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  no refutation is possible. For if a refutation were possible, a

  syllogism must be possible; although if a syllogism is possible it

  does not follow that a refutation is possible. Similarly refutation is

  not possible if nothing is conceded universally: since the fields of

  refutation and syllogism are defined in the same way.

  21

  It sometimes happens that just as we are deceived in the arrangement

  of the terms, so error may arise in our thought about them, e.g. if it

  is possible that the same predicate should belong to more than one

  subject immediately, but although knowing the one, a man may forget

  the other and think the opposite true. Suppose that A belongs to B and

  to C in virtue of their nature, and that B and C belong to all D in

  the same way. If then a man thinks that A belongs to all B, and B to

  D, but A to no C, and C to all D, he will both know and not know the

  same thing in respect of the same thing. Again if a man were to make a

  mistake about the members of a single series; e.g. suppose A belongs

  to B, B to C, and C to D, but some one thinks that A belongs to all B,

  but to no C: he will both know that A belongs to D, and think that

  it does not. Does he then maintain after this simply that what he

  knows, he does not think? For he knows in a way that A belongs to C

  through B, since the part is included in the whole; so that what he

  knows in a way, this he maintains he does not think at all: but that

  is impossible.

  In the former case, where the middle term does not belong to the

  same series, it is not possible to think both the premisses with

  reference to each of the two middle terms: e.g. that A belongs to

  all B, but to no C, and both B and C belong to all D. For it turns out

  that the first premiss of the one syllogism is either wholly or

  partially contrary to the first premiss of the other. For if he thinks

  that A belongs to everything to which B belongs, and he knows that B

  belongs to D, then he knows that A belongs to D. Consequently if again

  he thinks that A belongs to nothing to which C belongs, he thinks that

  A does not belong to some of that to which B belongs; but if he thinks

  that A belongs to everything to which B belongs, and again thinks that

  A does not belong to some of that to which B belongs, these beliefs

  are wholly or partially contrary. In this way then it is not

  possible to think; but nothing prevents a man thinking one premiss

  of each syllogism of both premisses of one of the two syllogisms: e.g.

  A belongs to all B, and B to D, and again A belongs to no C. An

  error of this kind is similar to the error into which we fall

  concerning particulars: e.g. if A belongs to all B, and B to all C,

  A will belong to all C. If then a man knows that A belongs to

  everything to which B belongs, he knows that A belongs to C. But

  nothing prevents his being ignorant that C exists; e.g. let A stand

  for two right angles, B for triangle, C for a particular diagram of

  a triangle. A man might think that C did not exist, though he knew

  that every triangle contains two right angles; consequently he will

  know and not know the same thing at the same time. For the

  expression 'to know that every triangle has its angles equal to two

  right angles' is ambiguous, meaning to have the knowledge either of

  the universal or of the particulars. Thus then he knows that C

  contains two right angles with a knowledge of the universal, but not

  with a knowledge of the particulars; consequently his knowledge will

  not be contrary to his ignorance. The argument in the Meno that

  learning is recollection may be criticized in a similar way. For it

  never happens that a man starts with a foreknowledge of the

  particular, but along with the process of being led to see the general

  principle he receives a knowledge of the particulars, by an act (as it

  were) of recognition. For we know some things directly; e.g. that

  the angles are equal to two right angles, if we know that the figure

  is a triangle. Similarly in all other cases.

