or simple premiss is assumed.
13
Perhaps enough has been said about the proof of necessity, how it
comes about and how it differs from the proof of a simple statement.
We proceed to discuss that which is possible, when and how and by what
means it can be proved. I use the terms 'to be possible' and 'the
possible' of that which is not necessary but, being assumed, results
in nothing impossible. We say indeed ambiguously of the necessary that
it is possible. But that my definition of the possible is correct is
clear from the phrases by which we deny or on the contrary affirm
possibility. For the expressions 'it is not possible to belong', 'it
is impossible to belong', and 'it is necessary not to belong' are
either identical or follow from one another; consequently their
opposites also, 'it is possible to belong', 'it is not impossible to
belong', and 'it is not necessary not to belong', will either be
identical or follow from one another. For of everything the
affirmation or the denial holds good. That which is possible then will
be not necessary and that which is not necessary will be possible.
It results that all premisses in the mode of possibility are
convertible into one another. I mean not that the affirmative are
convertible into the negative, but that those which are affirmative in
form admit of conversion by opposition, e.g. 'it is possible to
belong' may be converted into 'it is possible not to belong', and
'it is possible for A to belong to all B' into 'it is possible for A
to belong to no B' or 'not to all B', and 'it is possible for A to
belong to some B' into 'it is possible for A not to belong to some B'.
And similarly the other propositions in this mode can be converted.
For since that which is possible is not necessary, and that which is
not necessary may possibly not belong, it is clear that if it is
possible that A should belong to B, it is possible also that it should
not belong to B: and if it is possible that it should belong to all,
it is also possible that it should not belong to all. The same holds
good in the case of particular affirmations: for the proof is
identical. And such premisses are affirmative and not negative; for
'to be possible' is in the same rank as 'to be', as was said above.
Having made these distinctions we next point out that the expression
'to be possible' is used in two ways. In one it means to happen
generally and fall short of necessity, e.g. man's turning grey or
growing or decaying, or generally what naturally belongs to a thing
(for this has not its necessity unbroken, since man's existence is not
continuous for ever, although if a man does exist, it comes about
either necessarily or generally). In another sense the expression
means the indefinite, which can be both thus and not thus, e.g. an
animal's walking or an earthquake's taking place while it is
walking, or generally what happens by chance: for none of these
inclines by nature in the one way more than in the opposite.
That which is possible in each of its two senses is convertible into
its opposite, not however in the same way: but what is natural is
convertible because it does not necessarily belong (for in this
sense it is possible that a man should not grow grey) and what is
indefinite is convertible because it inclines this way no more than
that. Science and demonstrative syllogism are not concerned with
things which are indefinite, because the middle term is uncertain; but
they are concerned with things that are natural, and as a rule
arguments and inquiries are made about things which are possible in
this sense. Syllogisms indeed can be made about the former, but it
is unusual at any rate to inquire about them.
These matters will be treated more definitely in the sequel; our
business at present is to state the moods and nature of the
syllogism made from possible premisses. The expression 'it is possible
for this to belong to that' may be understood in two senses: 'that'
may mean either that to which 'that' belongs or that to which it may
belong; for the expression 'A is possible of the subject of B' means
that it is possible either of that of which B is stated or of that
of which B may possibly be stated. It makes no difference whether we
say, A is possible of the subject of B, or all B admits of A. It is
clear then that the expression 'A may possibly belong to all B'
might be used in two senses. First then we must state the nature and
characteristics of the syllogism which arises if B is possible of
the subject of C, and A is possible of the subject of B. For thus both
premisses are assumed in the mode of possibility; but whenever A is
possible of that of which B is true, one premiss is a simple
assertion, the other a problematic. Consequently we must start from
premisses which are similar in form, as in the other cases.
14
Whenever A may possibly belong to all B, and B to all C, there
will be a perfect syllogism to prove that A may possibly belong to all
C. This is clear from the definition: for it was in this way that we
explained 'to be possible for one term to belong to all of another'.
