Aristotle

by Various Works [lit]

  contradictories.

  In the first figure no syllogism whether affirmative or negative can

  be made out of opposed premisses: no affirmative syllogism is possible

  because both premisses must be affirmative, but opposites are, the one

  affirmative, the other negative: no negative syllogism is possible

  because opposites affirm and deny the same predicate of the same

  subject, and the middle term in the first figure is not predicated

  of both extremes, but one thing is denied of it, and it is affirmed of

  something else: but such premisses are not opposed.

  In the middle figure a syllogism can be made both

  oLcontradictories and of contraries. Let A stand for good, let B and C

  stand for science. If then one assumes that every science is good, and

  no science is good, A belongs to all B and to no C, so that B

  belongs to no C: no science then is a science. Similarly if after

  taking 'every science is good' one took 'the science of medicine is

  not good'; for A belongs to all B but to no C, so that a particular

  science will not be a science. Again, a particular science will not be

  a science if A belongs to all C but to no B, and B is science, C

  medicine, and A supposition: for after taking 'no science is

  supposition', one has assumed that a particular science is

  supposition. This syllogism differs from the preceding because the

  relations between the terms are reversed: before, the affirmative

  statement concerned B, now it concerns C. Similarly if one premiss

  is not universal: for the middle term is always that which is stated

  negatively of one extreme, and affirmatively of the other.

  Consequently it is possible that contradictories may lead to a

  conclusion, though not always or in every mood, but only if the

  terms subordinate to the middle are such that they are either

  identical or related as whole to part. Otherwise it is impossible: for

  the premisses cannot anyhow be either contraries or contradictories.

  In the third figure an affirmative syllogism can never be made out

  of opposite premisses, for the reason given in reference to the

  first figure; but a negative syllogism is possible whether the terms

  are universal or not. Let B and C stand for science, A for medicine.

  If then one should assume that all medicine is science and that no

  medicine is science, he has assumed that B belongs to all A and C to

  no A, so that a particular science will not be a science. Similarly if

  the premiss BA is not assumed universally. For if some medicine is

  science and again no medicine is science, it results that some science

  is not science, The premisses are contrary if the terms are taken

  universally; if one is particular, they are contradictory.

  We must recognize that it is possible to take opposites in the way

  we said, viz. 'all science is good' and 'no science is good' or

  'some science is not good'. This does not usually escape notice. But

  it is possible to establish one part of a contradiction through

  other premisses, or to assume it in the way suggested in the Topics.

  Since there are three oppositions to affirmative statements, it

  follows that opposite statements may be assumed as premisses in six

  ways; we may have either universal affirmative and negative, or

  universal affirmative and particular negative, or particular

  affirmative and universal negative, and the relations between the

  terms may be reversed; e.g. A may belong to all B and to no C, or to

  all C and to no B, or to all of the one, not to all of the other; here

  too the relation between the terms may be reversed. Similarly in the

  third figure. So it is clear in how many ways and in what figures a

  syllogism can be made by means of premisses which are opposed.

  It is clear too that from false premisses it is possible to draw a

  true conclusion, as has been said before, but it is not possible if

  the premisses are opposed. For the syllogism is always contrary to the

  fact, e.g. if a thing is good, it is proved that it is not good, if an

  animal, that it is not an animal because the syllogism springs out

  of a contradiction and the terms presupposed are either identical or

  related as whole and part. It is evident also that in fallacious

  reasonings nothing prevents a contradiction to the hypothesis from

  resulting, e.g. if something is odd, it is not odd. For the

  syllogism owed its contrariety to its contradictory premisses; if we

  assume such premisses we shall get a result that contradicts our

  hypothesis. But we must recognize that contraries cannot be inferred

  from a single syllogism in such a way that we conclude that what is

  not good is good, or anything of that sort unless a self-contradictory

  premiss is at once assumed, e.g. 'every animal is white and not

  white', and we proceed 'man is an animal'. Either we must introduce

  the contradiction by an additional assumption, assuming, e.g., that

  every science is supposition, and then assuming 'Medicine is a

  science, but none of it is supposition' (which is the mode in which

  refutations are made), or we must argue from two syllogisms. In no

  other way than this, as was said before, is it possible that the

  premisses should be really contrary.

