary follows at all, but if one is affirmative, the other negative,
and if the negative is stated universally, a syllogism always results relating
the minor to the major term, e.g. if A belongs to all or some B, and B
belongs to no C: for if the premisses are converted it is necessary that
C does not belong to some A. Similarly also in the other figures: a syllogism
always results by means of conversion. It is evident also that the substitution
of an indefinite for a particular affirmative will effect the same syllogism
in all the figures.
It is clear too that all the imperfect syllogisms are made perfect
by means of the first figure. For all are brought to a conclusion either
ostensively or per impossibile. In both ways the first figure is formed:
if they are made perfect ostensively, because (as we saw) all are brought
to a conclusion by means of conversion, and conversion produces the first
figure: if they are proved per impossibile, because on the assumption of
the false statement the syllogism comes about by means of the first figure,
e.g. in the last figure, if A and B belong to all C, it follows that A
belongs to some B: for if A belonged to no B, and B belongs to all C, A
would belong to no C: but (as we stated) it belongs to all C. Similarly
also with the rest.
It is possible also to reduce all syllogisms to the universal syllogisms
in the first figure. Those in the second figure are clearly made perfect
by these, though not all in the same way; the universal syllogisms are
made perfect by converting the negative premiss, each of the particular
syllogisms by reductio ad impossibile. In the first figure particular syllogisms
are indeed made perfect by themselves, but it is possible also to prove
them by means of the second figure, reducing them ad impossibile, e.g.
if A belongs to all B, and B to some C, it follows that A belongs to some
C. For if it belonged to no C, and belongs to all B, then B will belong
to no C: this we know by means of the second figure. Similarly also demonstration
will be possible in the case of the negative. For if A belongs to no B,
and B belongs to some C, A will not belong to some C: for if it belonged
to all C, and belongs to no B, then B will belong to no C: and this (as
we saw) is the middle figure. Consequently, since all syllogisms in the
middle figure can be reduced to universal syllogisms in the first figure,
and since particular syllogisms in the first figure can be reduced to syllogisms
in the middle figure, it is clear that particular syllogisms can be reduced
to universal syllogisms in the first figure. Syllogisms in the third figure,
if the terms are universal, are directly made perfect by means of those
syllogisms; but, when one of the premisses is particular, by means of the
particular syllogisms in the first figure: and these (we have seen) may
be reduced to the universal syllogisms in the first figure: consequently
also the particular syllogisms in the third figure may be so reduced. It
is clear then that all syllogisms may be reduced to the universal syllogisms
in the first figure.
We have stated then how syllogisms which prove that something belongs
or does not belong to something else are constituted, both how syllogisms
of the same figure are constituted in themselves, and how syllogisms of
different figures are related to one another.
Part 8
Since there is a difference according as something belongs, necessarily
belongs, or may belong to something else (for many things belong indeed,
but not necessarily, others neither necessarily nor indeed at all, but
it is possible for them to belong), it is clear that there will be different
syllogisms to prove each of these relations, and syllogisms with differently
related terms, one syllogism concluding from what is necessary, another
from what is, a third from what is possible.
There is hardly any difference between syllogisms from necessary
premisses and syllogisms from premisses which merely assert. When the terms
are put in the same way, then, whether something belongs or necessarily
belongs (or does not belong) to something else, a syllogism will or will
not result alike in both cases, the only difference being the addition
of the expression 'necessarily' to the terms. For the negative statement
is convertible alike in both cases, and we should give the same account
of the expressions 'to be contained in something as in a whole' and 'to
be predicated of all of something'. With the exceptions to be made below,
the conclusion will be proved to be necessary by means of conversion, in
the same manner as in the case of simple predication. But in the middle
figure when the universal statement is affirmative, and the particular
negative, and again in the third figure when the universal is affirmative
and the particular negative, the demonstration will not take the same form,
but it is necessary by the 'exposition' of a part of the subject of the
particular negative proposition, to which the predicate does not belong,
to make the syllogism in reference to this: with terms so chosen the conclusion
will necessarily follow. But if the relation is necessary in respect of
the part taken, it must hold of some of that term in which this part is
included: for the part taken is just some of that. And each of the resulting
syllogisms is in the appropriate figure.
Part 9
It happens sometimes also that when one premiss is necessary the
conclusion is necessary, not however when either premiss is necessary,
but only when the major is, e.g. if A is taken as necessarily belonging
or not belonging to B, but B is taken as simply belonging to C: for if
the premisses are taken in this way, A will necessarily belong or not belong
to C. For since necessarily belongs, or does not belong, to every B, and
since C is one of the Bs, it is clear that for C also the positive or the
negative relation to A will hold necessarily. But if the major premiss
is not necessary, but the minor is necessary, the conclusion will not be
necessary. For if it were, it would result both through the first figure
and through the third that A belongs necessarily to some B. But this is
false; for B may be such that it is possible that A should belong to none
of it. Further, an example also makes it clear that the conclusion not
be necessary, e.g. if A were movement, B animal, C man: man is an animal
necessarily, but an animal does not move necessarily, nor does man. Similarly
also if the major premiss is negative; for the proof is the
same.
In particular syllogisms, if the universal premiss is necessary,
then the conclusion will be necessary; but if the particular, the conclusion
will not be necessary, whether the universal premiss is negative or affirmative.
First let the universal be necessary, and let A belong to all B necessarily,
but let B simply belong to some C: it is necessary then that A belongs
to some C necessarily: for C falls under B, and A was assumed to belong
necessarily to all B. Similarly also if the syllogism should be negative:
for the proof will be the same. But if the particular premiss is necessary,
the conclusion will not be necessary: for from the denial of such a conclusion
nothing impossible results, just as it does not in the universal syllogisms.
The same is true of negative syllogisms. Try the terms movement, animal,
white.
Part 10
In the second figure, if the negative premiss is necessary, then
the conclusion will be necessary, but if the affirmative, not necessary.
First let the negative be necessary; let A be possible of no B, and simply
belong to C. Since then the negative statement is convertible, B is possible
of no A. But A belongs to all C; consequently B is possible of no C. For
C falls under A. The same result would be obtained if the minor premiss
were negative: for if A is possible be of no C, C is possible of no A:
but A belongs to all B, consequently C is possible of none of the Bs: for
again we have obtained the first figure. Neither then is B possible of
C: for conversion is possible without modifying the
relation.
But if the affirmative premiss is necessary, the conclusion will
not be necessary. Let A belong to all B necessarily, but to no C simply.
If then the negative premiss is converted, the first figure results. But
it has been proved in the case of the first figure that if the negative
major premiss is not necessary the conclusion will not be necessary either.
Therefore the same result will obtain here. Further, if the conclusion
is necessary, it follows that C necessarily does not belong to some A.
For if B necessarily belongs to no C, C will necessarily belong to no B.
But B at any rate must belong to some A, if it is true (as was assumed)
that A necessarily belongs to all B. Consequently it is necessary that
C does not belong to some A. But nothing prevents such an A being taken
that it is possible for C to belong to all of it. Further one might show
by an exposition of terms that the conclusion is not necessary without
qualification, though it is a necessary conclusion from the premisses.
For example let A be animal, B man, C white, and let the premisses be assumed
to correspond to what we had before: it is possible that animal should
belong to nothing white. Man then will not belong to anything white, but
not necessarily: for it is possible for man to be born white, not however
so long as animal belongs to nothing white. Consequently under these conditions
the conclusion will be necessary, but it is not necessary without
qualification.
Similar results will obtain also in particular syllogisms. For
whenever the negative premiss is both universal and necessary, then the
conclusion will be necessary: but whenever the affirmative premiss is universal,
the negative particular, the conclusion will not be necessary. First then
let the negative premiss be both universal and necessary: let it be possible
for no B that A should belong to it, and let A simply belong to some C.
Since the negative statement is convertible, it will be possible for no
A that B should belong to it: but A belongs to some C; consequently B necessarily
does not belong to some of the Cs. Again let the affirmative premiss be
both universal and necessary, and let the major premiss be affirmative.
If then A necessarily belongs to all B, but does not belong to some C,
it is clear that B will not belong to some C, but not necessarily. For
the same terms can be used to demonstrate the point, which were used in
the universal syllogisms. Nor again, if the negative statement is necessary
but particular, will the conclusion be necessary. The point can be demonstrated
by means of the same terms.
Part 11
In the last figure when the terms are related universally to the
middle, and both premisses are affirmative, if one of the two is necessary,
then the conclusion will be necessary. But if one is negative, the other
affirmative, whenever the negative is necessary the conclusion also will
be necessary, but whenever the affirmative is necessary the conclusion
will not be necessary. First let both the premisses be affirmative, and
let A and B belong to all C, and let Ac be necessary. Since then B belongs
to all C, C also will belong to some B, because the universal is convertible
into the particular: consequently if A belongs necessarily to all C, and
C belongs to some B, it is necessary that A should belong to some B also.
For B is under C. The first figure then is formed. A similar proof will
be given also if BC is necessary. For C is convertible with some A: consequently
if B belongs necessarily to all C, it will belong necessarily also to some
A.
Again let AC be negative, BC affirmative, and let the negative
premiss be necessary. Since then C is convertible with some B, but A necessarily
belongs to no C, A will necessarily not belong to some B either: for B
is under C. But if the affirmative is necessary, the conclusion will not
be necessary. For suppose BC is affirmative and necessary, while AC is
negative and not necessary. Since then the affirmative is convertible,
C also will belong to some B necessarily: consequently if A belongs to
none of the Cs, while C belongs to some of the Bs, A will not belong to
some of the Bs-but not of necessity; for it has been proved, in the case
of the first figure, that if the negative premiss is not necessary, neither
will the conclusion be necessary. Further, the point may be made clear
by considering the terms. Let the term A be 'good', let that which B signifies
be 'animal', let the term C be 'horse'. It is possible then that the term
good should belong to no horse, and it is necessary that the term animal
should belong to every horse: but it is not necessary that some animal
should not be good, since it is possible for every animal to be good. Or
if that is not possible, take as the term 'awake' or 'asleep': for every
animal can accept these.
If, then, the premisses are universal, we have stated when the
conclusion will be necessary. But if one premiss is universal, the other
particular, and if both are affirmative, whenever the universal is necessary
the conclusion also must be necessary. The demonstration is the same as
before; for the particular affirmative also is convertible. If then it
is necessary that B should belong to all C, and A falls under C, it is
necessary that B should belong to some A. But if B must belong to some
A, then A must belong to some B: for conversion is possible. Similarly
also if AC should be necessary and universal: for B falls under C. But
if the particular premiss is necessary, the conclusion will not be necessary.
Let the premiss BC be both particular and necessary, and let A belong to
all C, not however necessarily. If the proposition BC is converted the
first figure is formed, and the universal premiss is not necessary, but
the particular is necessary. But when the premisses were thus, the conclusion
(as we proved was not necessary: consequently it is not here either. Further,
the point is clear if we look at the terms. Let A be waking, B biped, and
C animal. It is necessary that B should belong to some C, but it is possible
for A to belong to C, and that A should belong to B is not necessary. For
there is no necessity that some biped should be asleep or awake. Similarly
and by means of the same terms proof can be made, should the proposition
Ac be both particular and necessary.
But if one premiss is affirmative, the other negative, whenever
the universal is both negative and necessary the conclusion also will be
necessary. For if it is not possible that A should belong to any C, but
B belongs to some C, it is necessary that A should not belong to some B.
But whenever the affirmative proposition is necessary, whether universal
or particular, or the negative is particular, the conclusion will not be
necessary. The proof of this by reduction will be the same as before; but
if terms are wanted, when the universal affirmative is necessary, take
the terms 'waking'-'animal'-'man', 'man' being middle, and when the affirmative
is particular and necessary, take the terms 'waking'-'animal'-'white':
for it is necessary that animal should belong to some white thing, but
it is possible that waking should belong to none, and it is not necessary
that waking should not belong to some animal. But when the negative proposition
being particular is necessary, take the terms 'biped', 'moving', 'animal',
'animal' being middle.
Part 12
It is clear then that a simple conclusion is not reached unless
both premisses are simple assertions, but a necessary conclusion is possible
although one only of the premisses is necessary. But in both cases, whether
the syllogisms are affirmative or negative, it is necessary that one premiss
should be similar to the conclusion. I mean by 'similar', if the conclusion
is a simple assertion, the premiss must be simple; if the conclusion is
necessary, the premiss must be necessary. Consequently this also is clear,
that the conclusion will be neither necessary nor simple unless a necessary
or simple premiss is assumed.
Part 13
Perhaps enough has been said about the proof of necessity, how
it comes about and how it differs from the proof of a simple statement.