Similarly if it is possible for A to belong no B, and for B to
belong to all C, then it is possible for A to belong to no C. For
the statement that it is possible for A not to belong to that of which
B may be true means (as we saw) that none of those things which can
possibly fall under the term B is left out of account. But whenever
A may belong to all B, and B may belong to no C, then indeed no
syllogism results from the premisses assumed, but if the premiss BC is
converted after the manner of problematic propositions, the same
syllogism results as before. For since it is possible that B should
belong to no C, it is possible also that it should belong to all C.
This has been stated above. Consequently if B is possible for all C,
and A is possible for all B, the same syllogism again results.
Similarly if in both the premisses the negative is joined with 'it
is possible': e.g. if A may belong to none of the Bs, and B to none of
the Cs. No syllogism results from the assumed premisses, but if they
are converted we shall have the same syllogism as before. It is
clear then that if the minor premiss is negative, or if both premisses
are negative, either no syllogism results, or if one it is not
perfect. For the necessity results from the conversion.
But if one of the premisses is universal, the other particular, when
the major premiss is universal there will be a perfect syllogism.
For if A is possible for all B, and B for some C, then A is possible
for some C. This is clear from the definition of being possible. Again
if A may belong to no B, and B may belong to some of the Cs, it is
necessary that A may possibly not belong to some of the Cs. The
proof is the same as above. But if the particular premiss is negative,
and the universal is affirmative, the major still being universal
and the minor particular, e.g. A is possible for all B, B m
ay possibly
not belong to some C, then a clear syllogism does not result from
the assumed premisses, but if the particular premiss is converted
and it is laid down that B possibly may belong to some C, we shall
have the same conclusion as before, as in the cases given at the
beginning.
But if the major premiss is the minor universal, whether both are
affirmative, or negative, or different in quality, or if both are
indefinite or particular, in no way will a syllogism be possible.
For nothing prevents B from reaching beyond A, so that as predicates
cover unequal areas. Let C be that by which B extends beyond A. To C
it is not possible that A should belong-either to all or to none or to
some or not to some, since premisses in the mode of possibility are
convertible and it is possible for B to belong to more things than A
can. Further, this is obvious if we take terms; for if the premisses
are as assumed, the major term is both possible for none of the
minor and must belong to all of it. Take as terms common to all the
cases under consideration 'animal'-'white'-'man', where the major
belongs necessarily to the minor; 'animal'-'white'-'garment', where it
is not possible that the major should belong to the minor. It is clear
then that if the terms are related in this manner, no syllogism
results. For every syllogism proves that something belongs either
simply or necessarily or possibly. It is clear that there is no
proof of the first or of the second. For the affirmative is
destroyed by the negative, and the negative by the affirmative.
There remains the proof of possibility. But this is impossible. For it
has been proved that if the terms are related in this manner it is
both necessary that the major should belong to all the minor and not
possible that it should belong to any. Consequently there cannot be
a syllogism to prove the possibility; for the necessary (as we stated)
is not possible.
It is clear that if the terms are universal in possible premisses
a syllogism always results in the first figure, whether they are
affirmative or negative, only a perfect syllogism results in the first
case, an imperfect in the second. But possibility must be understood
according to the definition laid down, not as covering necessity. This
is sometimes forgotten.
15
If one premiss is a simple proposition, the other a problematic,
whenever the major premiss indicates possibility all the syllogisms
will be perfect and establish possibility in the sense defined; but
whenever the minor premiss indicates possibility all the syllogisms
will be imperfect, and those which are negative will establish not
possibility according to the definition, but that the major does not
necessarily belong to any, or to all, of the minor. For if this is so,
we say it is possible that it should belong to none or not to all. Let
A be possible for all B, and let B belong to all C. Since C falls
under B, and A is possible for all B, clearly it is possible for all C
also. So a perfect syllogism results. Likewise if the premiss AB is
negative, and the premiss BC is affirmative, the former stating
possible, the latter simple attribution, a perfect syllogism results
proving that A possibly belongs to no C.
It is clear that perfect syllogisms result if the minor premiss
states simple belonging: but that syllogisms will result if the
modality of the premisses is reversed, must be proved per impossibile.
