Posterior Analytics   

predicated of another ad infinitum, since you have an infinity of
terms between you and F; and equally, if you ascend from F, there
are infinite terms between you and A. It follows that if these
processes are impossible there cannot be an infinity of
intermediates between A and F. Nor is it of any effect to urge that
some terms of the series AB...F are contiguous so as to exclude
intermediates, while others cannot be taken into the argument at
all: whichever terms of the series B...I take, the number of
intermediates in the direction either of A or of F must be finite or
infinite: where the infinite series starts, whether from the first
term or from a later one, is of no moment, for the succeeding terms in
any case are infinite in number.

21

Further, if in affirmative demonstration the series terminates in
both directions, clearly it will terminate too in negative
demonstration. Let us assume that we cannot proceed to infinity either
by ascending from the ultimate term (by 'ultimate term' I mean a
term such as was, not itself attributable to a subject but itself
the subject of attributes), or by descending towards an ultimate
from the primary term (by 'primary term' I mean a term predicable of a
subject but not itself a subject). If this assumption is justified,
the series will also terminate in the case of negation. For a negative
conclusion can be proved in all three figures. In the first figure
it is proved thus: no B is A, all C is B. In packing the interval
B-C we must reach immediate propositions--as is always the case with
the minor premiss--since B-C is affirmative. As regards the other
premiss it is plain that if the major term is denied of a term D prior
to B, D will have to be predicable of all B, and if the major is
denied of yet another term prior to D, this term must be predicable of
all D. Consequently, since the ascending series is finite, the descent
will also terminate and there will be a subject of which A is
primarily non-predicable. In the second figure the syllogism is, all A
is B, no C is B,..no C is A. If proof of this is required, plainly
it may be shown either in the first figure as above, in the second
as here, or in the third. The first figure has been discussed, and
we will proceed to display the second, proof by which will be as
follows: all B is D, no C is D..., since it is required that B
should be a subject of which a predicate is affirmed. Next, since D is
to be proved not to belong to C, then D has a further predicate
which is denied of C. Therefore, since the succession of predicates
affirmed of an ever higher universal terminates, the succession of
predicates denied terminates too.
The third figure shows it as follows: all B is A, some B is not C.
Therefore some A is not C. This premiss, i.e. C-B, will be proved
either in the same figure or in one of the two figures discussed
above. In the first and second figures the series terminates. If we
use the third figure, we shall take as premisses, all E is B, some E
is not C, and this premiss again will be proved by a similar
prosyllogism. But since it is assumed that the series of descending
subjects also terminates, plainly the series of more universal
non-predicables will terminate also. Even supposing that the proof
is not confined to one method, but employs them all and is now in
the first figure, now in the second or third-even so the regress
will terminate, for the methods are finite in number, and if finite
things are combined in a finite number of ways, the result must be
finite.
Thus it is plain that the regress of middles terminates in the
case of negative demonstration, if it does so also in the case of
affirmative demonstration. That in fact the regress terminates in both
these cases may be made clear by the following dialectical
considerations.