  16

  To beg and assume the original question is a species of failure to

  demonstrate the problem proposed; but this happens in many ways. A man

  may not reason syllogistically at all, or he may argue from

  premisses which are less known or equally unknown, or he may establish

  the antecedent by means of its consequents; for demonstration proceeds

  from what is more certain and is prior. Now begging the question is

  none of these: but since we get to know some things naturally

  through themselves, and other things by means of something else (the

  first principles through themselves, what is subordinate to them

  through something else), whenever a man tries to prove what is not

  self-evident by means of itself, then he begs the original question.

  This may be done by assuming what is in question at once; it is also

  possible to make a transition to other things which would naturally be

  proved through the thesis proposed, and demonstrate it through them,

  e.g. if A should be proved through B, and B through C, though it was

  natural that C should be proved through A: for it turns out that those

  who reason thus are proving A by means of itself. This is what those

  persons do who suppose that they are constructing parallel straight

  lines: for they fail to see that they are assuming facts which it is

  impossible to demonstrate unless the parallels exist. So it turns

  out that those who reason thus merely say a particular thing is, if it

  is: in this way everything will be self-evident. But that is

  impossible.

  If then it is uncertain whether A belongs to C, and also whether A

  belongs to B, and if one should assume that A does belong to B, it

  is not yet clear whether he begs the original question, but it is

  evident that he is not demonstrating: for what is as uncertain as

  the question to be answered cannot be a principle of a

  demonstration. If
however B is so related to C that they are

  identical, or if they are plainly convertible, or the one belongs to

  the other, the original question is begged. For one might equally well

  prove that A belongs to B through those terms if they are convertible.

  But if they are not convertible, it is the fact that they are not that

  prevents such a demonstration, not the method of demonstrating. But if

  one were to make the conversion, then he would be doing what we have

  described and effecting a reciprocal proof with three propositions.

  Similarly if he should assume that B belongs to C, this being as

  uncertain as the question whether A belongs to C, the question is

  not yet begged, but no demonstration is made. If however A and B are

  identical either because they are convertible or because A follows

  B, then the question is begged for the same reason as before. For we

  have explained the meaning of begging the question, viz. proving

  that which is not self-evident by means of itself.

  If then begging the question is proving what is not self-evident

  by means of itself, in other words failing to prove when the failure

  is due to the thesis to be proved and the premiss through which it

  is proved being equally uncertain, either because predicates which are

  identical belong to the same subject, or because the same predicate

  belongs to subjects which are identical, the question may be begged in

  the middle and third figures in both ways, though, if the syllogism is

  affirmative, only in the third and first figures. If the syllogism

  is negative, the question is begged when identical predicates are

  denied of the same subject; and both premisses do not beg the question

  indifferently (in a similar way the question may be begged in the

  middle figure), because the terms in negative syllogisms are not

  convertible. In scientific demonstrations the question is begged

  when the terms are really related in the manner described, in

  dialectical arguments when they are according to common opinion so

  related.

  17

  The objection that 'this is not the reason why the result is false',

  which we frequently make in argument, is made primarily in the case of

  a reductio ad impossibile, to rebut the proposition which was being

  proved by the reduction. For unless a man has contradicted this

  proposition he will not say, 'False cause', but urge that something

  false has been assumed in the earlier parts of the argument; nor

  will he use the formula in the case of an ostensive proof; for here

  what one denies is not assumed as a premiss. Further when anything

  is refuted ostensively by the terms ABC, it cannot be objected that

  the syllogism does not depend on the assumption laid down. For we

  use the expression 'false cause', when the syllogism is concluded in

  spite of the refutation of this position; but that is not possible

  in ostensive proofs: since if an assumption is refuted, a syllogism

  can no longer be drawn in reference to it. It is clear then that the

  expression 'false cause' can only be used in the case of a reductio ad

  impossibile, and when the original hypothesis is so related to the

  impossible conclusion, that the conclusion results indifferently

  whether the hypothesis is made or not. The most obvious case of the

  irrelevance of an assumption to a conclusion which is false is when

  a syllogism drawn from middle terms to an impossible conclusion is

  independent of the hypothesis, as we have explained in the Topics. For

  to put that which is not the cause as the cause, is just this: e.g. if

  a man, wishing to prove that the diagonal of the square is

  incommensurate with the side, should try to prove Zeno's theorem

  that motion is impossible, and so establish a reductio ad impossibile:

  for Zeno's false theorem has no connexion at all with the original

  assumption. Another case is where the impossible conclusion is

  connected with the hypothesis, but does not result from it. This may

  happen whether one traces the connexion upwards or downwards, e.g.