We proceed to discuss that which is possible, when and how and by what
means it can be proved. I use the terms 'to be possible' and 'the possible'
of that which is not necessary but, being assumed, results in nothing impossible.
We say indeed ambiguously of the necessary that it is possible. But that
my definition of the possible is correct is clear from the phrases by which
we deny or on the contrary affirm possibility. For the expressions 'it
is not possible to belong', 'it is impossible to belong', and 'it is necessary
not to belong' are either identical or follow from one another; consequently
their opposites also, 'it is possible to belong', 'it is not impossible
to belong', and 'it is not necessary not to belong', will either be identical
or follow from one another. For of everything the affirmation or the denial
holds good. That which is possible then will be not necessary and that
which is not necessary will be possible. It results that all premisses
in the mode of possibility are convertible into one another. I mean not
that the affirmative are convertible into the negative, but that those
which are affirmative in form admit of conversion by opposition, e.g. 'it
is possible to belong' may be converted into 'it is possible not to belong',
and 'it is possible for A to belong to all B' into 'it is possible for
A to belong to no B' or 'not to all B', and 'it is possible for A to belong
to some B' into 'it is possible for A not to belong to some B'. And similarly
the other propositions in this mode can be converted. For since that which
is possible is not necessary, and that which is not necessary may possibly
not belong, it is clear that if it is possible that A should belong to
B, it is possible also that it should not belong to B: and if it is possible
that it should belong to all, it is also possible that it should not belong
to all. The same holds good in the case of particular affirmations: for
the proof is identical. And such premisses are affirmative and not negative;
for 'to be possible' is in the same rank as 'to be', as was said
above.
Having made these distinctions we next point out that the expression
'to be possible' is used in two ways. In one it means to happen generally
and fall short of necessity, e.g. man's turning grey or growing or decaying,
or generally what naturally belongs to a thing (for this has not its necessity
unbroken, since man's existence is not continuous for ever, although if
a man does exist, it comes about either necessarily or generally). In another
sense the expression means the indefinite, which can be both thus and not
thus, e.g. an animal's walking or an earthquake's taking place while it
is walking, or generally what happens by chance: for none of these inclines
by nature in the one way more than in the opposite.
That which is possible in each of its two senses is convertible
into its opposite, not however in the same way: but what is natural is
convertible because it does not necessarily belong (for in this sense it
is possible that a man should not grow grey) and what is indefinite is
convertible because it inclines this way no more than that. Science and
demonstrative syllogism are not concerned with things which are indefinite,
because the middle term is uncertain; but they are concerned with things
that are natural, and as a rule arguments and inquiries are made about
things which are possible in this sense. Syllogisms indeed can be made
about the former, but it is unusual at any rate to inquire about
them.
These matters will be treated more definitely in the sequel; our
business at present is to state the moods and nature of the syllogism made
from possible premisses. The expression 'it is possible for this to belong
to that' may be understood in two senses: 'that' may mean either that to
which 'that' belongs or that to which it may belong; for the expression
'A is possible of the subject of B' means that it is possible either of
that of which B is stated or of that of which B may possibly be stated.
It makes no difference whether we say, A is possible of the subject of
B, or all B admits of A. It is clear then that the expression 'A may possibly
belong to all B' might be used in two senses. First then we must state
the nature and characteristics of the syllogism which arises if B is possible
of the subject of C, and A is possible of the subject of B. For thus both
premisses are assumed in the mode of possibility; but whenever A is possible
of that of which B is true, one premiss is a simple assertion, the other
a problematic. Consequently we must start from premisses which are similar
in form, as in the other cases.
Part 14
Whenever A may possibly belong to all B, and B to all C, there
will be a perfect syllogism to prove that A may possibly belong to all
C. This is clear from the definition: for it was in this way that we explained
'to be possible for one term to belong to all of another'. Similarly if
it is possible for A to belong no B, and for B to belong to all C, then
it is possible for A to belong to no C. For the statement that it is possible
for A not to belong to that of which B may be true means (as we saw) that
none of those things which can possibly fall under the term B is left out
of account. But whenever A may belong to all B, and B may belong to no
C, then indeed no syllogism results from the premisses assumed, but if
the premiss BC is converted after the manner of problematic propositions,
the same syllogism results as before. For since it is possible that B should
belong to no C, it is possible also that it should belong to all C. This
has been stated above. Consequently if B is possible for all C, and A is
possible for all B, the same syllogism again results. Similarly if in both
the premisses the negative is joined with 'it is possible': e.g. if A may
belong to none of the Bs, and B to none of the Cs. No syllogism results
from the assumed premisses, but if they are converted we shall have the
same syllogism as before. It is clear then that if the minor premiss is
negative, or if both premisses are negative, either no syllogism results,
or if one it is not perfect. For the necessity results from the
conversion.
But if one of the premisses is universal, the other particular,
when the major premiss is universal there will be a perfect syllogism.
For if A is possible for all B, and B for some C, then A is possible for
some C. This is clear from the definition of being possible. Again if A
may belong to no B, and B may belong to some of the Cs, it is necessary
that A may possibly not belong to some of the Cs. The proof is the same
as above. But if the particular premiss is negative, and the universal
is affirmative, the major still being universal and the minor particular,
e.g. A is possible for all B, B may possibly not belong to some C, then
a clear syllogism does not result from the assumed premisses, but if the
particular premiss is converted and it is laid down that B possibly may
belong to some C, we shall have the same conclusion as before, as in the
cases given at the beginning.
But if the major premiss is the minor universal, whether both are
affirmative, or negative, or different in quality, or if both are indefinite
or particular, in no way will a syllogism be possible. For nothing prevents
B from reaching beyond A, so that as predicates cover unequal areas. Let
C be that by which B extends beyond A. To C it is not possible that A should
belong-either to all or to none or to some or not to some, since premisses
in the mode of possibility are convertible and it is possible for B to
belong to more things than A can. Further, this is obvious if we take terms;
for if the premisses are as assumed, the major term is both possible for
none of the minor and must belong to all of it. Take as terms common to
all the cases under consideration 'animal'-'white'-'man', where the major
belongs necessarily to the minor; 'animal'-'white'-'garment', where it
is not possible that the major should belong to the minor. It is clear
then that if the terms are related in this manner, no syllogism results.
For every syllogism proves that something belongs either simply or necessarily
or possibly. It is clear that there is no proof of the first or of the
second. For the affirmative is destroyed by the negative, and the negative
by the affirmative. There remains the proof of possibility. But this is
impossible. For it has been proved that if the terms are related in this
manner it is both necessary that the major should belong to all the minor
and not possible that it should belong to any. Consequently there cannot
be a syllogism to prove the possibility; for the necessary (as we stated)
is not possible.
It is clear that if the terms are universal in possible premisses
a syllogism always results in the first figure, whether they are affirmative
or negative, only a perfect syllogism results in the first case, an imperfect
in the second. But possibility must be understood according to the definition
laid down, not as covering necessity. This is sometimes
forgotten.
Part 15
If one premiss is a simple proposition, the other a problematic,
whenever the major premiss indicates possibility all the syllogisms will
be perfect and establish possibility in the sense defined; but whenever
the minor premiss indicates possibility all the syllogisms will be imperfect,
and those which are negative will establish not possibility according to
the definition, but that the major does not necessarily belong to any,
or to all, of the minor. For if this is so, we say it is possible that
it should belong to none or not to all. Let A be possible for all B, and
let B belong to all C. Since C falls under B, and A is possible for all
B, clearly it is possible for all C also. So a perfect syllogism results.
Likewise if the premiss AB is negative, and the premiss BC is affirmative,
the former stating possible, the latter simple attribution, a perfect syllogism
results proving that A possibly belongs to no C.
It is clear that perfect syllogisms result if the minor premiss
states simple belonging: but that syllogisms will result if the modality
of the premisses is reversed, must be proved per impossibile. At the same
time it will be evident that they are imperfect: for the proof proceeds
not from the premisses assumed. First we must state that if B's being follows
necessarily from A's being, B's possibility will follow necessarily from
A's possibility. Suppose, the terms being so related, that A is possible,
and B is impossible. If then that which is possible, when it is possible
for it to be, might happen, and if that which is impossible, when it is
impossible, could not happen, and if at the same time A is possible and
B impossible, it would be possible for A to happen without B, and if to
happen, then to be. For that which has happened, when it has happened,
is. But we must take the impossible and the possible not only in the sphere
of becoming, but also in the spheres of truth and predicability, and the
various other spheres in which we speak of the possible: for it will be
alike in all. Further we must understand the statement that B's being depends
on A's being, not as meaning that if some single thing A is, B will be:
for nothing follows of necessity from the being of some one thing, but
from two at least, i.e. when the premisses are related in the manner stated
to be that of the syllogism. For if C is predicated of D, and D of F, then
C is necessarily predicated of F. And if each is possible, the conclusion
also is possible. If then, for example, one should indicate the premisses
by A, and the conclusion by B, it would not only result that if A is necessary
B is necessary, but also that if A is possible, B is
possible.
Since this is proved it is evident that if a false and not impossible
assumption is made, the consequence of the assumption will also be false
and not impossible: e.g. if A is false, but not impossible, and if B is
the consequence of A, B also will be false but not impossible. For since
it has been proved that if B's being is the consequence of A's being, then
B's possibility will follow from A's possibility (and A is assumed to be
possible), consequently B will be possible: for if it were impossible,
the same thing would at the same time be possible and
impossible.
Since we have defined these points, let A belong to all B, and
B be possible for all C: it is necessary then that should be a possible
attribute for all C. Suppose that it is not possible, but assume that B
belongs to all C: this is false but not impossible. If then A is not possible
for C but B belongs to all C, then A is not possible for all B: for a syllogism
is formed in the third degree. But it was assumed that A is a possible
attribute for all B. It is necessary then that A is possible for all C.
For though the assumption we made is false and not impossible, the conclusion
is impossible. It is possible also in the first figure to bring about the
impossibility, by assuming that B belongs to C. For if B belongs to all
C, and A is possible for all B, then A would be possible for all C. But
the assumption was made that A is not possible for all
C.
We must understand 'that which belongs to all' with no limitation
in respect of time, e.g. to the present or to a particular period, but
simply without qualification. For it is by the help of such premisses that
we make syllogisms, since if the premiss is understood with reference to
the present moment, there cannot be a syllogism. For nothing perhaps prevents
'man' belonging at a particular time to everything that is moving, i.e.
if nothing else were moving: but 'moving' is possible for every horse;
yet 'man' is possible for no horse. Further let the major term be 'animal',
the middle 'moving', the the minor 'man'. The premisses then will be as
before, but the conclusion necessary, not possible. For man is necessarily
animal. It is clear then that the universal must be understood simply,
without limitation in respect of time.
Again let the premiss AB be universal and negative, and assume
that A belongs to no B, but B possibly belongs to all C. These propositions
being laid down, it is necessary that A possibly belongs to no C. Suppose
that it cannot belong, and that B belongs to C, as above. It is necessary
then that A belongs to some B: for we have a syllogism in the third figure:
but this is impossible. Thus it will be possible for A to belong to no
C; for if at is supposed false, the consequence is an impossible one. This
syllogism then does not establish that which is possible according to the
definition, but that which does not necessarily belong to any part of the
subject (for this is the contradictory of the assumption which was made:
for it was supposed that A necessarily belongs to some C, but the syllogism
per impossibile establishes the contradictory which is opposed to this).
Further, it is clear also from an example that the conclusion will not
establish possibility. Let A be 'raven', B 'intelligent', and C 'man'.
A then belongs to no B: for no intelligent thing is a raven. But B is possible
for all C: for every man may possibly be intelligent. But A necessarily
belongs to no C: so the conclusion does not establish possibility. But
neither is it always necessary. Let A be 'moving', B 'science', C 'man'.
A then will belong to no B; but B is possible for all C. And the conclusion
will not be necessary. For it is not necessary that no man should move;
rather it is not necessary that any man should move. Clearly then the conclusion
establishes that one term does not necessarily belong to any instance of
another term. But we must take our terms better.
If the minor premiss is negative and indicates possibility, from
the actual premisses taken there can be no syllogism, but if the problematic
premiss is converted, a syllogism will be possible, as before. Let A belong
to all B, and let B possibly belong to no C. If the terms are arranged
thus, nothing necessarily follows: but if the proposition BC is converted
and it is assumed that B is possible for all C, a syllogism results as
before: for the terms are in the same relative positions. Likewise if both
the relations are negative, if the major premiss states that A does not
belong to B, and the minor premiss indicates that B may possibly belong
to no C. Through the premisses actually taken nothing necessary results
in any way; but if the problematic premiss is converted, we shall have
a syllogism. Suppose that A belongs to no B, and B may possibly belong
to no C. Through these comes nothing necessary. But if B is assumed to
be possible for all C (and this is true) and if the premiss AB remains
as before, we shall again have the same syllogism. But if it be assumed
that B does not belong to any C, instead of possibly not belonging, there
cannot be a syllogism anyhow, whether the premiss AB is negative or affirmative.
