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
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