The Complete Aristotle (eng.)

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.
22
    In the case of predicates constituting the essential nature of a
thing, it clearly terminates, seeing that if definition is
possible, or in other words, if essential form is knowable, and an
infinite series cannot be traversed, predicates constituting a
thing’s essential nature must be finite in number. But as regards
predicates generally we have the following prefatory remarks to
make. (1) We can affirm without falsehood ‘the white (thing) is
walking’, and that big (thing) is a log’; or again, ‘the log is
big’, and ‘the man walks’. But the affirmation differs in the two
cases. When I affirm ‘the white is a log’, I mean that something
which happens to be white is a log-not that white is the substratum
in which log inheres, for it was not qua white or qua a species of
white that the white (thing) came to be a log, and the white
(thing) is consequently not a log except incidentally. On the other
hand, when I affirm ‘the log is white’, I do not mean that
something else, which happens also to be a log, is white (as I
should if I said ‘the musician is white,’ which would mean ‘the man
who happens also to be a musician is white’); on the contrary, log
is here the substratum-the substratum which actually came to be
white, and did so qua wood or qua a species of wood and qua nothing
else.
    If we must lay down a rule, let us entitle the latter kind of
statement predication, and the former not predication at all, or
not strict but accidental predication. ‘White’ and ‘log’ will thus
serve as types respectively of predicate and subject.
    We shall assume, then, that the predicate is invariably
predicated strictly and not accidentally of the subject, for on
such predication demonstrations depend for their force. It follows
from this that when a single attribute is predicated of a single
subject, the predicate must affirm of the subject either some
element constituting its essential nature, or that it is in some
way qualified, quantified, essentially related, active, passive,
placed, or dated.
    (2) Predicates which signify substance signify that the subject
is identical with the predicate or with a species of the predicate.
Predicates not signifying substance which are predicated of a
subject not identical with themselves or with a species of
themselves are accidental or coincidental; e.g. white is a
coincident of man, seeing that man is not identical with white or a
species of white, but rather with animal, since man is identical
with a species of animal. These predicates which do not signify
substance must be predicates of some other subject, and nothing can
be white which is not also other than white. The Forms we can
dispense with, for they are mere sound without sense; and even if
there are such things, they are not relevant to our discussion,
since demonstrations are concerned with predicates such as we have
defined.
    (3) If A is a quality of B, B cannot be a quality of A-a quality
of a quality. Therefore A and B cannot be predicated reciprocally
of one another in strict predication: they can be affirmed without
falsehood of one another, but not genuinely predicated of each
other. For one alternative is that they should be substantially
predicated of one another, i.e. B would become the genus or
differentia of A-the predicate now become subject. But it has been
shown that in these substantial predications neither the ascending
predicates nor the descending subjects form an infinite series;
e.g. neither the series, man is biped, biped is animal, &c.,
nor the series predicating animal of man, man of Callias, Callias
of a further. subject as an element of its essential nature, is
infinite. For all such substance is definable, and an infinite
series cannot be traversed in thought: consequently neither the
ascent nor the descent is infinite, since a substance whose
predicates were infinite would not be definable. Hence they will
not be predicated each as the genus of the other; for this would
equate a genus with one of its own species. Nor (the other
alternative) can a quale be reciprocally predicated of a quale, nor
any term belonging to an adjectival category of another such term,
except by accidental predication; for all such predicates