  By a knowledge of the universal then we see the particulars, but

  we do not know them by the kind of knowledge which is proper to

  them; consequently it is possible that we may make mistakes about

  them, but not that we should have the knowledge and error that are

  contrary to one another: rather we have the knowledge of the universal

  but make a mistake in apprehending the particular. Similarly in the

  cases stated above. The error in respect of the middle term is not

  contrary to the knowledge obtained through the syllogism, nor is the

  thought in respect of one middle term contrary to that in respect of

  the other. Nothing prevents a man who knows both that A belongs to the

  whole of B, and that B again belongs to C, thinking that A does not

  belong to C, e.g. knowing that every mule is sterile and that this

  is a mule, and thinking that this animal is with foal: for he does not

  know that A belongs to C, unless he considers the two propositions

  together. So it is evident that if he knows the one and does not

  know the other, he will fall into error. And this is the relation of

  knowledge of the universal to knowledge of the particular. For we know

  no sensible thing, once it has passed beyond the range of our

  senses, even if we happen to have perceived it, except by means of the

  universal and the possession of the knowledge which is proper to the

  particular, but without the actual exercise of that knowledge. For

  to know is used in three senses: it may mean either to have

  knowledge of the universal or to have knowledge proper to the matter

  in hand or to exercise such knowledge: consequently three kinds of

  error also are possible. Nothing then prevents a man both knowing

  and being mistaken about the same thing, provided that his knowledge

  and his error are not contrary. And this happens also to the man whose

  knowledge is limited to each of the premisses and who has not

  previously considered the particular question. For when he thinks that

  the mule is with foal he has not the knowledge in the sense of its

  actual exercise, nor on the other hand has his thought caused an error

  contrary to his knowledge: for the error contrary to the knowledge

  of the universal would be a syllogism.

  But he who thinks the essence of good is the essence of bad will

  think the same thing to be the essence of good and the essence of bad.

  Let A stand for the essence of good and B for the essence of bad,

  and again C for the essence of good. Since then he thinks B and C

  identical, he will think that C is B, and similarly that B is A,

  consequently that C is A. For just as we saw that if B is true of

  all of which C is true, and A is true of all of which B is true, A

  is true of C, similarly with the word 'think'. Similarly also with the

  word 'is'; for we saw that if C is the same as B, and B as A, C is the

  same as A. Similarly therefore with 'opine'. Perhaps then this is

  necessary if a man will grant the first point. But presumably that

  is false, that any one could suppose the essence of good to be the
<
br />   essence of bad, save incidentally. For it is possible to think this in

  many different ways. But we must consider this matter better.

  22

  Whenever the extremes are convertible it is necessary that the

  middle should be convertible with both. For if A belongs to C

  through B, then if A and C are convertible and C belongs everything to

  which A belongs, B is convertible with A, and B belongs to

  everything to which A belongs, through C as middle, and C is

  convertible with B through A as middle. Similarly if the conclusion is

  negative, e.g. if B belongs to C, but A does not belong to B,

  neither will A belong to C. If then B is convertible with A, C will be

  convertible with A. Suppose B does not belong to A; neither then

  will C: for ex hypothesi B belonged to all C. And if C is

  convertible with B, B is convertible also with A, for C is said of

  that of all of which B is said. And if C is convertible in relation to

  A and to B, B also is convertible in relation to A. For C belongs to

  that to which B belongs: but C does not belong to that to which A

  belongs. And this alone starts from the conclusion; the preceding

  moods do not do so as in the affirmative syllogism. Again if A and B

  are convertible, and similarly C and D, and if A or C must belong to

  anything whatever, then B and D will be such that one or other belongs

  to anything whatever. For since B belongs to that to which A

  belongs, and D belongs to that to which C belongs, and since A or C

  belongs to everything, but not together, it is clear that B or D

  belongs to everything, but not together. For example if that which

  is uncreated is incorruptible and that which is incorruptible is

  uncreated, it is necessary that what is created should be

  corruptible and what is corruptible should have been created. For

  two syllogisms have been put together. Again if A or B belongs to

  everything and if C or D belongs to everything, but they cannot belong

  together, then when A and C are convertible B and D are convertible.

  For if B does not belong to something to which D belongs, it is

  clear that A belongs to it. But if A then C: for they are convertible.

  Therefore C and D belong together. But this is impossible. When A

  belongs to the whole of B and to C and is affirmed of nothing else,

  and B also belongs to all C, it is necessary that A and B should be

  convertible: for since A is said of B and C only, and B is affirmed

  both of itself and of C, it is clear that B will be said of everything

  of which A is said, except A itself. Again when A and B belong to

  the whole of C, and C is convertible with B, it is necessary that A

  should belong to all B: for since A belongs to all C, and C to B by

  conversion, A will belong to all B.

