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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。
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 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