As common instances of a necessary and positive relation we may take the
terms white-animal-snow: of a necessary and negative relation, white-animal-pitch.
Clearly then if the terms are universal, and one of the premisses is assertoric,
the other problematic, whenever the minor premiss is problematic a syllogism
always results, only sometimes it results from the premisses that are taken,
sometimes it requires the conversion of one premiss. We have stated when
each of these happens and the reason why. But if one of the relations is
universal, the other particular, then whenever the major premiss is universal
and problematic, whether affirmative or negative, and the particular is
affirmative and assertoric, there will be a perfect syllogism, just as
when the terms are universal. The demonstration is the same as before.
But whenever the major premiss is universal, but assertoric, not problematic,
and the minor is particular and problematic, whether both premisses are
negative or affirmative, or one is negative, the other affirmative, in
all cases there will be an imperfect syllogism. Only some of them will
be proved per impossibile, others by the conversion of the problematic
premiss, as has been shown above. And a syllogism will be possible by means
of conversion when the major premiss is universal and assertoric, whether
positive or negative, and the minor particular, negative, and problematic,
e.g. if A belongs to all B or to no B, and B may possibly not belong to
some C. For if the premiss BC is converted in respect of possibility, a
syllogism results. But whenever the particular premiss is assertoric and
negative, there cannot be a syllogism. As instances of the positive relation
we may take the terms white-animal-snow; of the negative, white-animal-pitch.
For the demonstration must be made through the indefinite nature of the
particular premiss. But if the minor premiss is universal, and the major
particular, whether either premiss is negative or affirmative, problematic
or assertoric, nohow is a syllogism possible. Nor is a syllogism possible
when the premisses are particular or indefinite, whether problematic or
assertoric, or the one problematic, the other assertoric. The demonstration
is the same as above. As instances of the necessary and positive relation
we may take the terms animal-white-man; of the necessary and negative relation,
animal-white-garment. It is evident then that if the major premiss is universal,
a syllogism always results, but if the minor is universal nothing at all
can ever be proved.
Part 16
Whenever one premiss is necessary, the other problematic, there
will be a syllogism when the terms are related as before; and a perfect
syllogism when the minor premiss is necessary. If the premisses are affirmative
the conclusion will be problematic, not assertoric, whether the premisses
are universal or not: but if one is affirmative, the other negative, when
the affirmative is necessary the conclusion will be problematic, not negative
assertoric; but when the negative is necessary the conclusion will be problematic
negative, and assertoric negative, whether the premisses are universal
or not. Possibility in the conclusion must be understood in the same manner
as before. There cannot be an inference to the necessary negative proposition:
for 'not necessarily to belong' is different from 'necessarily not to
belong'.
If the premisses are affirmative, clearly the conclusion which
follows is not necessary. Suppose A necessarily belongs to all B, and let
B be possible for all C. We shall have an imperfect syllogism to prove
that A may belong to all C. That it is imperfect is clear from the proof:
for it will be proved in the same manner as above. Again, let A be possible
for all B, and let B necessarily belong to all C. We shall then have a
syllogism to prove that A may belong to all C, not that A does belong to
all C: and it is perfect, not imperfect: for it is completed directly through
the original premisses.
But if the premisses are not similar in quality, suppose first
that the negative premiss is necessary, and let necessarily A not be possible
for any B, but let B be possible for all C. It is necessary then that A
belongs to no C. For suppose A to belong to all C or to some C. Now we
assumed that A is not possible for any B. Since then the negative proposition
is convertible, B is not possible for any A. But A is supposed to belong
to all C or to some C. Consequently B will not be possible for any C or
for all C. But it was originally laid down that B is possible for all C.
And it is clear that the possibility of belonging can be inferred, since
the fact of not belonging is inferred. Again, let the affirmative premiss
be necessary, and let A possibly not belong to any B, and let B necessarily
belong to all C. The syllogism will be perfect, but it will establish a
problematic negative, not an assertoric negative. For the major premiss
was problematic, and further it is not possible to prove the assertoric
conclusion per impossibile. For if it were supposed that A belongs to some
C, and it is laid down that A possibly does not belong to any B, no impossible
relation between B and C follows from these premisses. But if the minor
premiss is negative, when it is problematic a syllogism is possible by
conversion, as above; but when it is necessary no syllogism can be formed.
Nor again when both premisses are negative, and the minor is necessary.
The same terms as before serve both for the positive relation-white-animal-snow,
and for the negative relation-white-animal-pitch.
The same relation will obtain in particular syllogisms. Whenever
the negative proposition is necessary, the conclusion will be negative
assertoric: e.g. if it is not possible that A should belong to any B, but
B may belong to some of the Cs, it is necessary that A should not belong
to some of the Cs. For if A belongs to all C, but cannot belong to any
B, neither can B belong to any A. So if A belongs to all C, to none of
the Cs can B belong. But it was laid down that B may belong to some C.
But when the particular affirmative in the negative syllogism, e.g. BC
the minor premiss, or the universal proposition in the affirmative syllogism,
e.g. AB the major premiss, is necessary, there will not be an assertoric
conclusion. The demonstration is the same as before. But if the minor premiss
is universal, and problematic, whether affirmative or negative, and the
major premiss is particular and necessary, there cannot be a syllogism.
Premisses of this kind are possible both where the relation is positive
and necessary, e.g. animal-white-man, and where it is necessary and negative,
e.g. animal-white-garment. But when the universal is necessary, the particular
problematic, if the universal is negative we may take the terms animal-white-raven
to illustrate the positive relation, or animal-white-pitch to illustrate
the negative; and if the universal is affirmative we may take the terms
animal-white-swan to illustrate the positive relation, and animal-white-snow
to illustrate the negative and necessary relation. Nor again is a syllogism
possible when the premisses are indefinite, or both particular. Terms applicable
in either case to illustrate the positive relation are animal-white-man:
to illustrate the negative, animal-white-inanimate. For the relation of
animal to some white, and of white to some inanimate, is both necessary
and positive and necessary and negative. Similarly if the relation is problematic:
so the terms may be used for all cases.
Clearly then from what has been said a syllogism results or not
from similar relations of the terms whether we are dealing with simple
existence or necessity, with this exception, that if the negative premiss
is assertoric the conclusion is problematic, but if the negative premiss
is necessary the conclusion is both problematic and negative assertoric.
[It is clear also that all the syllogisms are imperfect and are perfected
by means of the figures above mentioned.]
Part 17
In the second figure whenever both premisses are problematic, no
syllogism is possible, whether the premisses are affirmative or negative,
universal or particular. But when one premiss is assertoric, the other
problematic, if the affirmative is assertoric no syllogism is possible,
but if the universal negative is assertoric a conclusion can always be
drawn. Similarly when one premiss is necessary, the other problematic.
Here also we must understand the term 'possible' in the conclusion, in
the same sense as before.
First we must point out that the negative problematic proposition
is not convertible, e.g. if A may belong to no B, it does not follow that
B may belong to no A. For suppose it to follow and assume that B may belong
to no A. Since then problematic affirmations are convertible with negations,
whether they are contraries or contradictories, and since B may belong
to no A, it is clear that B may belong to all A. But this is false: for
if all this can be that, it does not follow that all that can be this:
consequently the negative proposition is not convertible. Further, these
propositions are not incompatible, 'A may belong to no B', 'B necessarily
does not belong to some of the As'; e.g. it is possible that no man should
be white (for it is also possible that every man should be white), but
it is not true to say that it is possible that no white thing should be
a man: for many white things are necessarily not men, and the necessary
(as we saw) other than the possible.
Moreover it is not possible to prove the convertibility of these
propositions by a reductio ad absurdum, i.e. by claiming assent to the
following argument: 'since it is false that B may belong to no A, it is
true that it cannot belong to no A, for the one statement is the contradictory
of the other. But if this is so, it is true that B necessarily belongs
to some of the As: consequently A necessarily belongs to some of the Bs.
But this is impossible.' The argument cannot be admitted, for it does not
follow that some A is necessarily B, if it is not possible that no A should
be B. For the latter expression is used in two senses, one if A some is
necessarily B, another if some A is necessarily not B. For it is not true
to say that that which necessarily does not belong to some of the As may
possibly not belong to any A, just as it is not true to say that what necessarily
belongs to some A may possibly belong to all A. If any one then should
claim that because it is not possible for C to belong to all D, it necessarily
does not belong to some D, he would make a false assumption: for it does
belong to all D, but because in some cases it belongs necessarily, therefore
we say that it is not possible for it to belong to all. Hence both the
propositions 'A necessarily belongs to some B' and 'A necessarily does
not belong to some B' are opposed to the proposition 'A belongs to all
B'. Similarly also they are opposed to the proposition 'A may belong to
no B'. It is clear then that in relation to what is possible and not possible,
in the sense originally defined, we must assume, not that A necessarily
belongs to some B, but that A necessarily does not belong to some B. But
if this is assumed, no absurdity results: consequently no syllogism. It
is clear from what has been said that the negative proposition is not
convertible.
This being proved, suppose it possible that A may belong to no
B and to all C. By means of conversion no syllogism will result: for the
major premiss, as has been said, is not convertible. Nor can a proof be
obtained by a reductio ad absurdum: for if it is assumed that B can belong
to all C, no false consequence results: for A may belong both to all C
and to no C. In general, if there is a syllogism, it is clear that its
conclusion will be problematic because neither of the premisses is assertoric;
and this must be either affirmative or negative. But neither is possible.
Suppose the conclusion is affirmative: it will be proved by an example
that the predicate cannot belong to the subject. Suppose the conclusion
is negative: it will be proved that it is not problematic but necessary.
Let A be white, B man, C horse. It is possible then for A to belong to
all of the one and to none of the other. But it is not possible for B to
belong nor not to belong to C. That it is not possible for it to belong,
is clear. For no horse is a man. Neither is it possible for it not to belong.
For it is necessary that no horse should be a man, but the necessary we
found to be different from the possible. No syllogism then results. A similar
proof can be given if the major premiss is negative, the minor affirmative,
or if both are affirmative or negative. The demonstration can be made by
means of the same terms. And whenever one premiss is universal, the other
particular, or both are particular or indefinite, or in whatever other
way the premisses can be altered, the proof will always proceed through
the same terms. Clearly then, if both the premisses are problematic, no
syllogism results.
Part 18
But if one premiss is assertoric, the other problematic, if the
affirmative is assertoric and the negative problematic no syllogism will
be possible, whether the premisses are universal or particular. The proof
is the same as above, and by means of the same terms. But when the affirmative
premiss is problematic, and the negative assertoric, we shall have a syllogism.
Suppose A belongs to no B, but can belong to all C. If the negative proposition
is converted, B will belong to no A. But ex hypothesi can belong to all
C: so a syllogism is made, proving by means of the first figure that B
may belong to no C. Similarly also if the minor premiss is negative. But
if both premisses are negative, one being assertoric, the other problematic,
nothing follows necessarily from these premisses as they stand, but if
the problematic premiss is converted into its complementary affirmative
a syllogism is formed to prove that B may belong to no C, as before: for
we shall again have the first figure. But if both premisses are affirmative,
no syllogism will be possible. This arrangement of terms is possible both
when the relation is positive, e.g. health, animal, man, and when it is
negative, e.g. health, horse, man.
The same will hold good if the syllogisms are particular. Whenever
the affirmative proposition is assertoric, whether universal or particular,
no syllogism is possible (this is proved similarly and by the same examples
as above), but when the negative proposition is assertoric, a conclusion
can be drawn by means of conversion, as before. Again if both the relations
are negative, and the assertoric proposition is universal, although no
conclusion follows from the actual premisses, a syllogism can be obtained
by converting the problematic premiss into its complementary affirmative
as before. But if the negative proposition is assertoric, but particular,
no syllogism is possible, whether the other premiss is affirmative or negative.
Nor can a conclusion be drawn when both premisses are indefinite, whether
affirmative or negative, or particular. The proof is the same and by the
same terms.
Part 19
If one of the premisses is necessary, the other problematic, then
if the negative is necessary a syllogistic conclusion can be drawn, not
merely a negative problematic but also a negative assertoric conclusion;
but if the affirmative premiss is necessary, no conclusion is possible.
