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  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.

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  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.

  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.

  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.

  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 cannotr />
  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,