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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