Suppose that A necessarily belongs to no B, but may belong to all C. If
the negative premiss is converted B will belong to no A: but A ex hypothesi
is capable of belonging to all C: so once more a conclusion is drawn by
the first figure that B may belong to no C. But at the same time it is
clear that B will not belong to any C. For assume that it does: then if
A cannot belong to any B, and B belongs to some of the Cs, A cannot belong
to some of the Cs: but ex hypothesi it may belong to all. A similar proof
can be given if the minor premiss is negative. Again let the affirmative
proposition be necessary, and the other problematic; i.e. suppose that
A may belong to no B, but necessarily belongs to all C. When the terms
are arranged in this way, no syllogism is possible. For (1) it sometimes
turns out that B necessarily does not belong to C. Let A be white, B man,
C swan. White then necessarily belongs to swan, but may belong to no man;
and man necessarily belongs to no swan; Clearly then we cannot draw a problematic
conclusion; for that which is necessary is admittedly distinct from that
which is possible. (2) Nor again can we draw a necessary conclusion: for
that presupposes that both premisses are necessary, or at any rate the
negative premiss. (3) Further it is possible also, when the terms are so
arranged, that B should belong to C: for nothing prevents C falling under
B, A being possible for all B, and necessarily belonging to C; e.g. if
C stands for 'awake', B for 'animal', A for 'motion'. For motion necessarily
belongs to what is awake, and is possible for every animal: and everything
that is awake is animal. Clearly then the conclusion cannot be the negative
assertion, if the relation must be positive when the terms are related
as above. Nor can the opposite affirmations be established: consequently
no syllogism is possible. A similar proof is possible if the major premiss
is affirmative.
But if the premisses are similar in quality, when they are negative
a syllogism can always be formed by converting the problematic premiss
into its complementary affirmative as before. Suppose A necessarily does
not belong to B, and possibly may not belong to C: if the premisses are
converted B belongs to no A, and A may possibly belong to all C: thus we
have the first figure. Similarly if the minor premiss is negative. But
if the premisses are affirmative there cannot be a syllogism. Clearly the
conclusion cannot be a negative assertoric or a negative necessary proposition
because no negative premiss has been laid down either in the assertoric
or in the necessary mode. Nor can the conclusion be a problematic negative
proposition. For if the terms are so related, there are cases in which
B necessarily will not belong to C; e.g. suppose that A is white, B swan,
C man. Nor can the opposite affirmations be established, since we have
shown a case in which B necessarily does not belong to C. A syllogism then
is not possible at all.
Similar relations will obtain in particular syllogisms. For whenever
the negative proposition is universal and necessary, a syllogism will always
be possible to prove both a problematic and a negative assertoric proposition
(the proof proceeds by conversion); but when the affirmative proposition
is universal and necessary, no syllogistic conclusion can be drawn. This
can be proved in the same way as for universal propositions, and by the
same terms. Nor is a syllogistic conclusion possible when both premisses
are affirmative: this also may be proved as above. But when both premisses
are negative, and the premiss that definitely disconnects two terms is
universal and necessary, though nothing follows necessarily from the premisses
as they are stated, a conclusion can be drawn as above if the problematic
premiss is converted into its complementary affirmative. But if both are
indefinite or particular, no syllogism can be formed. The same proof will
serve, and the same terms.
It is clear then from what has been said that if the universal
and negative premiss is necessary, a syllogism is always possible, proving
not merely a negative problematic, but also a negative assertoric proposition;
but if the affirmative premiss is necessary no conclusion can be drawn.
It is clear too that a syllogism is possible or not under the same conditions
whether the mode of the premisses is assertoric or necessary. And it is
clear that all the syllogisms are imperfect, and are completed by means
of the figures mentioned.
Part 20
In the last figure a syllogism is possible whether both or only one
of the premisses is problematic. When the premisses are problematic the
conclusion will be problematic; and also when one premiss is problematic,
the other assertoric. But when the other premiss is necessary, if it is
affirmative the conclusion will be neither necessary or assertoric; but
if it is negative the syllogism will result in a negative assertoric proposition,
as above. In these also we must understand the expression 'possible' in
the conclusion in the same way as before.
First let the premisses be problematic and suppose that both A
and B may possibly belong to every C. Since then the affirmative proposition
is convertible into a particular, and B may possibly belong to every C,
it follows that C may possibly belong to some B. So, if A is possible for
every C, and C is possible for some of the Bs, then A is possible for some
of the Bs. For we have got the first figure. And A if may possibly belong
to no C, but B may possibly belong to all C, it follows that A may possibly
not belong to some B: for we shall have the first figure again by conversion.
But if both premisses should be negative no necessary consequence will
follow from them as they are stated, but if the premisses are converted
into their corresponding affirmatives there will be a syllogism as before.
For if A and B may possibly not belong to C, if 'may possibly belong' is
substituted we shall again have the first figure by means of conversion.
But if one of the premisses is universal, the other particular, a syllogism
will be possible, or not, under the arrangement of the terms as in the
case of assertoric propositions. Suppose that A may possibly belong to
all C, and B to some C. We shall have the first figure again if the particular
premiss is converted. For if A is possible for all C, and C for some of
the Bs, then A is possible for some of the Bs. Similarly if the proposition
BC is universal. Likewise also if the proposition AC is negative, and the
proposition BC affirmative: for we shall again have the first figure by
conversion. But if both premisses should be negative-the one universal
and the other particular-although no syllogistic conclusion will follow
from the premisses as they are put, it will follow if they are converted,
as above. But when both premisses are indefinite or particular, no syllogism
can be formed: for A must belong sometimes to all B and sometimes to no
B. To illustrate the affirmative relation take the terms animal-man-white;
to illustrate the negative, take the terms horse-man-white--white being
the middle term.
Part 21
If one premiss is pure, the other problematic, the conclusion will
be problematic, not pure; and a syllogism will be possible under the same
arrangement of the terms as before. First let the premisses be affirmative:
suppose that A belongs to all C, and B may possibly belong to all C. If
the proposition BC is converted, we shall have the first figure, and the
conclusion that A may possibly belong to some of the Bs. For when one of
the premisses in the first figure is problematic, the conclusion also (as
we saw) is problematic. Similarly if the proposition BC is pure, AC problematic;
or if AC is negative, Bc affirmative, no matter which of the two is pure;
in both cases the conclusion will be problematic: for the first figure
is obtained once more, and it has been proved that if one premiss is problematic
in that figure the conclusion also will be problematic. But if the minor
premiss BC is negative, or if both premisses are negative, no syllogistic
conclusion can be drawn from the premisses as they stand, but if they are
converted a syllogism is obtained as before.
If one of the premisses is universal, the other particular, then
when both are affirmative, or when the universal is negative, the particular
affirmative, we shall have the same sort of syllogisms: for all are completed
by means of the first figure. So it is clear that we shall have not a pure
but a problematic syllogistic conclusion. But if the affirmative premiss
is universal, the negative particular, the proof will proceed by a reductio
ad impossibile. Suppose that B belongs to all C, and A may possibly not
belong to some C: it follows that may possibly not belong to some B. For
if A necessarily belongs to all B, and B (as has been assumed) belongs
to all C, A will necessarily belong to all C: for this has been proved
before. But it was assumed at the outset that A may possibly not belong
to some C.
Whenever both premisses are indefinite or particular, no syllogism
will be possible. The demonstration is the same as was given in the case
of universal premisses, and proceeds by means of the same
terms.
Part 22
If one of the premisses is necessary, the other problematic, when
the premisses are affirmative a problematic affirmative conclusion can
always be drawn; when one proposition is affirmative, the other negative,
if the affirmative is necessary a problematic negative can be inferred;
but if the negative proposition is necessary both a problematic and a pure
negative conclusion are possible. But a necessary negative conclusion will
not be possible, any more than in the other figures. Suppose first that
the premisses are affirmative, i.e. that A necessarily belongs to all C,
and B may possibly belong to all C. Since then A must belong to all C,
and C may belong to some B, it follows that A may (not does) belong to
some B: for so it resulted in the first figure. A similar proof may be
given if the proposition BC is necessary, and AC is problematic. Again
suppose one proposition is affirmative, the other negative, the affirmative
being necessary: i.e. suppose A may possibly belong to no C, but B necessarily
belongs to all C. We shall have the first figure once more: and-since the
negative premiss is problematic-it is clear that the conclusion will be
problematic: for when the premisses stand thus in the first figure, the
conclusion (as we found) is problematic. But if the negative premiss is
necessary, the conclusion will be not only that A may possibly not belong
to some B but also that it does not belong to some B. For suppose that
A necessarily does not belong to C, but B may belong to all C. If the affirmative
proposition BC is converted, we shall have the first figure, and the negative
premiss is necessary. But when the premisses stood thus, it resulted that
A might possibly not belong to some C, and that it did not belong to some
C; consequently here it follows that A does not belong to some B. But when
the minor premiss is negative, if it is problematic we shall have a syllogism
by altering the premiss into its complementary affirmative, as before;
but if it is necessary no syllogism can be formed. For A sometimes necessarily
belongs to all B, and sometimes cannot possibly belong to any B. To illustrate
the former take the terms sleep-sleeping horse-man; to illustrate the latter
take the terms sleep-waking horse-man.
Similar results will obtain if one of the terms is related universally
to the middle, the other in part. If both premisses are affirmative, the
conclusion will be problematic, not pure; and also when one premiss is
negative, the other affirmative, the latter being necessary. But when the
negative premiss is necessary, the conclusion also will be a pure negative
proposition; for the same kind of proof can be given whether the terms
are universal or not. For the syllogisms must be made perfect by means
of the first figure, so that a result which follows in the first figure
follows also in the third. But when the minor premiss is negative and universal,
if it is problematic a syllogism can be formed by means of conversion;
but if it is necessary a syllogism is not possible. The proof will follow
the same course as where the premisses are universal; and the same terms
may be used.
It is clear then in this figure also when and how a syllogism can
be formed, and when the conclusion is problematic, and when it is pure.
It is evident also that all syllogisms in this figure are imperfect, and
that they are made perfect by means of the first figure.
Part 23
It is clear from what has been said that the syllogisms in these
figures are made perfect by means of universal syllogisms in the first
figure and are reduced to them. That every syllogism without qualification
can be so treated, will be clear presently, when it has been proved that
every syllogism is formed through one or other of these
figures.
It is necessary that every demonstration and every syllogism should
prove either that something belongs or that it does not, and this either
universally or in part, and further either ostensively or hypothetically.
One sort of hypothetical proof is the reductio ad impossibile. Let us speak
first of ostensive syllogisms: for after these have been pointed out the
truth of our contention will be clear with regard to those which are proved
per impossibile, and in general hypothetically.
If then one wants to prove syllogistically A of B, either as an
attribute of it or as not an attribute of it, one must assert something
of something else. If now A should be asserted of B, the proposition originally
in question will have been assumed. But if A should be asserted of C, but
C should not be asserted of anything, nor anything of it, nor anything
else of A, no syllogism will be possible. For nothing necessarily follows
from the assertion of some one thing concerning some other single thing.
Thus we must take another premiss as well. If then A be asserted of something
else, or something else of A, or something different of C, nothing prevents
a syllogism being formed, but it will not be in relation to B through the
premisses taken. Nor when C belongs to something else, and that to something
else and so on, no connexion however being made with B, will a syllogism
be possible concerning A in its relation to B. For in general we stated
that no syllogism can establish the attribution of one thing to another,
unless some middle term is taken, which is somehow related to each by way
of predication. For the syllogism in general is made out of premisses,
and a syllogism referring to this out of premisses with the same reference,
and a syllogism relating this to that proceeds through premisses which
relate this to that. But it is impossible to take a premiss in reference
to B, if we neither affirm nor deny anything of it; or again to take a
premiss relating A to B, if we take nothing common, but affirm or deny
peculiar attributes of each. So we must take something midway between the
two, which will connect the predications, if we are to have a syllogism
relating this to that. If then we must take something common in relation
to both, and this is possible in three ways (either by predicating A of
C, and C of B, or C of both, or both of C), and these are the figures of
which we have spoken, it is clear that every syllogism must be made in
one or other of these figures. The argument is the same if several middle
terms should be necessary to establish the relation to B; for the figure
will be the same whether there is one middle term or
many.
It is clear then that the ostensive syllogisms are effected by
means of the aforesaid figures; these considerations will show that reductiones
ad also are effected in the same way. For all who effect an argument per
impossibile infer syllogistically what is false, and prove the original
conclusion hypothetically when something impossible results from the assumption
of its contradictory; e.g. that the diagonal of the square is incommensurate
with the side, because odd numbers are equal to evens if it is supposed
to be commensurate. One infers syllogistically that odd numbers come out
equal to evens, and one proves hypothetically the incommensurability of
the diagonal, since a falsehood results through contradicting this. For
this we found to be reasoning per impossibile, viz. proving something impossible
by means of an hypothesis conceded at the beginning. Consequently, since
the falsehood is established in reductions ad impossibile by an ostensive
syllogism, and the original conclusion is proved hypothetically, and we
have already stated that ostensive syllogisms are effected by means of
these figures, it is evident that syllogisms per impossibile also will
be made through these figures. Likewise all the other hypothetical syllogisms:
for in every case the syllogism leads up to the proposition that is substituted
for the original thesis; but the original thesis is reached by means of
a concession or some other hypothesis. But if this is true, every demonstration
and every syllogism must be formed by means of the three figures mentioned
above. But when this has been shown it is clear that every syllogism is
perfected by means of the first figure and is reducible to the universal
syllogisms in this figure.
