Handbook of the History of Logic: - Fordham University Faculty

696 Russell Wahl
For axioms:
3. Require in axioms only those things which are perfectly evident.
4. Accept as evident that which needs only a little attention to be recognized
as true.
For demonstrations:
5. Prove all propositions which are even a little obscure, employing in the proofs
only prior definitionsm axioms agreed upon, or propositions which have already
been demonstrated.
6. Never exploit the equivocation of terms by failing to substitute mentally the
definitions which restrict them and explicate them.
For method:
7. Treat things, as much as possible, in their natural order, beginning with
the most general and simplest and explaining everything that belongs to the
nature of the genus before going on to the particular species.
8. Divide, as much as possible, each genus into all its species, each whole into
all its parts and each difficulty into all its cases. (334)
The major difference between this set of rules and that given for the method of
analysis simply involves the rules for demonstrations, requiring the explicit use
of definitions and axioms. The rules for method closely follow remarks Descartes
himself made with respect to analysis, and the rules for definitions and axioms can
be seen as a further explication of Descartes’ first rule in the Discourse on Method.
Given the general claim that we need no rules for thinking, these rules are cautionary
rules about not paying enough care. This final statement of the rules is
formulated after a discussion of the defects, or flaws, of the geometers’ method.
These defects are not said to turn the arguments away from their purpose, but
instead make arguments circuitous and therefore less likely to lead to understanding.
The list includes, among other things, proving things that do not need to be
proven, focusing on conviction rather than understanding, and proving things by
reductio ad absurdum (démonstrations par l’impossible). This last is criticized for
showing that something is the case without showing why it is so. Arnauld and
Nicole criticize Euclid for using this method of demonstration when things could
be proved positively, but do leave the suggestion that there are things that can be
proved only by this means (329).
11 PROBABILITY AND CONTINGENT REASONING
At the end of the fourth part of the Logic, there is an interesting discussion of
reasoning about contingent things in a setting of imperfect knowledge. This section