is not-white or all are not-white is false. Similarly also 'every
animal is not-white' is not the negation of 'every animal is white'
(for both are false): the proper negation is 'every animal is not
white'. Since it is clear that 'it is not-white' and 'it is not white'
mean different things, and one is an affirmation, the other a
denial, it is evident that the method of proving each cannot be the
same, e.g. that whatever is an animal is not white or may not be
white, and that it is true to call it not-white; for this means that
it is not-white. But we may prove that it is true to call it white
or not-white in the same way for both are proved constructively by
means of the first figure. For the expression 'it is true' stands on a
similar footing to 'it is'. For the negation of 'it is true to call it
white' is not 'it is true to call it not-white' but 'it is not true to
call it white'. If then it is to be true to say that whatever is a man
is musical or is not-musical, we must assume that whatever is an
animal either is musical or is not-musical; and the proof has been
made. That whatever is a man is not musical is proved destructively in
the three ways mentioned.
In general whenever A and B are such that they cannot belong at
the same time to the same thing, and one of the two necessarily
belongs to everything, and again C and D are related in the same
way, and A follows C but the relation cannot be reversed, then D
must follow B and the relation cannot be reversed. And A and D may
belong to the same thing, but B and C cannot. First it is clear from
the following consideration that D follows B. For since either C or
D necessarily belongs to everything; and since C cannot belong to that
to which B belongs, because it carries A along with it and A and B
cannot belong to the same thing; it is clear that D must follow B.
Again since C does not reciprocate with but A, but C or D belongs to
everything, it is possible that A and D should belong to the same
thing. But B and C cannot belong to the same thing, because A
follows C; and so something impossible results. It is clear then
that B does not reciprocate with D either, since it is possible that D
and A should belong at the same time to the same thing.
It results sometimes even in such an arrangement of terms that one
is deceived through not apprehending the opposites rightly, one of
which must belong to everything, e.g. we may reason that 'if A and B
cannot belong at the same time to the same thing, but it is
necessary that one of them should belong to whatever the other does
not belong to: and again C and D are related in the same way, and
follows everything which C follows: it will result that B belongs
necessarily to everything to which D belongs': but this is false.
'Assume that F stands for the negation of A and B, and again that H
stands for the negation of C and D. It is necessary then that either A
or F should belong to everything: for either the affirmation or the
denial must belong. And again either C or H must belong to everything:
for they are related as affirmation and denial. And ex hypothesi A
belongs to everything ever thing to which C belongs. Therefore H
belongs to everything to which F belongs. Again since either F or B
belongs to everything, and similarly either H or D, and since H
follows F, B must follow D: for we know this. If then A follows C, B
must follow D'. But this is false: for as we proved the sequence is
reversed in terms so constituted. The fallacy arises because perhaps
it is not necessary that A or F should belong to everything, or that F
or B should belong to everything: for F is not the denial of A. For
not good is the negation of good: and not-good is not identical with
'neither good nor not-good'. Similarly also with C and D. For two
negations have been assumed in respect to one term.
Book II
1
WE have already explained the number of the figures, the character
and number of the premisses, when and how a syllogism is formed;
further what we must look for when a refuting and establishing
propositions, and how we should investigate a given problem in any
branch of inquiry, also by what means we shall obtain principles
appropriate to each subject. Since some syllogisms are universal,
others particular, all the universal syllogisms give more than one
result, and of particular syllogisms the affirmative yield more than
one, the negative yield only the stated conclusion. For all
propositions are convertible save only the particular negative: and
the conclusion states one definite thing about another definite thing.
Consequently all syllogisms save the particular negative yield more
than one conclusion, e.g. if A has been proved to to all or to some B,
then B must belong to some A: and if A has been proved to belong to no
B, then B belongs to no A. This is a different conclusion from the
former. But if A does not belong to some B, it is not necessary that B
should not belong to some A: for it may possibly belong to all A.
