Various Works Page 40

  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