The Complete Aristotle (eng.)

be contended that the pupil’s hearing is also
an hypothesis required by the teacher. Hypotheses, on the contrary,
postulate facts on the being of which depends the being of the fact
inferred. Nor are the geometer’s hypotheses false, as some have
held, urging that one must not employ falsehood and that the
geometer is uttering falsehood in stating that the line which he
draws is a foot long or straight, when it is actually neither. The
truth is that the geometer does not draw any conclusion from the
being of the particular line of which he speaks, but from what his
diagrams symbolize. A further distinction is that all hypotheses
and illegitimate postulates are either universal or particular,
whereas a definition is neither.
11
    So demonstration does not necessarily imply the being of Forms
nor a One beside a Many, but it does necessarily imply the
possibility of truly predicating one of many; since without this
possibility we cannot save the universal, and if the universal
goes, the middle term goes witb. it, and so demonstration becomes
impossible. We conclude, then, that there must be a single
identical term unequivocally predicable of a number of
individuals.
    The law that it is impossible to affirm and deny simultaneously
the same predicate of the same subject is not expressly posited by
any demonstration except when the conclusion also has to be
expressed in that form; in which case the proof lays down as its
major premiss that the major is truly affirmed of the middle but
falsely denied. It makes no difference, however, if we add to the
middle, or again to the minor term, the corresponding negative. For
grant a minor term of which it is true to predicate man-even if it
be also true to predicate not-man of it—still grant simply that man
is animal and not not-animal, and the conclusion follows: for it
will still be true to say that Callias—even if it be also true to
say that not-Callias—is animal and not not-animal. The reason is
that the major term is predicable not only of the middle, but of
something other than the middle as well, being of wider
application; so that the conclusion is not affected even if the
middle is extended to cover the original middle term and also what
is not the original middle term.
    The law that every predicate can be either truly affirmed or
truly denied of every subject is posited by such demonstration as
uses reductio ad impossibile, and then not always universally, but
so far as it is requisite; within the limits, that is, of the
genus-the genus, I mean (as I have already explained), to which the
man of science applies his demonstrations. In virtue of the common
elements of demonstration-I mean the common axioms which are used
as premisses of demonstration, not the subjects nor the attributes
demonstrated as belonging to them-all the sciences have communion
with one another, and in communion with them all is dialectic and
any science which might attempt a universal proof of axioms such as
the law of excluded middle, the law that the subtraction of equals
from equals leaves equal remainders, or other axioms of the same
kind. Dialectic has no definite sphere of this kind, not being
confined to a single genus. Otherwise its method would not be
interrogative; for the interrogative method is barred to the
demonstrator, who cannot use the opposite facts to prove the same
nexus. This was shown in my work on the syllogism.
12
    If a syllogistic question is equivalent to a proposition
embodying one of the two sides of a contradiction, and if each
science has its peculiar propositions from which its peculiar
conclusion is developed, then there is such a thing as a
distinctively scientific question, and it is the interrogative form
of the premisses from which the ‘appropriate’ conclusion of each
science is developed. Hence it is clear that not every question
will be relevant to geometry, nor to medicine, nor to any other
science: only those questions will be geometrical which form
premisses for the proof of the theorems of geometry or of any other
science, such as optics, which uses the same basic truths as
geometry. Of the other sciences the like is true. Of these
questions the geometer is bound to give his account, using the
basic truths of geometry in conjunction with his previous
conclusions; of the basic truths the geometer, as such, is not
bound to give any account. The like is true of the other sciences.
There is a limit, then, to the questions which we may put to each
man