  When, of two opposites A and B, A is preferable to B, and

  similarly D is preferable to C, then if A and C together are

  preferable to B and D together, A must be preferable to D. For A is an

  object of desire to the same extent as B is an object of aversion,

  since they are opposites: and C is similarly related to D, since

  they also are opposites. If then A is an object of desire to the

  same extent as D, B is an object of aversion to the same extent as C

  (since each is to the same extent as each-the one an object of

  aversion, the other an object of desire). Therefore both A and C

  together, and B and D together, will be equally objects of desire or

  aversion. But since A and C are preferable to B and D, A cannot be

  equally desirable with D; for then B along with D would be equally

  desirable with A along with C. But if D is preferable to A, then B

  must be less an object of aversion than C: for the less is opposed

  to the less. But the greater good and lesser evil are preferable to

  the lesser good and greater evil: the whole BD then is preferable to

  the whole AC. But ex hypothesi this is not so. A then is preferable to

  D, and C consequently is less an object of aversion than B. If then

  every lover in virtue of his love would prefer A, viz. that the

  beloved should be such as to grant a favour, and yet should not

  grant it (for which C stands), to the beloved's granting the favour

  (represented by D) without being such as to grant it (represented by

  B), it is clear that A (being of such a nature) is preferable to

  granting the favour. To receive affection then is preferable in love

  to sexual intercourse. Love then is more dependent on friendship

  than on intercourse. And if it is most dependent on receiving

  affection, then this is its end. Intercourse then either is not an end

  at all or is an end relative to the further end, the receiving of

  affection. And indeed the same is true of the other desires and arts.

  23

  It is clear then how the terms are related in conversion, and in

  respect of being in a higher degree objects of aversion or of

  desire. We must now state that not only dialectical and

  demonstrative syllogisms are formed by means of the aforesaid figures,

  but also rhetorical syllogisms and in general any form of

  persuasion, however it may be presented. For every belief comes either

  through syllogism or from induction.

  Now induction, or rather the syllogism which springs out of

  induction, consists in establishing syllogistically a relation between

  one extreme and the middle by means of the other extreme, e.g. if B is

  the middle term between A and C, it consists in proving through C that

  A belongs to B. For this is the manner in which we make inductions.

  For example let A stand for long-lived, B for bileless, and C for

  the particular long-lived animals, e.g. man, horse, mule. A then

  belongs to the whole of C: for whatever is bileless is long-lived. But

  B also ('not possessing bile') belongs to all C. If then C is

  convertible with B, and the middle term is not wider in extension,

  it is necessary that A should belong to B. For it has already been

  proved that if two things belong to the same thing, and the extreme is

  convertible with one of them, then the other predicate will belong

  to the predicate that is converted. But we must apprehend C as made up

  of all the particulars. For induction proceeds through an

  enumeration of all the cases.

  Such is the syllogism which establishes the first and immediate

  premiss: for where there is a middle term the syllogism proceeds

  through the middle term; when there is no middle term, through

  induction. And in a way induction is opposed to syllogism: for the

  latter proves the major term to belong to the third term by means of

  the middle, the former proves the major to belong to the middle by

  means of the third. In the order of nature, syllogism through the

  middle term is prior and better known, but syllogism through induction

  is clearer to us.

  24

  We have an 'example' when the major term is proved to belong to

  the middle by means of a term which resembles the third. It ought to

  be known both that the middle belongs to the third term, and that

  the first belongs to that which resembles the third. For example let A

  be evil
, B making war against neighbours, C Athenians against Thebans,

  D Thebans against Phocians. If then we wish to prove that to fight

  with the Thebans is an evil, we must assume that to fight against

  neighbours is an evil. Evidence of this is obtained from similar

  cases, e.g. that the war against the Phocians was an evil to the

  Thebans. Since then to fight against neighbours is an evil, and to

  fight against the Thebans is to fight against neighbours, it is

  clear that to fight against the Thebans is an evil. Now it is clear

  that B belongs to C and to D (for both are cases of making war upon

  one's neighbours) and that A belongs to D (for the war against the

  Phocians did not turn out well for the Thebans): but that A belongs to

  B will be proved through D. Similarly if the belief in the relation of

  the middle term to the extreme should be produced by several similar

  cases. Clearly then to argue by example is neither like reasoning from

  part to whole, nor like reasoning from whole to part, but rather

  reasoning from part to part, when both particulars are subordinate

  to the same term, and one of them is known. It differs from induction,

  because induction starting from all the particular cases proves (as we

  saw) that the major term belongs to the middle, and does not apply the

  syllogistic conclusion to the minor term, whereas argument by

  example does make this application and does not draw its proof from

  all the particular cases.

  25

  By reduction we mean an argument in which the first term clearly

  belongs to the middle, but the relation of the middle to the last term

  is uncertain though equally or more probable than the conclusion; or

  again an argument in which the terms intermediate between the last

  term and the middle are few. For in any of these cases it turns out

  that we approach more nearly to knowledge. For example let A stand for

  what can be taught, B for knowledge, C for justice. Now it is clear

  that knowledge can be taught: but it is uncertain whether virtue is

  knowledge. If now the statement BC is equally or more probable than

  AC, we have a reduction: for we are nearer to knowledge, since we have