Various Works Read online

  number is taken as middle, and it is assumed that A belongs to no B,

  and B to some C, then A will not belong to some C, which ex

  hypothesi is true. And the premiss AB is true, the premiss BC false.

  (10) Also if the premiss AB is partially false, and the premiss BC

  is false too, the conclusion may be true. For nothing prevents A

  belonging to some B and to some C, though B belongs to no C, e.g. if B

  is the contrary of C, and both are accidents of the same genus: for

  animal belongs to some white things and to some black things, but

  white belongs to no black thing. If then it is assumed that A

  belongs to all B, and B to some C, the conclusion will be true.

  Similarly if the premiss AB is negative: for the same terms arranged

  in the same way will serve for the proof.

  (11) Also though both premisses are false the conclusion may be

  true. For it is possible that A may belong to no B and to some C,

  while B belongs to no C, e.g. a genus in relation to the species of

  another genus, and to the accident of its own species: for animal

  belongs to no number, but to some white things, and number to

  nothing white. If then it is assumed that A belongs to all B and B

  to some C, the conclusion will be true, though both premisses are

  false. Similarly also if the premiss AB is negative. For nothing

  prevents A belonging to the whole of B, and not to some C, while B

  belongs to no C, e.g. animal belongs to every swan, and not to some

  black things, and swan belongs to nothing black. Consequently if it is

  assumed that A belongs to no B, and B to some C, then A does not

  belong to some C. The conclusion then is true, but the premisses arc

  false.

  3

  In the middle figure it is possible in every way to reach a true

  conclusion through false premisses, whether the syllogisms are

  universal or particular, viz. when both premisses are wholly false;

  when each is partially false; when one is true, the other wholly false

  (it does not matter which of the two premisses is false); if both

  premisses are partially false; if one is quite true, the other

  partially false; if one is wholly false, the other partially true. For

  (1) if A belongs to no B and to all C, e.g. animal to no stone and

  to every horse, then if the premisses are stated contrariwise and it

  is assumed that A belongs to all B and to no C, though the premisses

  are wholly false they will yield a true conclusion. Similarly if A

  belongs to all B and to no C: for we shall have the same syllogism.

  (2) Again if one premiss is wholly false, the other wholly true: for

  nothing prevents A belonging to all B and to all C, though B belongs

  to no C, e.g. a genus to its co-ordinate species. For animal belongs

  to every horse and man, and no man is a horse. If then it is assumed

  that animal belongs to all of the one, and none of the other, the

  one premiss will be wholly false, the other wholly true, and the

  conclusion will be true whichever term the negative statement

  concerns.

  (3) Also if one premiss is partially false, the other wholly true.

  For it is possible that A should belong to some B and to all C, though

  B belongs to no C, e.g. animal to some white things and to every

  raven, though white belongs to no raven. If then it is assumed that

  A belongs to no B, but to the whole of C, the premiss AB is

  partially false, the premiss AC wholly true, and the conclusion

  true. Similarly if the negative statement is transposed: the proof can

  be made by means of the same terms. Also if the affirmative premiss is

  partially false, the negative wholly true, a true conclusion is

  possible. For nothing prevents A belonging to some B, but not to C

  as a whole, while B belongs to no C, e.g. animal belongs to some white

  things, but to no pitch, and white belongs to no pitch. Consequently

  if it is assumed that A belongs to the whole of B, but to no C, the

  premiss AB is partially false, the premiss AC is wholly true, and

  the conclusion is true.

  (4) And if both the premisses are partially false, the conclusion

  may be true. For it is possible that A should belong to some B and

  to some C, and B to no C, e.g. animal to some white things and to some

  black things, though white belongs to nothing black. If then it is

  assumed that A belongs to all B and to no C, both premisses are

  partially false, but the conclusion is true. Similarly, if the

  negative premiss is transposed, the proof can be made by means of

  the same terms.

  It is clear also that our thesis holds in particular syllogisms. For

  (5) nothing prevents A belonging to all B and to some C, though B does

  not belong to some C, e.g. animal to every man and to some white

  things, though man will not belong to some white things. If then it is

  stated that A belongs to no B and to some C, the universal premiss

  is wholly false, the particular premiss is true, and the conclusion is

  true. Similarly if the premiss AB is affirmative: for it is possible

  that A should belong to no B, and not to some C, though B does not

  belong to some C, e.g. animal belongs to nothing lifeless, and does

  not belong to some white things, and lifeless will not belong to

  some white things. If then it is stated that A belongs to all B and

  not to some C, the premiss AB which is universal is wholly false,

  the premiss AC is true, and the conclusion is true. Also a true

  conclusion is possible when the universal premiss is true, and the

  particular is false. For nothing prevents A following neither B nor

  C at all, while B does not belong to some C, e.g. animal belongs to no

  number nor to anything lifeless, and number does not follow some

  lifeless things. If then it is stated that A belongs to no B and to

  some C, the conclusion will be true, and the universal premiss true,

  but the particular false. Similarly if the premiss which is stated

  universally is affirmative. For it is possible that should A belong

  both to B and to C as wholes, though B does not follow some C, e.g.