At the same time it will be evident that they are imperfect: for the
proof proceeds not from the premisses assumed. First we must state
that if B's being follows necessarily from A's being, B's
possibility will follow necessarily from A's possibility. Suppose, the
terms being so related, that A is possible, and B is impossible. If
then that which is possible, when it is possible for it to be, might
happen, and if that which is impossible, when it is impossible,
could not happen, and if at the same time A is possible and B
impossible, it would be possible for A to happen without B, and if
to happen, then to be. For that which has happened, when it has
happened, is. But we must take the impossible and the possible not
only in the sphere of becoming, but also in the spheres of truth and
predicability, and the various other spheres in which we speak of
the possible: for it will be alike in all. Further we must
understand the statement that B's being depends on A's being, not as
meaning that if some single thing A is, B will be: for nothing follows
of necessity from the being of some one thing, but from two at
least, i.e. when the premisses are related in the manner stated to
be that of the syllogism. For if C is predicated of D, and D of F,
then C is necessarily predicated of F. And if each is possible, the
conclusion also is possible. If then, for example, one should indicate
the premisses by A, and the conclusion by B, it would not only
result that if A is necessary B is necessary, but also that if A is
possible, B is possible.
Since this is proved it is evident that if a false and not
impossible assumption is made, the consequence of the assumption
will also be false and not impossible: e.g. if A is false, but not
impossible, and if B is the consequence of A, B also will be false but
not impossible. For since it has been proved that if B's being is
the consequence of A's being, then B's possibility will follow from
A's possibility (and A is assumed to be possible), consequently B will
be possible: for if it were impossible, the same thing would at the
same time be possible and impossible.
Since we have defined these points, let A belong to all B, and B
be possible for all C: it is necessary then that should be a
possible attribute for all C. Suppose that it is not possible, but
assume that B belongs to all C: this is false but not impossible. If
then A is not possible for C but B belongs to all C, then A is not
possible for all B: for a syllogism is formed in the third degree. But
it was assumed that A is a possible attribute for all B. It is
necessary then that A is possible for all C. For though the assumption
we made is false and not impossible, the conclusion is impossible.
It is possible also in the first figure to bring about the
impossibility, by assuming that B belongs to C. For if B belongs to
all C, and A is possible for all B, then A would be possible for all
C. But the assumption was made that A is not possible for all C.
We must understand 'that which belongs to all' with no limitation in
respect of time, e.g. to the present or to a particular period, but
simply without qualification. For it is by the help of such
premisses that we make syllogisms, since if the premiss is
understood with reference to the present moment, there cannot be a
syllogism. For nothing perhaps prevents 'man' belonging at a
particular time to everything that is moving, i.e. if nothing else
were moving: but 'moving' is possible for every horse; yet 'man' is
possible
for no horse. Further let the major term be 'animal', the
middle 'moving', the the minor 'man'. The premisses then will be as
before, but the conclusion necessary, not possible. For man is
necessarily animal. It is clear then that the universal must be
understood simply, without limitation in respect of time.
Again let the premiss AB be universal and negative, and assume
that A belongs to no B, but B possibly belongs to all C. These
propositions being laid down, it is necessary that A possibly
belongs to no C. Suppose that it cannot belong, and that B belongs
to C, as above. It is necessary then that A belongs to some B: for
we have a syllogism in the third figure: but this is impossible.
Thus it will be possible for A to belong to no C; for if at is
supposed false, the consequence is an impossible one. This syllogism
then does not establish that which is possible according to the
definition, but that which does not necessarily belong to any part
of the subject (for this is the contradictory of the assumption
which was made: for it was supposed that A necessarily belongs to some
C, but the syllogism per impossibile establishes the contradictory
which is opposed to this). Further, it is clear also from an example
that the conclusion will not establish possibility. Let A be
'raven', B 'intelligent', and C 'man'. A then belongs to no B: for
no intelligent thing is a raven. But B is possible for all C: for
every man may possibly be intelligent. But A necessarily belongs to no
C: so the conclusion does not establish possibility. But neither is it
always necessary. Let A be 'moving', B 'science', C 'man'. A then will
belong to no B; but B is possible for all C. And the conclusion will
not be necessary. For it is not necessary that no man should move;
rather it is not necessary that any man should move. Clearly then
the conclusion establishes that one term does not necessarily belong
to any instance of another term. But we must take our terms better.
If the minor premiss is negative and indicates possibility, from the
actual premisses taken there can be no syllogism, but if the
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