  if it is laid down that A belongs to B, B to C, and C to D, and it

  should be false that B belongs to D: for if we eliminated A and

  assumed all the same that B belongs to C and C to D, the false

  conclusion would not depend on the original hypothesis. Or again trace

  the connexion upwards; e.g. suppose that A belongs to B, E to A and

  F to E, it being false that F belongs to A. In this way too the

  impossible conclusion would result, though the original hypothesis

  were eliminated. But the impossible conclusion ought to be connected

  with the original terms: in this way it will depend on the hypothesis,

  e.g. when one traces the connexion downwards, the impossible

  conclusion must be connected with that term which is predicate in

  the hypothesis: for if it is impossible that A should belong to D, the

  false conclusion will no longer result after A has been eliminated. If

  one traces the connexion upwards, the impossible conclusion must be

  connected with that term which is subject in the hypothesis: for if it

  is impossible that F should belong to B, the impossible conclusion

  will disappear if B is eliminated. Similarly when the syllogisms are

  negative.

  It is clear then that when the impossibility is not related to the

  original terms, the false conclusion does not result on account of the

  assumption. Or perhaps even so it may sometimes be independent. For if

  it were laid down that A belongs not to B but to K, and that K belongs

  to C and C to D, the impossible conclusion would still stand.

  Similarly if one takes the terms in an ascending series.

  Consequently since the impossibility results whether the first

  assumption is suppressed or not, it would appear to be independent

  of that assumption. Or perhaps we ought not to understand the

  statement that the false conclusion results independently of the

  assumption, in the sense that if something else were supposed the

  impossibility would result; but rather we mean that when the first

  assumption is eliminated, the same impossibility results through the

  remaining premisses; since it is not perhaps absurd that the same

  false result should follow from several hypotheses, e.g. that

  parallels meet, both on the assumption that the interior angle is

  greater than the exterior and on the assumption that a triangle

  contains more than two right angles.

  18

  A false argument depends on the first false statement in it. Every

  syllogism is made out of two or more premisses. If then the false

  conclusion is drawn from two premisses, one or both of them must be

  false: for (as we proved) a false syllogism cannot be drawn from two

  premisses. But if the premisses are more than two, e.g. if C is

  established through A and B, and these through D, E, F, and G, one

  of these higher propositions must be false, and on this the argument

  depends: for A and B are inferred by means of D, E, F, and G.

  Therefore the conclusion and the error results from one of them.

  19

  In order to avoid having a syllogism drawn against us we must take />
  care, whenever an opponent asks us to admit the reason without the

  conclusions, not to grant him the same term twice over in his

  premisses, since we know that a syllogism cannot be drawn without a

  middle term, and that term which is stated more than once is the

  middle. How we ought to watch the middle in reference to each

  conclusion, is evident from our knowing what kind of thesis is

  proved in each figure. This will not escape us since we know how we

  are maintaining the argument.

  That which we urge men to beware of in their admissions, they

  ought in attack to try to conceal. This will be possible first, if,

  instead of drawing the conclusions of preliminary syllogisms, they

  take the necessary premisses and leave the conclusions in the dark;

  secondly if instead of inviting assent to propositions which are

  closely connected they take as far as possible those that are not

  connected by middle terms. For example suppose that A is to be

  inferred to be true of F, B, C, D, and E being middle terms. One ought

  then to ask whether A belongs to B, and next whether D belongs to E,

  instead of asking whether B belongs to C; after that he may ask

  whether B belongs to C, and so on. If the syllogism is drawn through

  one middle term, he ought to begin with that: in this way he will most

  likely deceive his opponent.

  20

  Since we know when a syllogism can be formed and how its terms

  must be related, it is clear when refutation will be possible and when

  impossible. A refutation is possible whether everything is conceded,

  or the answers alternate (one, I mean, being affirmative, the other

  negative). For as has been shown a syllogism is possible whether the

  terms are related in affirmative propositions or one proposition is

  affirmative, the other negative: consequently, if what is laid down is

  contrary to the conclusion, a refutation must take place: for a

  refutation is a syllogism which establishes the contradictory. But

  if nothing is conceded, a refutation is impossible: for no syllogism

  is possible (as we saw) when all the terms are negative: therefore