  a genus in relation to its species and difference: for animal

  follows every man and footed things as a whole, but man does not

  follow every footed thing. Consequently if it is assumed that A

  belongs to the whole of B, but does not belong to some C, the

  universal premiss is true, the particular false, and the conclusion

  true.

  (6) It is clear too that though both premisses are false they may

  yield a true conclusion, since it is possible that A should belong

  both to B and to C as wholes, though B does not follow some C. For

  if it is assumed that A belongs to no B and to some C, the premisses

  are both false, but the conclusion is true. Similarly if the universal

  premiss is affirmative and the particular negative. For it is possible

  that A should follow no B and all C, though B does not belong to

  some C, e.g. animal follows no science but every man, though science

  does not follow every man. If then A is assumed to belong to the whole

  of B, and not to follow some C, the premisses are false but the

  conclusion is true.

  4


  In the last figure a true conclusion may come through what is false,

  alike when both premisses are wholly false, when each is partly false,

  when one premiss is wholly true, the other false, when one premiss

  is partly false, the other wholly true, and vice versa, and in every

  other way in which it is possible to alter the premisses. For (1)

  nothing prevents neither A nor B from belonging to any C, while A

  belongs to some B, e.g. neither man nor footed follows anything

  lifeless, though man belongs to some footed things. If then it is

  assumed that A and B belong to all C, the premisses will be wholly

  false, but the conclusion true. Similarly if one premiss is

  negative, the other affirmative. For it is possible that B should

  belong to no C, but A to all C, and that should not belong to some

  B, e.g. black belongs to no swan, animal to every swan, and animal not

  to everything black. Consequently if it is assumed that B belongs to

  all C, and A to no C, A will not belong to some B: and the

  conclusion is true, though the premisses are false.

  (2) Also if each premiss is partly false, the conclusion may be

  true. For nothing prevents both A and B from belonging to some C while

  A belongs to some B, e.g. white and beautiful belong to some

  animals, and white to some beautiful things. If then it is stated that

  A and B belong to all C, the premisses are partially false, but the

  conclusion is true. Similarly if the premiss AC is stated as negative.

  For nothing prevents A from not belonging, and B from belonging, to

  some C, while A does not belong to all B, e.g. white does not belong

  to some animals, beautiful belongs to some animals, and white does not

  belong to everything beautiful. Consequently if it is assumed that A

  belongs to no C, and B to all C, both premisses are partly false,

  but the conclusion is true.

  (3) Similarly if one of the premisses assumed is wholly false, the

  other wholly true. For it is possible that both A and B should

  follow all C, though A does not belong to some B, e.g. animal and

  white follow every swan, though animal does not belong to everything

  white. Taking these then as terms, if one assumes that B belongs to

  the whole of C, but A does not belong to C at all, the premiss BC will

  be wholly true, the premiss AC wholly false, and the conclusion

  true. Similarly if the statement BC is false, the statement AC true,

  the conclusion may be true. The same terms will serve for the proof.

  Also if both the premisses assumed are affirmative, the conclusion may

  be true. For nothing prevents B from following all C, and A from not

  belonging to C at all, though A belongs to some B, e.g. animal belongs

  to every swan, black to no swan, and black to some animals.

  Consequently if it is assumed that A and B belong to every C, the

  premiss BC is wholly true, the premiss AC is wholly false, and the

  conclusion is true. Similarly if the premiss AC which is assumed is

  true: the proof can be made through the same terms.

  (4) Again if one premiss is wholly true, the other partly false, the

  conclusion may be true. For it is possible that B should belong to all

  C, and A to some C, while A belongs to some B, e.g. biped belongs to

  every man, beautiful not to every man, and beautiful to some bipeds.

