The Complete Aristotle (eng.)

are
coincidents and are predicated of substances. On the other hand-in
proof of the impossibility of an infinite ascending series-every
predication displays the subject as somehow qualified or quantified
or as characterized under one of the other adjectival categories,
or else is an element in its substantial nature: these latter are
limited in number, and the number of the widest kinds under which
predications fall is also limited, for every predication must
exhibit its subject as somehow qualified, quantified, essentially
related, acting or suffering, or in some place or at some time.
    I assume first that predication implies a single subject and a
single attribute, and secondly that predicates which are not
substantial are not predicated of one another. We assume this
because such predicates are all coincidents, and though some are
essential coincidents, others of a different type, yet we maintain
that all of them alike are predicated of some substratum and that a
coincident is never a substratum-since we do not class as a
coincident anything which does not owe its designation to its being
something other than itself, but always hold that any coincident is
predicated of some substratum other than itself, and that another
group of coincidents may have a different substratum. Subject to
these assumptions then, neither the ascending nor the descending
series of predication in which a single attribute is predicated of
a single subject is infinite. For the subjects of which coincidents
are predicated are as many as the constitutive elements of each
individual substance, and these we have seen are not infinite in
number, while in the ascending series are contained those
constitutive elements with their coincidents-both of which are
finite. We conclude that there is a given subject (D) of which some
attribute (C) is primarily predicable; that there must be an
attribute (B) primarily predicable of the first attribute, and that
the series must end with a term (A) not predicable of any term
prior to the last subject of which it was predicated (B), and of
which no term prior to it is predicable.
    The argument we have given is one of the so-called proofs; an
alternative proof follows. Predicates so related to their subjects
that there are other predicates prior to them predicable of those
subjects are demonstrable; but of demonstrable propositions one
cannot have something better than knowledge, nor can one know them
without demonstration. Secondly, if a consequent is only known
through an antecedent (viz. premisses prior to it) and we neither
know this antecedent nor have something better than knowledge of
it, then we shall not have scientific knowledge of the consequent.
Therefore, if it is possible through demonstration to know anything
without qualification and not merely as dependent on the acceptance
of certain premisses-i.e. hypothetically-the series of intermediate
predications must terminate. If it does not terminate, and beyond
any predicate taken as higher than another there remains another
still higher, then every predicate is demonstrable. Consequently,
since these demonstrable predicates are infinite in number and
therefore cannot be traversed, we shall not know them by
demonstration. If, therefore, we have not something better than
knowledge of them, we cannot through demonstration have unqualified
but only hypothetical science of anything.
    As dialectical proofs of our contention these may carry
conviction, but an analytic process will show more briefly that
neither the ascent nor the descent of predication can be infinite
in the demonstrative sciences which are the object of our
investigation. Demonstration proves the inherence of essential
attributes in things. Now attributes may be essential for two
reasons: either because they are elements in the essential nature
of their subjects, or because their subjects are elements in their
essential nature. An example of the latter is odd as an attribute
of number-though it is number’s attribute, yet number itself is an
element in the definition of odd; of the former, multiplicity or
the indivisible, which are elements in the definition of number. In
neither kind of attribution can the terms be infinite. They are not
infinite where each is related to the term below it as odd is to
number, for this would mean the inherence in odd of another
attribute of odd in whose nature odd was an essential element: but
then number will be an ultimate subject of the whole infinite