Part 24
Further in every syllogism one of the premisses must be affirmative,
and universality must be present: unless one of the premisses is universal
either a syllogism will not be possible, or it will not refer to the subject
proposed, or the original position will be begged. Suppose we have to prove
that pleasure in music is good. If one should claim as a premiss that pleasure
is good without adding 'all', no syllogism will be possible; if one should
claim that some pleasure is good, then if it is different from pleasure
in music, it is not relevant to the subject proposed; if it is this very
pleasure, one is assuming that which was proposed at the outset to be proved.
This is more obvious in geometrical proofs, e.g. that the angles at the
base of an isosceles triangle are equal. Suppose the lines A and B have
been drawn to the centre. If then one should assume that the angle AC is
equal to the angle BD, without claiming generally that angles of semicircles
are equal; and again if one should assume that the angle C is equal to
the angle D, without the additional assumption that every angle of a segment
is equal to every other angle of the same segment; and further if one should
assume that when equal angles are taken from the whole angles, which are
themselves equal, the remainders E and F are equal, he will beg the thing
to be proved, unless he also states that when equals are taken from equals
the remainders are equal.
It is clear then that in every syllogism there must be a universal
premiss, and that a universal statement is proved only when all the premisses
are universal, while a particular statement is proved both from two universal
premisses and from one only: consequently if the conclusion is universal,
the premisses also must be universal, but if the premisses are universal
it is possible that the conclusion may not be universal. And it is clear
also that in every syllogism either both or one of the premisses must be
like the conclusion. I mean not only in being affirmative or negative,
but also in being necessary, pure, problematic. We must consider also the
other forms of predication.
It is clear also when a syllogism in general can be made and when
it cannot; and when a valid, when a perfect syllogism can be formed; and
that if a syllogism is formed the terms must be arranged in one of the
ways that have been mentioned.
Part 25
It is clear too that every demonstration will proceed through three
terms and no more, unless the same conclusion is established by different
pairs of propositions; e.g. the conclusion E may be established through
the propositions A and B, and through the propositions C and D, or through
the propositions A and B, or A and C, or B and C. For nothing prevents
there being several middles for the same terms. But in that case there
is not one but several syllogisms. Or again when each of the propositions
A and B is obtained by syllogistic inference, e.g. by means of D and E,
and again B by means of F and G. Or one may be obtained by syllogistic,
the other by inductive inference. But thus also the syllogisms are many;
for the conclusions are many, e.g. A and B and C. But if this can be called
one syllogism, not many, the same conclusion may be reached by more than
three terms in this way, but it cannot be reached as C is established by
means of A and B. Suppose that the proposition E is inferred from the premisses
A, B, C, and D. It is necessary then that of these one should be related
to another as whole to part: for it has already been proved that if a syllogism
is formed some of its terms must be related in this way. Suppose then that
A stands in this relation to B. Some conclusion then follows from them.
It must either be E or one or other of C and D, or something other than
these.
(1) If it is E the syllogism will have A and B for its sole premisses.
But if C and D are so related that one is whole, the other part, some conclusion
will follow from them also; and it must be either E, or one or other of
the propositions A and B, or something other than these. And if it is (i)
E, or (ii) A or B, either (i) the syllogisms will be more than one, or
(ii) the same thing happens to be inferred by means of several terms only
in the sense which we saw to be possible. But if (iii) the conclusion is
other than E or A or B, the syllogisms will be many, and unconnected with
one another. But if C is not so related to D as to make a syllogism, the
propositions will have been assumed to no purpose, unless for the sake
of induction or of obscuring the argument or something of the
sort.
(2) But if from the propositions A and B there follows not E but
some other conclusion, and if from C and D either A or B follows or something
else, then there are several syllogisms, and they do not establish the
conclusion proposed: for we assumed that the syllogism proved E. And if
no conclusion follows from C and D, it turns out that these propositions
have been assumed to no purpose, and the syllogism does not prove the original
proposition.
So it is clear that every demonstration and every syllogism will
proceed through three terms only.
This being evident, it is clear that a syllogistic conclusion follows
from two premisses and not from more than two. For the three terms make
two premisses, unless a new premiss is assumed, as was said at the beginning,
to perfect the syllogisms. It is clear therefore that in whatever syllogistic
argument the premisses through which the main conclusion follows (for some
of the preceding conclusions must be premisses) are not even in number,
this argument either has not been drawn syllogistically or it has assumed
more than was necessary to establish its thesis.
If then syllogisms are taken with respect to their main premisses,
every syllogism will consist of an even number of premisses and an odd
number of terms (for the terms exceed the premisses by one), and the conclusions
will be half the number of the premisses. But whenever a conclusion is
reached by means of prosyllogisms or by means of several continuous middle
terms, e.g. the proposition AB by means of the middle terms C and D, the
number of the terms will similarly exceed that of the premisses by one
(for the extra term must either be added outside or inserted: but in either
case it follows that the relations of predication are one fewer than the
terms related), and the premisses will be equal in number to the relations
of predication. The premisses however will not always be even, the terms
odd; but they will alternate-when the premisses are even, the terms must
be odd; when the terms are even, the premisses must be odd: for along with
one term one premiss is added, if a term is added from any quarter. Consequently
since the premisses were (as we saw) even, and the terms odd, we must make
them alternately even and odd at each addition. But the conclusions will
not follow the same arrangement either in respect to the terms or to the
premisses. For if one term is added, conclusions will be added less by
one than the pre-existing terms: for the conclusion is drawn not in relation
to the single term last added, but in relation to all the rest, e.g. if
to ABC the term D is added, two conclusions are thereby added, one in relation
to A, the other in relation to B. Similarly with any further additions.
And similarly too if the term is inserted in the middle: for in relation
to one term only, a syllogism will not be constructed. Consequently the
conclusions will be much more numerous than the terms or the
premisses.
Part 26
Since we understand the subjects with which syllogisms are concerned,
what sort of conclusion is established in each figure, and in how many
moods this is done, it is evident to us both what sort of problem is difficult
and what sort is easy to prove. For that which is concluded in many figures
and through many moods is easier; that which is concluded in few figures
and through few moods is more difficult to attempt. The universal affirmative
is proved by means of the first figure only and by this in only one mood;
the universal negative is proved both through the first figure and through
the second, through the first in one mood, through the second in two. The
particular affirmative is proved through the first and through the last
figure, in one mood through the first, in three moods through the last.
The particular negative is proved in all the figures, but once in the first,
in two moods in the second, in three moods in the third. It is clear then
that the universal affirmative is most difficult to establish, most easy
to overthrow. In general, universals are easier game for the destroyer
than particulars: for whether the predicate belongs to none or not to some,
they are destroyed: and the particular negative is proved in all the figures,
the universal negative in two. Similarly with universal negatives: the
original statement is destroyed, whether the predicate belongs to all or
to some: and this we found possible in two figures. But particular statements
can be refuted in one way only-by proving that the predicate belongs either
to all or to none. But particular statements are easier to establish: for
proof is possible in more figures and through more moods. And in general
we must not forget that it is possible to refute statements by means of
one another, I mean, universal statements by means of particular, and particular
statements by means of universal: but it is not possible to establish universal
statements by means of particular, though it is possible to establish particular
statements by means of universal. At the same time it is evident that it
is easier to refute than to establish.
The manner in which every syllogism is produced, the number of
the terms and premisses through which it proceeds, the relation of the
premisses to one another, the character of the problem proved in each figure,
and the number of the figures appropriate to each problem, all these matters
are clear from what has been said.
Part 27
We must now state how we may ourselves always have a supply of
syllogisms in reference to the problem proposed and by what road we may
reach the principles relative to the problem: for perhaps we ought not
only to investigate the construction of syllogisms, but also to have the
power of making them.
Of all the things which exist some are such that they cannot be
predicated of anything else truly and universally, e.g. Cleon and Callias,
i.e. the individual and sensible, but other things may be predicated of
them (for each of these is both man and animal); and some things are themselves
predicated of others, but nothing prior is predicated of them; and some
are predicated of others, and yet others of them, e.g. man of Callias and
animal of man. It is clear then that some things are naturally not stated
of anything: for as a rule each sensible thing is such that it cannot be
predicated of anything, save incidentally: for we sometimes say that that
white object is Socrates, or that that which approaches is Callias. We
shall explain in another place that there is an upward limit also to the
process of predicating: for the present we must assume this. Of these ultimate
predicates it is not possible to demonstrate another predicate, save as
a matter of opinion, but these may be predicated of other things. Neither
can individuals be predicated of other things, though other things can
be predicated of them. Whatever lies between these limits can be spoken
of in both ways: they may be stated of others, and others stated of them.
And as a rule arguments and inquiries are concerned with these things.
We must select the premisses suitable to each problem in this manner: first
we must lay down the subject and the definitions and the properties of
the thing; next we must lay down those attributes which follow the thing,
and again those which the thing follows, and those which cannot belong
to it. But those to which it cannot belong need not be selected, because
the negative statement implied above is convertible. Of the attributes
which follow we must distinguish those which fall within the definition,
those which are predicated as properties, and those which are predicated
as accidents, and of the latter those which apparently and those which
really belong. The larger the supply a man has of these, the more quickly
will he reach a conclusion; and in proportion as he apprehends those which
are truer, the more cogently will he demonstrate. But he must select not
those which follow some particular but those which follow the thing as
a whole, e.g. not what follows a particular man but what follows every
man: for the syllogism proceeds through universal premisses. If the statement
is indefinite, it is uncertain whether the premiss is universal, but if
the statement is definite, the matter is clear. Similarly one must select
those attributes which the subject follows as wholes, for the reason given.
But that which follows one must not suppose to follow as a whole, e.g.
that every animal follows man or every science music, but only that it
follows, without qualification, and indeed we state it in a proposition:
for the other statement is useless and impossible, e.g. that every man
is every animal or justice is all good. But that which something follows
receives the mark 'every'. Whenever the subject, for which we must obtain
the attributes that follow, is contained by something else, what follows
or does not follow the highest term universally must not be selected in
dealing with the subordinate term (for these attributes have been taken
in dealing with the superior term; for what follows animal also follows
man, and what does not belong to animal does not belong to man); but we
must choose those attributes which are peculiar to each subject. For some
things are peculiar to the species as distinct from the genus; for species
being distinct there must be attributes peculiar to each. Nor must we take
as things which the superior term follows, those things which the inferior
term follows, e.g. take as subjects of the predicate 'animal' what are
really subjects of the predicate 'man'. It is necessary indeed, if animal
follows man, that it should follow all these also. But these belong more
properly to the choice of what concerns man. One must apprehend also normal
consequents and normal antecedents-, for propositions which obtain normally
are established syllogistically from premisses which obtain normally, some
if not all of them having this character of normality. For the conclusion
of each syllogism resembles its principles. We must not however choose
attributes which are consequent upon all the terms: for no syllogism can
be made out of such premisses. The reason why this is so will be clear
in the sequel.
Part 28
If men wish to establish something about some whole, they must
look to the subjects of that which is being established (the subjects of
which it happens to be asserted), and the attributes which follow that
of which it is to be predicated. For if any of these subjects is the same
as any of these attributes, the attribute originally in question must belong
to the subject originally in question. But if the purpose is to establish
not a universal but a particular proposition, they must look for the terms
of which the terms in question are predicable: for if any of these are
identical, the attribute in question must belong to some of the subject
in question. Whenever the one term has to belong to none of the other,
one must look to the consequents of the subject, and to those attributes
which cannot possibly be present in the predicate in question: or conversely
to the attributes which cannot possibly be present in the subject, and
to the consequents of the predicate. If any members of these groups are
identical, one of the terms in question cannot possibly belong to any of
the other. For sometimes a syllogism in the first figure results, sometimes
a syllogism in the second. But if the object is to establish a particular
negative proposition, we must find antecedents of the subject in question
and attributes which cannot possibly belong to the predicate in question.
