Posterior Analytics   

either or both premisses are false, the conclusion will equally be
false.
In the second figure the premisses cannot both be wholly false;
for if all B is A, no middle term can be with truth universally
affirmed of one extreme and universally denied of the other: but
premisses in which the middle is affirmed of one extreme and denied of
the other are the necessary condition if one is to get a valid
inference at all. Therefore if, taken in this way, they are wholly
false, their contraries conversely should be wholly true. But this
is impossible. On the other hand, there is nothing to prevent both
premisses being partially false; e.g. if actually some A is C and some
B is C, then if it is premised that all A is C and no B is C, both
premisses are false, yet partially, not wholly, false. The same is
true if the major is made negative instead of the minor. Or one
premiss may be wholly false, and it may be either of them. Thus,
supposing that actually an attribute of all A must also be an
attribute of all B, then if C is yet taken to be a universal attribute
of all but universally non-attributable to B, C-A will be true but C-B
false. Again, actually that which is an attribute of no B will not
be an attribute of all A either; for if it be an attribute of all A,
it will also be an attribute of all B, which is contrary to
supposition; but if C be nevertheless assumed to be a universal
attribute of A, but an attribute of no B, then the premiss C-B is true
but the major is false. The case is similar if the major is made the
negative premiss. For in fact what is an attribute of no A will not be
an attribute of any B either; and if it be yet assumed that C is
universally non-attributable to A, but a universal attribute of B, the
premiss C-A is true but the minor wholly false. Again, in fact it is
false to assume that that which is an attribute of all B is an
attribute of no A, for if it be an attribute of all B, it must be an
attribute of some A. If then C is nevertheless assumed to be an
attribute of all B but of no A, C-B will be true but C-A false.
It is thus clear that in the case of atomic propositions erroneous
inference will be possible not only when both premisses are false
but also when only one is false.

17

In the case of attributes not atomically connected with or
disconnected from their subjects, (a) (i) as long as the false
conclusion is inferred through the 'appropriate' middle, only the
major and not both premisses can be false. By 'appropriate middle' I
mean the middle term through which the contradictory-i.e. the
true-conclusion is inferrible. Thus, let A be attributable to B
through a middle term C: then, since to produce a conclusion the
premiss C-B must be taken affirmatively, it is clear that this premiss
must always be true, for its quality is not changed. But the major A-C
is false, for it is by a change in the quality of A-C that the
conclusion becomes its contradictory-i.e. true. Similarly (ii) if
the middle is taken from another series of predication; e.g. suppose D
to be not only contained within A as a part within its whole but
also predicable of all B. Then the premiss D-B must remain
unchanged, but the quality of A-D must be changed; so that D-B is
always true, A-D always false. Such error is practically identical
with that which is inferred through the 'appropriate' middle. On the
other hand, (b) if the conclusion is not inferred through the
'appropriate' middle-(i) when the middle is subordinate to A but is
predicable of no B, both premisses must be false, because if there
is to be a conclusion both must be posited as asserting the contrary
of what is actually the fact, and so posited both become false: e.g.
suppose that actually all D is A but no B is D; then if these
premisses are changed in quality, a conclusion will follow and both of
the new premisses will be false. When, however, (ii) the middle D is
not subordinate to A, A-D will be true, D-B false-A-D true because A