Posterior Analytics   

affirmative; whereas, on the other hand, some conclusions are negative
and some are not universal; e.g. all in the second figure are
negative, none in the third are universal. And again, not even all
affirmative conclusions in the first figure are definable, e.g. 'every
triangle has its angles equal to two right angles'. An argument
proving this difference between demonstration and definition is that
to have scientific knowledge of the demonstrable is identical with
possessing a demonstration of it: hence if demonstration of such
conclusions as these is possible, there clearly cannot also be
definition of them. If there could, one might know such a conclusion
also in virtue of its definition without possessing the
demonstration of it; for there is nothing to stop our having the one
without the other.
Induction too will sufficiently convince us of this difference;
for never yet by defining anything-essential attribute or accident-did
we get knowledge of it. Again, if to define is to acquire knowledge of
a substance, at any rate such attributes are not substances.
It is evident, then, that not everything demonstrable can be
defined. What then? Can everything definable be demonstrated, or
not? There is one of our previous arguments which covers this too.
Of a single thing qua single there is a single scientific knowledge.
Hence, since to know the demonstrable scientifically is to possess the
demonstration of it, an impossible consequence will follow:-possession
of its definition without its demonstration will give knowledge of the
demonstrable.
Moreover, the basic premisses of demonstrations are definitions, and
it has already been shown that these will be found indemonstrable;
either the basic premisses will be demonstrable and will depend on
prior premisses, and the regress will be endless; or the primary
truths will be indemonstrable definitions.
But if the definable and the demonstrable are not wholly the same,
may they yet be partially the same? Or is that impossible, because
there can be no demonstration of the definable? There can be none,
because definition is of the essential nature or being of something,
and all demonstrations evidently posit and assume the essential
nature-mathematical demonstrations, for example, the nature of unity
and the odd, and all the other sciences likewise. Moreover, every
demonstration proves a predicate of a subject as attaching or as not
attaching to it, but in definition one thing is not predicated of
another; we do not, e.g. predicate animal of biped nor biped of
animal, nor yet figure of plane-plane not being figure nor figure
plane. Again, to prove essential nature is not the same as to prove
the fact of a connexion. Now definition reveals essential nature,
demonstration reveals that a given attribute attaches or does not
attach to a given subject; but different things require different
demonstrations-unless the one demonstration is related to the other as
part to whole. I add this because if all triangles have been proved to
possess angles equal to two right angles, then this attribute has been
proved to attach to isosceles; for isosceles is a part of which all
triangles constitute the whole. But in the case before us the fact and
the essential nature are not so related to one another, since the
one is not a part of the other.
So it emerges that not all the definable is demonstrable nor all the
demonstrable definable; and we may draw the general conclusion that
there is no identical object of which it is possible to possess both a
definition and a demonstration. It follows obviously that definition
and demonstration are neither identical nor contained either within
the other: if they were, their objects would be related either as
identical or as whole and part.

4

So much, then, for the first stage of our problem. The next step
is to raise the question whether syllogism-i.e. demonstration-of the