This then is the reason common to all syllogisms whether universal
or particular. But it is possible to give another reason concerning
those which are universal. For all the things that are subordinate
to the middle term or to the conclusion may be proved by the same
syllogism, if the former are placed in the middle, the latter in the
conclusion; e.g. if the conclusion AB is proved through C, whatever is
subordinate to B or C must accept the predicate A: for if D is
included in B as in a whole, and B is included in A, then D will be
included in A. Again if E is included in C as in a whole, and C is
included in A, then E will be included in A. Similarly if the
syllogism is negative. In the second figure it will be possible to
infer only that which is subordinate to the conclusion, e.g. if A
belongs to no B and to all C; we conclude that B belongs to no C. If
then D is subordinate to C, clearly B does not belong to it. But
that B does not belong to what is subordinate to A is not clear by
means of the syllogism. And yet B does not belong to E, if E is
subordinate to A. But while it has been proved through the syllogism
that B belongs to no C, it has been assumed without proof that B
does not belong to A, consequently it does not result through the
syllogism that B does not belong to E.
But in particular syllogisms there will be no necessity of inferring
what is subordinate to the conclusion (for a syllogism does not result
when this premiss is particular), but whatever is subordinate to the
middle term may be inferred, not however through the syllogism, e.g.
if A belongs to all B and B to some C. Nothing can be inferred about
that which is subordinate to C; something can be inferred about that
which is subordinate to B, but not through the preceding syllogism.
Similarly in the other figures. That which is subordinate to the
conclusion cannot be proved; the other subordinate can be proved, only
not through the syllogism, just as in the universal syllogisms what
is
subordinate to the middle term is proved (as we saw) from a premiss
which is not demonstrated: consequently either a conclusion is not
possible in the case of universal syllogisms or else it is possible
also in the case of particular syllogisms.
2
It is possible for the premisses of the syllogism to be true, or
to be false, or to be the one true, the other false. The conclusion is
either true or false necessarily. From true premisses it is not
possible to draw a false conclusion, but a true conclusion may be
drawn from false premisses, true however only in respect to the
fact, not to the reason. The reason cannot be established from false
premisses: why this is so will be explained in the sequel.
First then that it is not possible to draw a false conclusion from
true premisses, is made clear by this consideration. If it is
necessary that B should be when A is, it is necessary that A should
not be when B is not. If then A is true, B must be true: otherwise
it will turn out that the same thing both is and is not at the same
a single term, be supposed that it is possible, when a single fact
is given, that something should necessarily result. For that is not
possible. For what results necessarily is the conclusion, and the
means by which this comes about are at the least three terms, and
two relations of subject and predicate or premisses. If then it is
true that A belongs to all that to which B belongs, and that B belongs
to all that to which C belongs, it is necessary that A should belong
to all that to which C belongs, and this cannot be false: for then the
same thing will belong and not belong at the same time. So A is
posited as one thing, being two premisses taken together. The same
holds good of negative syllogisms: it is not possible to prove a false
conclusion from true premisses.
But from what is false a true conclusion may be drawn, whether
both the premisses are false or only one, provided that this is not
either of the premisses indifferently, if it is taken as wholly false:
but if the premiss is not taken as wholly false, it does not matter
which of the two is false. (1) Let A belong to the whole of C, but
to none of the Bs, neither let B belong to C. This is possible, e.g.
animal belongs to no stone, nor stone to any man. If then A is taken
to belong to all B and B to all C, A will belong to all C;
consequently though both the premisses are false the conclusion is
true: for every man is an animal. Similarly with the negative. For
it is possible that neither A nor B should belong to any C, although A
belongs to all B, e.g. if the same terms are taken and man is put as
middle: for neither animal nor man belongs to any stone, but animal
belongs to every man. Consequently if one term is taken to belong to
none of that to which it does belong, and the other term is taken to
belong to all of that to which it does not belong, though both the
premisses are false the conclusion will be true. (2) A similar proof
may be given if each premiss is partially false.
(3) But if one only of the premisses is false, when the first
premiss is wholly false, e.g. AB, the conclusion will not be true, but
if the premiss BC is wholly false, a true conclusion will be possible.
I mean by 'wholly false' the contrary of the truth, e.g. if what
belongs to none is assumed to belong to all, or if what belongs to all
is assumed to belong to none. Let A belong to no B, and B to all C. If
then the premiss BC which I take is true, and the premiss AB is wholly
false, viz. that A belongs to all B, it is impossible that the
conclusion should be true: for A belonged to none of the Cs, since A
belonged to nothing to which B belonged, and B belonged to all C.
