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premisses asserts one term of another； while the other denies one term
of another。 It is clear； then； that these are the fundamentals and
so…called hypotheses of syllogism。 Assume them as they have been
stated； and proof is bound to follow…proof that A inheres in C through
B； and again that A inheres in B through some other middle term； and
similarly that B inheres in C。 If our reasoning aims at gaining
credence and so is merely dialectical； it is obvious that we have only
to see that our inference is based on premisses as credible as
possible： so that if a middle term between A and B is credible
though not real； one can reason through it and complete a
dialectical syllogism。 If； however； one is aiming at truth； one must
be guided by the real connexions of subjects and attributes。 Thus：
since there are attributes which are predicated of a subject
essentially or naturally and not coincidentally…not； that is； in the
sense in which we say 'That white （thing） is a man'； which is not
the same mode of predication as when we say 'The man is white'： the
man is white not because he is something else but because he is man；
but the white is man because 'being white' coincides with 'humanity'
within one substratum…therefore there are terms such as are
naturally subjects of predicates。 Suppose； then； C such a term not
itself attributable to anything else as to a subject； but the
proximate subject of the attribute Bi。e。 so that B…C is immediate；
suppose further E related immediately to F； and F to B。 The first
question is； must this series terminate； or can it proceed to
infinity？ The second question is as follows： Suppose nothing is
essentially predicated of A； but A is predicated primarily of H and of
no intermediate prior term； and suppose H similarly related to G and G
to B； then must this series also terminate； or can it too proceed to
infinity？ There is this much difference between the questions： the
first is； is it possible to start from that which is not itself
attributable to anything else but is the subject of attributes； and
ascend to infinity？ The second is the problem whether one can start
from that which is a predicate but not itself a subject of predicates；
and descend to infinity？ A third question is； if the extreme terms are
fixed； can there be an infinity of middles？ I mean this： suppose for
example that A inheres in C and B is intermediate between them； but
between B and A there are other middles； and between these again fresh
middles； can these proceed to infinity or can they not？ This is the
equivalent of inquiring； do demonstrations proceed to infinity； i。e。
is everything demonstrable？ Or do ultimate subject and primary
attribute limit one another？
I hold that the same questions arise with regard to negative
conclusions and premisses： viz。 if A is attributable to no B； then
either this predication will be primary； or there will be an
intermediate term prior to B to which a is not attributable…G； let
us say； which is attributable to all B…and there may still be
another term H prior to G； which is attributable to all G。 The same
questions arise； I say； because in these cases too either the series
of prior terms to which a is not attributable is infinite or it
terminates。
One cannot ask the same questions in the case of reciprocating
terms； since when subject and 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 propositionsas is always the case with
the minor premisssince 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 figur