If any members of these two groups are identical, it follows that one of
the terms in question does not belong to some of the other. Perhaps each
of these statements will become clearer in the following way. Suppose the
consequents of A are designated by B, the antecedents of A by C, attributes
which cannot possibly belong to A by D. Suppose again that the attributes
of E are designated by F, the antecedents of E by G, and attributes which
cannot belong to E by H. If then one of the Cs should be identical with
one of the Fs, A must belong to all E: for F belongs to all E, and A to
all C, consequently A belongs to all E. If C and G are identical, A must
belong to some of the Es: for A follows C, and E follows all G. If F and
D are identical, A will belong to none of the Es by a prosyllogism: for
since the negative proposition is convertible, and F is identical with
D, A will belong to none of the Fs, but F belongs to all E. Again, if B
and H are identical, A will belong to none of the Es: for B will belong
to all A, but to no E: for it was assumed to be identical with H, and H
belonged to none of the Es. If D and G are identical, A will not belong
to some of the Es: for it will not belong to G, because it does not belong
to D: but G falls under E: consequently A will not belong to some of the
Es. If B is identical with G, there will be a converted syllogism: for
E will belong to all A since B belongs to A and E to B (for B was found
to be identical with G): but that A should belong to all E is not necessary,
but it must belong to some E because it is possible to convert the universal
statement into a particular.
It is clear then that in every proposition which requires proof
we must look to the aforesaid relations of the subject and predicate in
question: for all syllogisms proceed through these. But if we are seeking
consequents and antecedents we must look for those which are primary and
most universal, e.g. in reference to E we must look to Kf rather than to
F alone, and in reference to A we must look to KC rather than to C alone.
For if A belongs to KF, it belongs both to F and to E: but if it does not
follow KF, it may yet follow F. Similarly we must consider the antecedents
of A itself: for if a term follows the primary antecedents, it will follow
those also which are subordinate, but if it does not follow the former,
it may yet follow the latter.
It is clear too that the inquiry proceeds through the three terms
and the two premisses, and that all the syllogisms proceed through the
aforesaid figures. For it is proved that A belongs to all E, whenever an
identical term is found among the Cs and Fs. This will be the middle term;
A and E will be the extremes. So the first figure is formed. And A will
belong to some E, whenever C and G are apprehended to be the same. This
is the last figure: for G becomes the middle term. And A will belong to
no E, when D and F are identical. Thus we have both the first figure and
the middle figure; the first, because A belongs to no F, since the negative
statement is convertible, and F belongs to all E: the middle figure because
D belongs to no A, and to all E. And A will not belong to some E, whenever
D and G are identical. This is the last figure: for A will belong to no
G, and E will belong to all G. Clearly then all syllogisms proceed through
the aforesaid figures, and we must not select consequents of all the terms,
because no syllogism is produced from them. For (as we saw) it is not possible
at all to establish a proposition from consequents, and it is not possible
to refute by means of a consequent of both the terms in question: for the
middle term must belong to the one, and not belong to the
other.
It is clear too that other methods of inquiry by selection of middle
terms are useless to produce a syllogism, e.g. if the consequents of the
terms in question are identical, or if the antecedents of A are identical
with those attributes which cannot possibly belong to E, or if those attributes
are identical which cannot belong to either term: for no syllogism is produced
by means of these. For if the consequents are identical, e.g. B and F,
we have the middle figure with both premisses affirmative: if the antecedents
of A are identical with attributes which cannot belong to E, e.g. C with
H, we have the first figure with its minor premiss negative. If attributes
which cannot belong to either term are identical, e.g. C and H, both premisses
are negative, either in the first or in the middle figure. But no syllogism
is possible in this way.
It is evident too that we must find out which terms in this inquiry
are identical, not which are different or contrary, first because the object
of our investigation is the middle term, and the middle term must be not
diverse but identical. Secondly, wherever it happens that a syllogism results
from taking contraries or terms which cannot belong to the same thing,
all arguments can be reduced to the aforesaid moods, e.g. if B and F are
contraries or cannot belong to the same thing. For if these are taken,
a syllogism will be formed to prove that A belongs to none of the Es, not
however from the premisses taken but in the aforesaid mood. For B will
belong to all A and to no E. Consequently B must be identical with one
of the Hs. Again, if B and G cannot belong to the same thing, it follows
that A will not belong to some of the Es: for then too we shall have the
middle figure: for B will belong to all A and to no G. Consequently B must
be identical with some of the Hs. For the fact that B and G cannot belong
to the same thing differs in no way from the fact that B is identical with
some of the Hs: for that includes everything which cannot belong to
E.
It is clear then that from the inquiries taken by themselves no
syllogism results; but if B and F are contraries B must be identical with
one of the Hs, and the syllogism results through these terms. It turns
out then that those who inquire in this manner are looking gratuitously
for some other way than the necessary way because they have failed to observe
the identity of the Bs with the Hs.
Part 29
Syllogisms which lead to impossible conclusions are similar to
ostensive syllogisms; they also are formed by means of the consequents
and antecedents of the terms in question. In both cases the same inquiry
is involved. For what is proved ostensively may also be concluded syllogistically
per impossibile by means of the same terms; and what is proved per impossibile
may also be proved ostensively, e.g. that A belongs to none of the Es.
For suppose A to belong to some E: then since B belongs to all A and A
to some of the Es, B will belong to some of the Es: but it was assumed
that it belongs to none. Again we may prove that A belongs to some E: for
if A belonged to none of the Es, and E belongs to all G, A will belong
to none of the Gs: but it was assumed to belong to all. Similarly with
the other propositions requiring proof. The proof per impossibile will
always and in all cases be from the consequents and antecedents of the
terms in question. Whatever the problem the same inquiry is necessary whether
one wishes to use an ostensive syllogism or a reduction to impossibility.
For both the demonstrations start from the same terms, e.g. suppose it
has been proved that A belongs to no E, because it turns out that otherwise
B belongs to some of the Es and this is impossible-if now it is assumed
that B belongs to no E and to all A, it is clear that A will belong to
no E. Again if it has been proved by an ostensive syllogism that A belongs
to no E, assume that A belongs to some E and it will be proved per impossibile
to belong to no E. Similarly with the rest. In all cases it is necessary
to find some common term other than the subjects of inquiry, to which the
syllogism establishing the false conclusion may relate, so that if this
premiss is converted, and the other remains as it is, the syllogism will
be ostensive by means of the same terms. For the ostensive syllogism differs
from the reductio ad impossibile in this: in the ostensive syllogism both
remisses are laid down in accordance with the truth, in the reductio ad
impossibile one of the premisses is assumed falsely.
These points will be made clearer by the sequel, when we discuss
the reduction to impossibility: at present this much must be clear, that
we must look to terms of the kinds mentioned whether we wish to use an
ostensive syllogism or a reduction to impossibility. In the other hypothetical
syllogisms, I mean those which proceed by substitution, or by positing
a certain quality, the inquiry will be directed to the terms of the problem
to be proved-not the terms of the original problem, but the new terms introduced;
and the method of the inquiry will be the same as before. But we must consider
and determine in how many ways hypothetical syllogisms are
possible.
Each of the problems then can be proved in the manner described;
but it is possible to establish some of them syllogistically in another
way, e.g. universal problems by the inquiry which leads up to a particular
conclusion, with the addition of an hypothesis. For if the Cs and the Gs
should be identical, but E should be assumed to belong to the Gs only,
then A would belong to every E: and again if the Ds and the Gs should be
identical, but E should be predicated of the Gs only, it follows that A
will belong to none of the Es. Clearly then we must consider the matter
in this way also. The method is the same whether the relation is necessary
or possible. For the inquiry will be the same, and the syllogism will proceed
through terms arranged in the same order whether a possible or a pure proposition
is proved. We must find in the case of possible relations, as well as terms
that belong, terms which can belong though they actually do not: for we
have proved that the syllogism which establishes a possible relation proceeds
through these terms as well. Similarly also with the other modes of
predication.
It is clear then from what has been said not only that all syllogisms
can be formed in this way, but also that they cannot be formed in any other.
For every syllogism has been proved to be formed through one of the aforementioned
figures, and these cannot be composed through other terms than the consequents
and antecedents of the terms in question: for from these we obtain the
premisses and find the middle term. Consequently a syllogism cannot be
formed by means of other terms.
Part 30
The method is the same in all cases, in philosophy, in any art
or study. We must look for the attributes and the subjects of both our
terms, and we must supply ourselves with as many of these as possible,
and consider them by means of the three terms, refuting statements in one
way, confirming them in another, in the pursuit of truth starting from
premisses in which the arrangement of the terms is in accordance with truth,
while if we look for dialectical syllogisms we must start from probable
premisses. The principles of syllogisms have been stated in general terms,
both how they are characterized and how we must hunt for them, so as not
to look to everything that is said about the terms of the problem or to
the same points whether we are confirming or refuting, or again whether
we are confirming of all or of some, and whether we are refuting of all
or some. we must look to fewer points and they must be definite. We have
also stated how we must select with reference to everything that is, e.g.
about good or knowledge. But in each science the principles which are peculiar
are the most numerous. Consequently it is the business of experience to
give the principles which belong to each subject. I mean for example that
astronomical experience supplies the principles of astronomical science:
for once the phenomena were adequately apprehended, the demonstrations
of astronomy were discovered. Similarly with any other art or science.
Consequently, if the attributes of the thing are apprehended, our business
will then be to exhibit readily the demonstrations. For if none of the
true attributes of things had been omitted in the historical survey, we
should be able to discover the proof and demonstrate everything which admitted
of proof, and to make that clear, whose nature does not admit of
proof.
In general then we have explained fairly well how we must select
premisses: we have discussed the matter accurately in the treatise concerning
dialectic.
Part 31
It is easy to see that division into classes is a small part of
the method we have described: for division is, so to speak, a weak syllogism;
for what it ought to prove, it begs, and it always establishes something
more general than the attribute in question. First, this very point had
escaped all those who used the method of division; and they attempted to
persuade men that it was possible to make a demonstration of substance
and essence. Consequently they did not understand what it is possible to
prove syllogistically by division, nor did they understand that it was
possible to prove syllogistically in the manner we have described. In demonstrations,
when there is a need to prove a positive statement, the middle term through
which the syllogism is formed must always be inferior to and not comprehend
the first of the extremes. But division has a contrary intention: for it
takes the universal as middle. Let animal be the term signified by A, mortal
by B, and immortal by C, and let man, whose definition is to be got, be
signified by D. The man who divides assumes that every animal is either
mortal or immortal: i.e. whatever is A is all either B or C. Again, always
dividing, he lays it down that man is an animal, so he assumes A of D as
belonging to it. Now the true conclusion is that every D is either B or
C, consequently man must be either mortal or immortal, but it is not necessary
that man should be a mortal animal-this is begged: and this is what ought
to have been proved syllogistically. And again, taking A as mortal animal,
B as footed, C as footless, and D as man, he assumes in the same way that
A inheres either in B or in C (for every mortal animal is either footed
or footless), and he assumes A of D (for he assumed man, as we saw, to
be a mortal animal); consequently it is necessary that man should be either
a footed or a footless animal; but it is not necessary that man should
be footed: this he assumes: and it is just this again which he ought to
have demonstrated. Always dividing then in this way it turns out that these
logicians assume as middle the universal term, and as extremes that which
ought to have been the subject of demonstration and the differentiae. In
conclusion, they do not make it clear, and show it to be necessary, that
this is man or whatever the subject of inquiry may be: for they pursue
the other method altogether, never even suspecting the presence of the
rich supply of evidence which might be used. It is clear that it is neither
possible to refute a statement by this method of division, nor to draw
a conclusion about an accident or property of a thing, nor about its genus,
nor in cases in which it is unknown whether it is thus or thus, e.g. whether
the diagonal is incommensurate. For if he assumes that every length is
either commensurate or incommensurate, and the diagonal is a length, he
has proved that the diagonal is either incommensurate or commensurate.
But if he should assume that it is incommensurate, he will have assumed
what he ought to have proved. He cannot then prove it: for this is his
method, but proof is not possible by this method. Let A stand for 'incommensurate
or commensurate', B for 'length', C for 'diagonal'. It is clear then that
this method of investigation is not suitable for every inquiry, nor is
it useful in those cases in which it is thought to be most
suitable.
From what has been said it is clear from what elements demonstrations
are formed and in what manner, and to what points we must look in each
problem.
Part 32
Our next business is to state how we can reduce syllogisms to the
aforementioned figures: for this part of the inquiry still remains. If
we should investigate the production of the syllogisms and had the power
of discovering them, and further if we could resolve the syllogisms produced
into the aforementioned figures, our original problem would be brought
to a conclusion. It will happen at the same time that what has been already
said will be confirmed and its truth made clearer by what we are about
to say. For everything that is true must in every respect agree with itself
First then we must attempt to select the two premisses of the syllogism
(for it is easier to divide into large parts than into small, and the composite
parts are larger than the elements out of which they are made); next we
must inquire which are universal and which particular, and if both premisses
have not been stated, we must ourselves assume the one which is missing.