  If then it is assumed that both A and B belong to the whole of C,

  the premiss BC is wholly true, the premiss AC partly false, the

  conclusion true. Similarly if of the premisses assumed AC is true

  and BC partly false, a true conclusion is possible: this can be

  proved, if the same terms as before are transposed. Also the

  conclusion may be true if one premiss is negative, the other

  affirmative. For since it is possible that B should belong to the

  whole of C, and A to some C, and, when they are so, that A should

  not belong to all B, therefore it is assumed that B belongs to the

  whole of C, and A to no C, the negative premiss is partly false, the

  other premiss wholly true, and the conclusion is true. Again since

  it has been proved that if A belongs to no C and B to some C, it is

  possible that A should not belong to some C, it is clear that if the

  premiss AC is wholly true, and the premiss BC partly false, it is

  possible that the conclusion should be true. For if it is assumed that

  A belongs to no C, and B to all C, the premiss AC is wholly true,

  and the premiss BC is partly false.

  (5) It is clear also in the case of particular syllogisms that a

  true conclusion may come through what is false, in every possible way.

  For the same terms must be taken as have been taken when the premisses

  are universal, positive terms in positive syllogisms, negative terms

  in negative. For it makes no difference to the setting out of the

  terms, whether one assumes that what belongs to none belongs to all or

  that what belongs to some belongs to all. The same applies to negative

  statements.

  It is clear then that if the conclusion is false, the premisses of

  the argument must be false, either all or some of them; but when the

  conclusion is true, it is not necessary that the premisses should be

  true, either one or all, yet it is possible, though no part of the

  syllogism is true, that the conclusion may none the less be true;

  but it is not necessitated. The reason is that when two things are

  so related to one another, that if the one is, the other necessarily

  is, then if the latter is not, the former will not be either, but if

  the latter is, it is not necessary that the former should be. But it

  is impossible that the same thing should be necessitated by the

  being and by the not-being of the same thing. I mean, for example,

  that it is impossible that B should necessarily be great since A is

  white and that B should necessarily be great since A is not white. For

  whenever since this, A, is white it is necessary that that, B,

  should be great, and since B is great that C should not be white, then

  it is necessary if is white that C should not be white. And whenever

  it is necessary, since one of two things is, that the other should be,

  it is necessary, if the latter is not, that the former (viz. A) should

  not be. If then B is not great A cannot be white. But if, when A is

  not white, it is necessary that B should be great, it necessarily

  results that if B is not great, B itself is great. (But this is

  impossible.) For if B is not great, A will necessarily not be white.

  If then when this is not white B must be great, it results that if B

  is not great, it is great, just as if it were proved through three

  terms.

  5

  Circular and reciprocal proof means proof by means of the

  conclusion, i.e. by converting one of the premisses simply and

  inferring the premiss which was assumed in the original syllogism:

  e.g. suppose it has been necessary to prove that A belongs to all C,

  and it has been proved through B; suppose that A should now be

  proved to belong to B by assuming that A belongs to C, and C to B-so A

  belongs to B: but in the first syllogism the converse was assumed,

  viz. that B belongs to C. Or suppose it
is necessary to prove that B

  belongs to C, and A is assumed to belong to C, which was the

  conclusion of the first syllogism, and B to belong to A but the

  converse was assumed in the earlier syllogism, viz. that A belongs

  to B. In no other way is reciprocal proof possible. If another term is

  taken as middle, the proof is not circular: for neither of the

  propositions assumed is the same as before: if one of the accepted

  terms is taken as middle, only one of the premisses of the first

  syllogism can be assumed in the second: for if both of them are

  taken the same conclusion as before will result: but it must be

  different. If the terms are not convertible, one of the premisses from

  which the syllogism results must be undemonstrated: for it is not

  possible to demonstrate through these terms that the third belongs

  to the middle or the middle to the first. If the terms are

  convertible, it is possible to demonstrate everything reciprocally,

  e.g. if A and B and C are convertible with one another. Suppose the

  proposition AC has been demonstrated through B as middle term, and

  again the proposition AB through the conclusion and the premiss BC

  converted, and similarly the proposition BC through the conclusion and

  the premiss AB converted. But it is necessary to prove both the

  premiss CB, and the premiss BA: for we have used these alone without

  demonstrating them. If then it is assumed that B belongs to all C, and

  C to all A, we shall have a syllogism relating B to A. Again if it

  is assumed that C belongs to all A, and A to all B, C must belong to

  all B. In both these syllogisms the premiss CA has been assumed

  without being demonstrated: the other premisses had ex hypothesi

  been proved. Consequently if we succeed in demonstrating this premiss,

  all the premisses will have been proved reciprocally. If then it is

  assumed that C belongs to all B, and B to all A, both the premisses

  assumed have been proved, and C must belong to A. It is clear then

  that only if the terms are convertible is circular and reciprocal

  demonstration possible (if the terms are not convertible, the matter