The Complete Aristotle (eng.)

predicate are convertible there is
neither primary nor ultimate subject, seeing that all the
reciprocals qua subjects stand in the same relation to one another,
whether we say that the subject has an infinity of attributes or
that both subjects and attributes-and we raised the question in
both cases-are infinite in number. These questions then cannot be
asked-unless, indeed, the terms can reciprocate by two different
modes, by accidental predication in one relation and natural
predication in the other.
20
    Now, it is clear that if the predications terminate in both the
upward and the downward direction (by ‘upward’ I mean the ascent to
the more universal, by ‘downward’ the descent to the more
particular), the middle terms cannot be infinite in number. For
suppose that A is predicated of F, and that the intermediates-call
them BB’B”… -are infinite, then clearly you might descend from and
find one term 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