For sometimes men put forward the universal premiss, but do not posit the
premiss which is contained in it, either in writing or in discussion: or
men put forward the premisses of the principal syllogism, but omit those
through which they are inferred, and invite the concession of others to
no purpose. We must inquire then whether anything unnecessary has been
assumed, or anything necessary has been omitted, and we must posit the
one and take away the other, until we have reached the two premisses: for
unless we have these, we cannot reduce arguments put forward in the way
described. In some arguments it is easy to see what is wanting, but some
escape us, and appear to be syllogisms, because something necessary results
from what has been laid down, e.g. if the assumptions were made that substance
is not annihilated by the annihilation of what is not substance, and that
if the elements out of which a thing is made are annihilated, then that
which is made out of them is destroyed: these propositions being laid down,
it is necessary that any part of substance is substance; this has not however
been drawn by syllogism from the propositions assumed, but premisses are
wanting. Again if it is necessary that animal should exist, if man does,
and that substance should exist, if animal does, it is necessary that substance
should exist if man does: but as yet the conclusion has not been drawn
syllogistically: for the premisses are not in the shape we required. We
are deceived in such cases because something necessary results from what
is assumed, since the syllogism also is necessary. But that which is necessary
is wider than the syllogism: for every syllogism is necessary, but not
everything which is necessary is a syllogism. Consequently, though something
results when certain propositions are assumed, we must not try to reduce
it directly, but must first state the two premisses, then divide them into
their terms. We must take that term as middle which is stated in both the
remisses: for it is necessary that the middle should be found in both premisses
in all the figures.
If then the middle term is a predicate and a subject of predication,
or if it is a predicate, and something else is denied of it, we shall have
the first figure: if it both is a predicate and is denied of something,
the middle figure: if other things are predicated of it, or one is denied,
the other predicated, the last figure. For it was thus that we found the
middle term placed in each figure. It is placed similarly too if the premisses
are not universal: for the middle term is determined in the same way. Clearly
then, if the same term is not stated more than once in the course of an
argument, a syllogism cannot be made: for a middle term has not been taken.
Since we know what sort of thesis is established in each figure, and in
which the universal, in what sort the particular is described, clearly
we must not look for all the figures, but for that which is appropriate
to the thesis in hand. If the thesis is established in more figures than
one, we shall recognize the figure by the position of the middle
term.
Part 33
Men are frequently deceived about syllogisms because the inference
is necessary, as has been said above; sometimes they are deceived by the
similarity in the positing of the terms; and this ought not to escape our
notice. E.g. if A is stated of B, and B of C: it would seem that a syllogism
is possible since the terms stand thus: but nothing necessary results,
nor does a syllogism. Let A represent the term 'being eternal', B 'Aristomenes
as an object of thought', C 'Aristomenes'. It is true then that A belongs
to B. For Aristomenes as an object of thought is eternal. But B also belongs
to C: for Aristomenes is Aristomenes as an object of thought. But A does
not belong to C: for Aristomenes is perishable. For no syllogism was made
although the terms stood thus: that required that the premiss Ab should
be stated universally. But this is false, that every Aristomenes who is
an object of thought is eternal, since Aristomenes is perishable. Again
let C stand for 'Miccalus', B for 'musical Miccalus', A for 'perishing
to-morrow'. It is true to predicate B of C: for Miccalus is musical Miccalus.
Also A can be predicated of B: for musical Miccalus might perish to-morrow.
But to state A of C is false at any rate. This argument then is identical
with the former; for it is not true universally that musical Miccalus perishes
to-morrow: but unless this is assumed, no syllogism (as we have shown)
is possible.
This deception then arises through ignoring a small distinction.
For if we accept the conclusion as though it made no difference whether
we said 'This belong to that' or 'This belongs to all of
that'.
Part 34
Men will frequently fall into fallacies through not setting out
the terms of the premiss well, e.g. suppose A to be health, B disease,
C man. It is true to say that A cannot belong to any B (for health belongs
to no disease) and again that B belongs to every C (for every man is capable
of disease). It would seem to follow that health cannot belong to any man.
The reason for this is that the terms are not set out well in the statement,
since if the things which are in the conditions are substituted, no syllogism
can be made, e.g. if 'healthy' is substituted for 'health' and 'diseased'
for 'disease'. For it is not true to say that being healthy cannot belong
to one who is diseased. But unless this is assumed no conclusion results,
save in respect of possibility: but such a conclusion is not impossible:
for it is possible that health should belong to no man. Again the fallacy
may occur in a similar way in the middle figure: 'it is not possible that
health should belong to any disease, but it is possible that health should
belong to every man, consequently it is not possible that disease should
belong to any man'. In the third figure the fallacy results in reference
to possibility. For health and diseae and knowledge and ignorance, and
in general contraries, may possibly belong to the same thing, but cannot
belong to one another. This is not in agreement with what was said before:
for we stated that when several things could belong to the same thing,
they could belong to one another.
It is evident then that in all these cases the fallacy arises from
the setting out of the terms: for if the things that are in the conditions
are substituted, no fallacy arises. It is clear then that in such premisses
what possesses the condition ought always to be substituted for the condition
and taken as the term.
Part 35
We must not always seek to set out the terms a single word: for
we shall often have complexes of words to which a single name is not given.
Hence it is difficult to reduce syllogisms with such terms. Sometimes too
fallacies will result from such a search, e.g. the belief that syllogism
can establish that which has no mean. Let A stand for two right angles,
B for triangle, C for isosceles triangle. A then belongs to C because of
B: but A belongs to B without the mediation of another term: for the triangle
in virtue of its own nature contains two right angles, consequently there
will be no middle term for the proposition AB, although it is demonstrable.
For it is clear that the middle must not always be assumed to be an individual
thing, but sometimes a complex of words, as happens in the case
mentioned.
Part 36
That the first term belongs to the middle, and the middle to the
extreme, must not be understood in the sense that they can always be predicated
of one another or that the first term will be predicated of the middle
in the same way as the middle is predicated of the last term. The same
holds if the premisses are negative. But we must suppose the verb 'to belong'
to have as many meanings as the senses in which the verb 'to be' is used,
and in which the assertion that a thing 'is' may be said to be true. Take
for example the statement that there is a single science of contraries.
Let A stand for 'there being a single science', and B for things which
are contrary to one another. Then A belongs to B, not in the sense that
contraries are the fact of there being a single science of them, but in
the sense that it is true to say of the contraries that there is a single
science of them.
It happens sometimes that the first term is stated of the middle,
but the middle is not stated of the third term, e.g. if wisdom is knowledge,
and wisdom is of the good, the conclusion is that there is knowledge of
the good. The good then is not knowledge, though wisdom is knowledge. Sometimes
the middle term is stated of the third, but the first is not stated of
the middle, e.g. if there is a science of everything that has a quality,
or is a contrary, and the good both is a contrary and has a quality, the
conclusion is that there is a science of the good, but the good is not
science, nor is that which has a quality or is a contrary, though the good
is both of these. Sometimes neither the first term is stated of the middle,
nor the middle of the third, while the first is sometimes stated of the
third, and sometimes not: e.g. if there is a genus of that of which there
is a science, and if there is a science of the good, we conclude that there
is a genus of the good. But nothing is predicated of anything. And if that
of which there is a science is a genus, and if there is a science of the
good, we conclude that the good is a genus. The first term then is predicated
of the extreme, but in the premisses one thing is not stated of
another.
The same holds good where the relation is negative. For 'that does
not belong to this' does not always mean that 'this is not that', but sometimes
that 'this is not of that' or 'for that', e.g. 'there is not a motion of
a motion or a becoming of a becoming, but there is a becoming of pleasure:
so pleasure is not a becoming.' Or again it may be said that there is a
sign of laughter, but there is not a sign of a sign, consequently laughter
is not a sign. This holds in the other cases too, in which the thesis is
refuted because the genus is asserted in a particular way, in relation
to the terms of the thesis. Again take the inference 'opportunity is not
the right time: for opportunity belongs to God, but the right time does
not, since nothing is useful to God'. We must take as terms opportunity-right
time-God: but the premiss must be understood according to the case of the
noun. For we state this universally without qualification, that the terms
ought always to be stated in the nominative, e.g. man, good, contraries,
not in oblique cases, e.g. of man, of a good, of contraries, but the premisses
ought to be understood with reference to the cases of each term-either
the dative, e.g. 'equal to this', or the genitive, e.g. 'double of this',
or the accusative, e.g. 'that which strikes or sees this', or the nominative,
e.g. 'man is an animal', or in whatever other way the word falls in the
premiss.
Part 37
The expressions 'this belongs to that' and 'this holds true of
that' must be understood in as many ways as there are different categories,
and these categories must be taken either with or without qualification,
and further as simple or compound: the same holds good of the corresponding
negative expressions. We must consider these points and define them
better.
Part 38
A term which is repeated in the premisses ought to be joined to
the first extreme, not to the middle. I mean for example that if a syllogism
should be made proving that there is knowledge of justice, that it is good,
the expression 'that it is good' (or 'qua good') should be joined to the
first term. Let A stand for 'knowledge that it is good', B for good, C
for justice. It is true to predicate A of B. For of the good there is knowledge
that it is good. Also it is true to predicate B of C. For justice is identical
with a good. In this way an analysis of the argument can be made. But if
the expression 'that it is good' were added to B, the conclusion will not
follow: for A will be true of B, but B will not be true of C. For to predicate
of justice the term 'good that it is good' is false and not intelligible.
Similarly if it should be proved that the healthy is an object of knowledge
qua good, of goat-stag an object of knowledge qua not existing, or man
perishable qua an object of sense: in every case in which an addition is
made to the predicate, the addition must be joined to the
extreme.
The position of the terms is not the same when something is established
without qualification and when it is qualified by some attribute or condition,
e.g. when the good is proved to be an object of knowledge and when it is
proved to be an object of knowledge that it is good. If it has been proved
to be an object of knowledge without qualification, we must put as middle
term 'that which is', but if we add the qualification 'that it is good',
the middle term must be 'that which is something'. Let A stand for 'knowledge
that it is something', B stand for 'something', and C stand for 'good'.
It is true to predicate A of B: for ex hypothesi there is a science of
that which is something, that it is something. B too is true of C: for
that which C represents is something. Consequently A is true of C: there
will then be knowledge of the good, that it is good: for ex hypothesi the
term 'something' indicates the thing's special nature. But if 'being' were
taken as middle and 'being' simply were joined to the extreme, not 'being
something', we should not have had a syllogism proving that there is knowledge
of the good, that it is good, but that it is; e.g. let A stand for knowledge
that it is, B for being, C for good. Clearly then in syllogisms which are
thus limited we must take the terms in the way stated.
Part 39
We ought also to exchange terms which have the same value, word
for word, and phrase for phrase, and word and phrase, and always take a
word in preference to a phrase: for thus the setting out of the terms will
be easier. For example if it makes no difference whether we say that the
supposable is not the genus of the opinable or that the opinable is not
identical with a particular kind of supposable (for what is meant is the
same in both statements), it is better to take as the terms the supposable
and the opinable in preference to the phrase suggested.
Part 40
Since the expressions 'pleasure is good' and 'pleasure is the good'
are not identical, we must not set out the terms in the same way; but if
the syllogism is to prove that pleasure is the good, the term must be 'the
good', but if the object is to prove that pleasure is good, the term will
be 'good'. Similarly in all other cases.
Part 41
It is not the same, either in fact or in speech, that A belongs
to all of that to which B belongs, and that A belongs to all of that to
all of which B belongs: for nothing prevents B from belonging to C, though
not to all C: e.g. let B stand for beautiful, and C for white. If beauty
belongs to something white, it is true to say that beauty belongs to that
which is white; but not perhaps to everything that is white. If then A
belongs to B, but not to everything of which B is predicated, then whether
B belongs to all C or merely belongs to C, it is not necessary that A should
belong, I do not say to all C, but even to C at all. But if A belongs to
everything of which B is truly stated, it will follow that A can be said
of all of that of all of which B is said. If however A is said of that
of all of which B may be said, nothing prevents B belonging to C, and yet
A not belonging to all C or to any C at all. If then we take three terms
it is clear that the expression 'A is said of all of which B is said' means
this, 'A is said of all the things of which B is said'. And if B is said
of all of a third term, so also is A: but if B is not said of all of the
third term, there is no necessity that A should be said of all of
it.
We must not suppose that something absurd results through setting
out the terms: for we do not use the existence of this particular thing,
but imitate the geometrician who says that 'this line a foot long' or 'this
straight line' or 'this line without breadth' exists although it does not,
but does not use the diagrams in the sense that he reasons from them. For
in general, if two things are not related as whole to part and part to
whole, the prover does not prove from them, and so no syllogism a is formed.
