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Furthermore, the λόγοι are teachable and learnable. Aristotle is thinking
here primarily of natural speech, where there are two types of speaking.
When orators speak publicly in court or in the senate they appeal to the
common understanding of things which is shared by everyone. Such
speaking adduces no scientific proofs but does awaken a conviction among
the auditors. This occurs διὰ παραδειγμάτων, by introducing a striking
example. δεικνύντες τὸ καθόλου διὰ τὸ δῆλον εἶναι τὸ καθ᾽ ἕκαστον
(a8f.): "They show the universal, " which is supposed to be binding on
others, "through the obviousness of some particular case," i.e., through a
definite example. This is one way to produce a conviction in others. This
is the way of ἐπαγωγή (a6), which is a simple leading toward something
but not an arguing in the proper sense. One can also proceed in such a
way that what is binding and universal λαμβάνοντες ὡς παρὰ ξυνιέντων
(a7f.), is taken from the natural understanding: i.e., from what all people
know and agree upon. One takes into account definite cognitions which
the audience possesses, and these are not discussed further. On the basis
of these, one tries then to prove to the audience something by means of
συλλογισμός (a5). Συλλογισμός and ἐπαγωγή are the two ways to impart
to others a knowledge about definite things. The concluding ἐκ
προγιγνωσκομένων (cf. a6) "out of what is known at the outset" is the
mode in which ἐπιστήμη is communicated. Hence it is possible to impart
to someone a particular science without his having seen all the facts himself
or being able to see them, provided he possesses the required presuppositions.
This μάθησις is developed in the most pure way in mathematics.
The axioms of mathematics are such προγιγνωσκόμενα, from which the
separate deductions can be carried out, without the need of a genuine
understanding of those axioms. The mathematician does not himself discuss
the axioms; instead, he merely operates with them. To be sure, modern
mathematics contains a theory of axioms. But, as can be observed, mathematicians
attempt to treat even the axioms mathematically. They seek to
prove the axioms by means of deduction and the theory of relations, hence
in a way which itself has its ground in the axioms. This procedure will
never elucidate the axioms. To elucidate what is familiar already at the
outset is rather a matter of ἐπαγωγή, the mode of clarification proper to
straightforward perception. Ἐπαγωγή is hence clearly the beginning, i.e.,
that which discloses the ἀρχή; *it* is the more original, not ἐπιστήμη. It
indeed leads originally to the καθόλου, whereas ἐπιστήμη and
συλλογισμός are ἐκ τῶν καθόλου(*Nic. Eth.* VI, 3, 1139b29). In any case,
ἐπαγωγή is needed, whether it now simply stands on its own or whether
an actual proof results from it. Every ἐπιστήμη is διδασκαλία, i.e., it always
presupposes that which it cannot itself elucidate as ἐπιστήμη. It is
ἀπόδεξις, it shows something on the basis of that which is already familiar
and known.