Posterior Analytics   

demonstration which derives from fewer postulates or hypotheses-in
short from fewer premisses; for, given that all these are equally well
known, where they are fewer knowledge will be more speedily
acquired, and that is a desideratum. The argument implied in our
contention that demonstration from fewer assumptions is superior may
be set out in universal form as follows. Assuming that in both cases
alike the middle terms are known, and that middles which are prior are
better known than such as are posterior, we may suppose two
demonstrations of the inherence of A in E, the one proving it
through the middles B, C and D, the other through F and G. Then A-D is
known to the same degree as A-E (in the second proof), but A-D is
better known than and prior to A-E (in the first proof); since A-E
is proved through A-D, and the ground is more certain than the
conclusion.
Hence demonstration by fewer premisses is ceteris paribus
superior. Now both affirmative and negative demonstration operate
through three terms and two premisses, but whereas the former
assumes only that something is, the latter assumes both that something
is and that something else is not, and thus operating through more
kinds of premiss is inferior.
(2) It has been proved that no conclusion follows if both
premisses are negative, but that one must be negative, the other
affirmative. So we are compelled to lay down the following
additional rule: as the demonstration expands, the affirmative
premisses must increase in number, but there cannot be more than one
negative premiss in each complete proof. Thus, suppose no B is A,
and all C is B. Then if both the premisses are to be again expanded, a
middle must be interposed. Let us interpose D between A and B, and E
between B and C. Then clearly E is affirmatively related to B and C,
while D is affirmatively related to B but negatively to A; for all B
is D, but there must be no D which is A. Thus there proves to be a
single negative premiss, A-D. In the further prosyllogisms too it is
the same, because in the terms of an affirmative syllogism the
middle is always related affirmatively to both extremes; in a negative
syllogism it must be negatively related only to one of them, and so
this negation comes to be a single negative premiss, the other
premisses being affirmative. If, then, that through which a truth is
proved is a better known and more certain truth, and if the negative
proposition is proved through the affirmative and not vice versa,
affirmative demonstration, being prior and better known and more
certain, will be superior.
(3) The basic truth of demonstrative syllogism is the universal
immediate premiss, and the universal premiss asserts in affirmative
demonstration and in negative denies: and the affirmative
proposition is prior to and better known than the negative (since
affirmation explains denial and is prior to denial, just as being is
prior to not-being). It follows that the basic premiss of
affirmative demonstration is superior to that of negative
demonstration, and the demonstration which uses superior basic
premisses is superior.
(4) Affirmative demonstration is more of the nature of a basic
form of proof, because it is a sine qua non of negative demonstration.

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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,