its attributes in number
and number in them-as to be commensurate with the subject and not of
wider extent. Attributes which are essential elements in the nature of
their subjects are equally finite: otherwise definition would be
impossible. Hence, if all the attributes predicated are essential
and these cannot be infinite, the ascending series will terminate, and
consequently the descending series too.
If this is so, it follows that the intermediates between any two
terms are also always limited in number. An immediately obvious
consequence of this is that demonstrations necessarily involve basic
truths, and that the contention of some-referred to at the outset-that
all truths are demonstrable is mistaken. For if there are basic
truths, (a) not all truths are demonstrable, and (b) an infinite
regress is impossible; since if either (a) or (b) were not a fact,
it would mean that no interval was immediate and indivisible, but that
all intervals were divisible. This is true because a conclusion is
demonstrated by the interposition, not the apposition, of a fresh
term. If such interposition could continue to infinity there might
be an infinite number of terms between any two terms; but this is
impossible if both the ascending and descending series of
predication terminate; and of this fact, which before was shown
dialectically, analytic proof has now been given.
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