### Aristotle - The Organon ANALYTICA POSTERIORA Book 1 Part 26

## Negative demonstration and reductio ad impossibile

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1.
Since affirmative demonstration is superior to negative, it is
clearly superior also to reductio ad impossibile.
We must first make
certain what is the difference between negative demonstration and
reductio ad impossibile. Let us suppose that no B is A, and
that all C
is B: the conclusion necessarily follows that no C is A. If these
premisses are assumed, therefore, the negative demonstration
that no C
is A is direct. Reductio ad impossibile, on the other hand, proceeds
as follows. Supposing we are to prove that does not inhere in B, we
have to assume that it does inhere, and further that B inheres in C,
with the resulting inference that A inheres in C. This we have to
suppose a known and admitted impossibility; and we then infer that A
cannot inhere in B. Thus if the inherence of B in C is not
questioned,
A's inherence in B is impossible.

2.
The order of the terms is the same in both proofs:
they differ
according to which of the negative propositions is the better known,
the one denying A of B or the one denying A of C. When the falsity
of the conclusion is the better known, we use reductio ad
impossible; when the major premiss of the syllogism is the more
obvious, we use direct demonstration. All the same the proposition
denying A of B is, in the order of being, prior to that denying A of
C; for premisses are prior to the conclusion which follows from
them, and 'no C is A' is the conclusion, 'no B is A' one of its
premisses. For the destructive result of reductio ad impossibile is
not a proper conclusion, nor are its antecedents proper premisses.
On the contrary: the constituents of syllogism are premisses related
to one another as whole to part or part to whole, whereas the
premisses A-C and A-B are not thus related to one another. Now the
superior demonstration is that which proceeds from better known and
prior premisses, and while both these forms depend for
credence on the
not-being of something, yet the source of the one is prior to that
of the other. Therefore negative demonstration will have an
unqualified superiority to reductio ad impossibile, and affirmative
demonstration, being superior to negative, will consequently be
superior also to reductio ad impossibile.

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