The Complete Aristotle (eng.)

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.
26
    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.
    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.
27
    The science which is knowledge at once of the fact and of the
reasoned fact, not of the fact by itself without the reasoned fact,
is the more exact and the prior science.
    A science such as arithmetic, which is not a science of
properties qua inhering in a substratum, is more exact than and
prior to a science like harmonics, which is a science of
pr,operties inhering in a substratum; and similarly a science like
arithmetic, which is constituted of fewer basic elements, is more
exact than and prior to geometry, which requires additional
elements. What I mean by ‘additional elements’ is this: a unit is
substance without position, while a point is substance with
position; the latter contains an additional element.
28
    A single science is one whose domain is a single genus, viz. all
the subjects constituted out of the primary entities of the
genus-i.e. the parts of this total subject-and their essential
properties.
    One science differs from another when their basic truths have
neither a common source nor are derived those of the one science
from those the other. This is verified when we reach the
indemonstrable premisses of a science, for they must be within one
genus with its conclusions: and this again is verified if the
conclusions proved by means of them fall within one genus-i.e. are
homogeneous.
29
    One can have several demonstrations of the same connexion not
only by taking from the same series of predication middles which
are other than the immediately cohering term e.g. by taking C, D,
and F severally to prove A-B—but also by taking a middle from
another series. Thus let A be change, D alteration of a property, B
feeling pleasure, and G relaxation. We can then without falsehood
predicate D of B and A of D, for he who is pleased suffers
alteration of a property, and that