We (I mean the learner) use the process of setting out terms like perception
by sense, not as though it were impossible to demonstrate without these
illustrative terms, as it is to demonstrate without the premisses of the
syllogism.
Part 42
We should not forget that in the same syllogism not all conclusions
are reached through one figure, but one through one figure, another through
another. Clearly then we must analyse arguments in accordance with this.
Since not every problem is proved in every figure, but certain problems
in each figure, it is clear from the conclusion in what figure the premisses
should be sought.
Part 43
In reference to those arguments aiming at a definition which have
been directed to prove some part of the definition, we must take as a term
the point to which the argument has been directed, not the whole definition:
for so we shall be less likely to be disturbed by the length of the term:
e.g. if a man proves that water is a drinkable liquid, we must take as
terms drinkable and water.
Part 44
Further we must not try to reduce hypothetical syllogisms; for
with the given premisses it is not possible to reduce them. For they have
not been proved by syllogism, but assented to by agreement. For instance
if a man should suppose that unless there is one faculty of contraries,
there cannot be one science, and should then argue that not every faculty
is of contraries, e.g. of what is healthy and what is sickly: for the same
thing will then be at the same time healthy and sickly. He has shown that
there is not one faculty of all contraries, but he has not proved that
there is not a science. And yet one must agree. But the agreement does
not come from a syllogism, but from an hypothesis. This argument cannot
be reduced: but the proof that there is not a single faculty can. The latter
argument perhaps was a syllogism: but the former was an
hypothesis.
The same holds good of arguments which are brought to a conclusion
per impossibile. These cannot be analysed either; but the reduction to
what is impossible can be analysed since it is proved by syllogism, though
the rest of the argument cannot, because the conclusion is reached from
an hypothesis. But these differ from the previous arguments: for in the
former a preliminary agreement must be reached if one is to accept the
conclusion; e.g. an agreement that if there is proved to be one faculty
of contraries, then contraries fall under the same science; whereas in
the latter, even if no preliminary agreement has been made, men still accept
the reasoning, because the falsity is patent, e.g. the falsity of what
follows from the assumption that the diagonal is commensurate, viz. that
then odd numbers are equal to evens.
Many other arguments are brought to a conclusion by the help of
an hypothesis; these we ought to consider and mark out clearly. We shall
describe in the sequel their differences, and the various ways in which
hypothetical arguments are formed: but at present this much must be clear,
that it is not possible to resolve such arguments into the figures. And
we have explained the reason.
Part 45
Whatever problems are proved in more than one figure, if they have
been established in one figure by syllogism, can be reduced to another
figure, e.g. a negative syllogism in the first figure can be reduced to
the second, and a syllogism in the middle figure to the first, not all
however but some only. The point will be clear in the sequel. If A belongs
to no B, and B to all C, then A belongs to no C. Thus the first figure;
but if the negative statement is converted, we shall have the middle figure.
For B belongs to no A, and to all C. Similarly if the syllogism is not
universal but particular, e.g. if A belongs to no B, and B to some C. Convert
the negative statement and you will have the middle
figure.
The universal syllogisms in the second figure can be reduced to
the first, but only one of the two particular syllogisms. Let A belong
to no B and to all C. Convert the negative statement, and you will have
the first figure. For B will belong to no A and A to all C. But if the
affirmative statement concerns B, and the negative C, C must be made first
term. For C belongs to no A, and A to all B: therefore C belongs to no
B. B then belongs to no C: for the negative statement is
convertible.
But if the syllogism is particular, whenever the negative statement
concerns the major extreme, reduction to the first figure will be possible,
e.g. if A belongs to no B and to some C: convert the negative statement
and you will have the first figure. For B will belong to no A and A to
some C. But when the affirmative statement concerns the major extreme,
no resolution will be possible, e.g. if A belongs to all B, but not to
all C: for the statement AB does not admit of conversion, nor would there
be a syllogism if it did.
Again syllogisms in the third figure cannot all be resolved into
the first, though all syllogisms in the first figure can be resolved into
the third. Let A belong to all B and B to some C. Since the particular
affirmative is convertible, C will belong to some B: but A belonged to
all B: so that the third figure is formed. Similarly if the syllogism is
negative: for the particular affirmative is convertible: therefore A will
belong to no B, and to some C.
Of the syllogisms in the last figure one only cannot be resolved
into the first, viz. when the negative statement is not universal: all
the rest can be resolved. Let A and B be affirmed of all C: then C can
be converted partially with either A or B: C then belongs to some B. Consequently
we shall get the first figure, if A belongs to all C, and C to some of
the Bs. If A belongs to all C and B to some C, the argument is the same:
for B is convertible in reference to C. But if B belongs to all C and A
to some C, the first term must be B: for B belongs to all C, and C to some
A, therefore B belongs to some A. But since the particular statement is
convertible, A will belong to some B. If the syllogism is negative, when
the terms are universal we must take them in a similar way. Let B belong
to all C, and A to no C: then C will belong to some B, and A to no C; and
so C will be middle term. Similarly if the negative statement is universal,
the affirmative particular: for A will belong to no C, and C to some of
the Bs. But if the negative statement is particular, no resolution will
be possible, e.g. if B belongs to all C, and A not belong to some C: convert
the statement BC and both premisses will be particular.
It is clear that in order to resolve the figures into one another
the premiss which concerns the minor extreme must be converted in both
the figures: for when this premiss is altered, the transition to the other
figure is made.
One of the syllogisms in the middle figure can, the other cannot,
be resolved into the third figure. Whenever the universal statement is
negative, resolution is possible. For if A belongs to no B and to some
C, both B and C alike are convertible in relation to A, so that B belongs
to no A and C to some A. A therefore is middle term. But when A belongs
to all B, and not to some C, resolution will not be possible: for neither
of the premisses is universal after conversion.
Syllogisms in the third figure can be resolved into the middle
figure, whenever the negative statement is universal, e.g. if A belongs
to no C, and B to some or all C. For C then will belong to no A and to
some B. But if the negative statement is particular, no resolution will
be possible: for the particular negative does not admit of
conversion.
It is clear then that the same syllogisms cannot be resolved in
these figures which could not be resolved into the first figure, and that
when syllogisms are reduced to the first figure these alone are confirmed
by reduction to what is impossible.
It is clear from what we have said how we ought to reduce syllogisms,
and that the figures may be resolved into one another.
Part 46
In establishing or refuting, it makes some difference whether we
suppose the expressions 'not to be this' and 'to be not-this' are identical
or different in meaning, e.g. 'not to be white' and 'to be not-white'.
For they do not mean the same thing, nor is 'to be not-white' the negation
of 'to be white', but 'not to be white'. The reason for this is as follows.
The relation of 'he can walk' to 'he can not-walk' is similar to the relation
of 'it is white' to 'it is not-white'; so is that of 'he knows what is
good' to 'he knows what is not-good'. For there is no difference between
the expressions 'he knows what is good' and 'he is knowing what is good',
or 'he can walk' and 'he is able to walk': therefore there is no difference
between their contraries 'he cannot walk'-'he is not able to walk'. If
then 'he is not able to walk' means the same as 'he is able not to walk',
capacity to walk and incapacity to walk will belong at the same time to
the same person (for the same man can both walk and not-walk, and is possessed
of knowledge of what is good and of what is not-good), but an affirmation
and a denial which are opposed to one another do not belong at the same
time to the same thing. As then 'not to know what is good' is not the same
as 'to know what is not good', so 'to be not-good' is not the same as 'not
to be good'. For when two pairs correspond, if the one pair are different
from one another, the other pair also must be different. Nor is 'to be
not-equal' the same as 'not to be equal': for there is something underlying
the one, viz. that which is not-equal, and this is the unequal, but there
is nothing underlying the other. Wherefore not everything is either equal
or unequal, but everything is equal or is not equal. Further the expressions
'it is a not-white log' and 'it is not a white log' do not imply one another's
truth. For if 'it is a not-white log', it must be a log: but that which
is not a white log need not be a log at all. Therefore it is clear that
'it is not-good' is not the denial of 'it is good'. If then every single
statement may truly be said to be either an affirmation or a negation,
if it is not a negation clearly it must in a sense be an affirmation. But
every affirmation has a corresponding negation. The negation then of 'it
is not-good' is 'it is not not-good'. The relation of these statements
to one another is as follows. Let A stand for 'to be good', B for 'not
to be good', let C stand for 'to be not-good' and be placed under B, and
let D stand for not to be not-good' and be placed under A. Then either
A or B will belong to everything, but they will never belong to the same
thing; and either C or D will belong to everything, but they will never
belong to the same thing. And B must belong to everything to which C belongs.
For if it is true to say 'it is a not-white', it is true also to say 'it
is not white': for it is impossible that a thing should simultaneously
be white and be not-white, or be a not-white log and be a white log; consequently
if the affirmation does not belong, the denial must belong. But C does
not always belong to B: for what is not a log at all, cannot be a not-white
log either. On the other hand D belongs to everything to which A belongs.
For either C or D belongs to everything to which A belongs. But since a
thing cannot be simultaneously not-white and white, D must belong to everything
to which A belongs. For of that which is white it is true to say that it
is not not-white. But A is not true of all D. For of that which is not
a log at all it is not true to say A, viz. that it is a white log. Consequently
D is true, but A is not true, i.e. that it is a white log. It is clear
also that A and C cannot together belong to the same thing, and that B
and D may possibly belong to the same thing.
Privative terms are similarly related positive ter terms respect
of this arrangement. Let A stand for 'equal', B for 'not equal', C for
'unequal', D for 'not unequal'.
In many things also, to some of which something belongs which does
not belong to others, the negation may be true in a similar way, viz. that
all are not white or that each is not white, while that each is not-white
or all are not-white is false. Similarly also 'every animal is not-white'
is not the negation of 'every animal is white' (for both are false): the
proper negation is 'every animal is not white'. Since it is clear that
'it is not-white' and 'it is not white' mean different things, and one
is an affirmation, the other a denial, it is evident that the method of
proving each cannot be the same, e.g. that whatever is an animal is not
white or may not be white, and that it is true to call it not-white; for
this means that it is not-white. But we may prove that it is true to call
it white or not-white in the same way for both are proved constructively
by means of the first figure. For the expression 'it is true' stands on
a similar footing to 'it is'. For the negation of 'it is true to call it
white' is not 'it is true to call it not-white' but 'it is not true to
call it white'. If then it is to be true to say that whatever is a man
is musical or is not-musical, we must assume that whatever is an animal
either is musical or is not-musical; and the proof has been made. That
whatever is a man is not musical is proved destructively in the three ways
mentioned.
In general whenever A and B are such that they cannot belong at
the same time to the same thing, and one of the two necessarily belongs
to everything, and again C and D are related in the same way, and A follows
C but the relation cannot be reversed, then D must follow B and the relation
cannot be reversed. And A and D may belong to the same thing, but B and
C cannot. First it is clear from the following consideration that D follows
B. For since either C or D necessarily belongs to everything; and since
C cannot belong to that to which B belongs, because it carries A along
with it and A and B cannot belong to the same thing; it is clear that D
must follow B. Again since C does not reciprocate with but A, but C or
D belongs to everything, it is possible that A and D should belong to the
same thing. But B and C cannot belong to the same thing, because A follows
C; and so something impossible results. It is clear then that B does not
reciprocate with D either, since it is possible that D and A should belong
at the same time to the same thing.
It results sometimes even in such an arrangement of terms that
one is deceived through not apprehending the opposites rightly, one of
which must belong to everything, e.g. we may reason that 'if A and B cannot
belong at the same time to the same thing, but it is necessary that one
of them should belong to whatever the other does not belong to: and again
C and D are related in the same way, and follows everything which C follows:
it will result that B belongs necessarily to everything to which D belongs':
but this is false. 'Assume that F stands for the negation of A and B, and
again that H stands for the negation of C and D. It is necessary then that
either A or F should belong to everything: for either the affirmation or
the denial must belong. And again either C or H must belong to everything:
for they are related as affirmation and denial. And ex hypothesi A belongs
to everything ever thing to which C belongs. Therefore H belongs to everything
to which F belongs. Again since either F or B belongs to everything, and
similarly either H or D, and since H follows F, B must follow D: for we
know this. If then A follows C, B must follow D'. But this is false: for
as we proved the sequence is reversed in terms so constituted. The fallacy
arises because perhaps it is not necessary that A or F should belong to
everything, or that F or B should belong to everything: for F is not the
denial of A. For not good is the negation of good: and not-good is not
identical with 'neither good nor not-good'. Similarly also with C and D.
For two negations have been assumed in respect to one
term.