Similarly there cannot be a true conclusion if A belongs to all B, and
B to all C, but while the true premiss BC is assumed, the wholly false
premiss AB is also assumed, viz. that A belongs to nothing to which
B belongs: here the conclusion must be false. For A will belong to all
C, since A belongs to everything to which B belongs, and B to all C.
It is clear then that when the first premiss is wholly false,
whether affirmative or negative, and the other premiss is true, the
conclusion cannot be true.
(4) But if the premiss is not wholly false, a true conclusion is
possible. For if A belongs to all C and to some B, and if B belongs to
all C, e.g. animal to every swan and to some white thing, and white to
every swan, then if we take as premisses that A belongs to all B,
and B to all C, A will belong to all C truly: for every swan is an
animal. Similarly if the statement AB is negative. For it is
possible that A should belong to some B and to no C, and that B should
belong to all C, e.g. animal to some white thing, but to no snow,
and white to all snow. If then one should assume that A belongs to
no B, and B to all C, then will belong to no C.
(5) But if the premiss AB, which is assumed, is wholly true, and the
premiss BC is wholly false, a true syllogism will be possible: for
nothing prevents A belonging to all B and to all C, though B belongs
subordinate one to the other: for animal belongs both to horse and
to man, but horse to no man. If then it is assumed that A belongs to
all B and B to all C, the conclusion will be true, although the
premiss BC is wholly false. Similarly if the premiss AB is negative.
For it is possible that A should belong neither to any B nor to any C,
and that B should not belong to any C, e.g. a genus to species of
another genus: for animal belongs neither to music nor to the art of
healing, nor does music belong to the art of healing. If then it is
assumed that A belongs to no B, and B to all C, the conclusion will be
true.
(6) And if the premiss BC is not wholly false but in part only, even
so the conclusion may be true. For nothing prevents A belonging to the
whole of B and of C, while B belongs to some C, e.g. a genus to its
species and difference: for animal belongs to every man and to every
footed thing, and man to some footed things though not to all. If then
it is assumed that A belongs to all B, and B to all C, A will belong
to all C: and this ex hypothesi is true. Similarly if the premiss AB
is negative. For it is possible that A should neither belong to any
B nor to any C, though B belongs to some C, e.g. a genus to the
species of another genus and its difference: for animal neither
belongs to any wisdom nor to any instance of 'speculative', but wisdom
belongs to some instance of 'speculative'. If then it should be
assumed that A belongs to no B, and B to all C, will belong to no C:
and this ex hypothesi is true.
In particular syllogisms it is possible when the first premiss is
wholly false, and the other true, that the conclusion should be
true; also when the first premiss is false in part, and the other
true; and when th
e first is true, and the particular is false; and
when both are false. (7) For nothing prevents A belonging to no B, but
to some C, and B to some C, e.g. animal belongs to no snow, but to
some white thing, and snow to some white thing. If then snow is
taken as middle, and animal as first term, and it is assumed that A
belongs to the whole of B, and B to some C, then the premiss BC is
wholly false, the premiss BC true, and the conclusion true.
Similarly if the premiss AB is negative: for it is possible that A
should belong to the whole of B, but not to some C, although B belongs
to some C, e.g. animal belongs to every man, but does not follow
some white, but man belongs to some white; consequently if man be
taken as middle term and it is assumed that A belongs to no B but B
belongs to some C, the conclusion will be true although the premiss AB
is wholly false. (If the premiss AB is false in part, the conclusion
may be true. For nothing prevents A belonging both to B and to some C,
and B belonging to some C, e.g. animal to something beautiful and to
something great, and beautiful belonging to something great. If then A
is assumed to belong to all B, and B to some C, the a premiss AB
will be partially false, the premiss BC will be true, and the
conclusion true. Similarly if the premiss AB is negative. For the same
terms will serve, and in the same positions, to prove the point.
(9) Again if the premiss AB is true, and the premiss BC is false,
the conclusion may be true. For nothing prevents A belonging to the
whole of B and to some C, while B belongs to no C, e.g. animal to
every swan and to some black things, though swan belongs to no black
thing. Consequently if it should be assumed that A belongs to all B,
and B to some C, the conclusion will be true, although the statement
BC is false. Similarly if the premiss AB is negative. For it is
possible that A should belong to no B, and not to some C, while B
belongs to no C, e.g. a genus to the species of another genus and to
the accident of its own species: for animal belongs to no number and
not to some white things, and number belongs to nothing white. If then