### Aristotle - The Organon ANALYTICA POSTERIORA Book 1 Part 20

## Middle terms cannot be infinite in number

1.
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 B B'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.

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