Handbook of the History of Logic: - Fordham University Faculty

688 Russell Wahl
which is included in the extension of a universal idea applies to none of the subjects
of which the idea is denied, the attribute of a negative proposition being taken
throughout its entire extension . . . (197). Illustrating this with an instance of
Camestres, they point out that if “All P are M” is true, then P is included in the
extension of M. If“NoS are M” is true, then M is denied of each S, and thus P
also will be denied, since the denial, “No S are M,” denies the entire extension of
M to S. While the concern with reduction was dismissed as useless, it is interesting
that a vestige of reduction is included in these sections, for Cesare and Festino are
justified by pointing out, as Aristotle argued in the Prior Analytics (27a), that
legitimate conversions yield Celarent and Ferio. While the arguments concerning
Camestres and Baroco are not indirect in the manner of the Prior Analytics, the
reasoning is still quite close to Aristotle’s.
With respect to the third figure, these principles are given for the moods:
Affirmative moods: When two terms can be affirmed of the same thing,
they can also be affirmed of each other, taken particularly.
Negative Moods: When of two terms one can be denied and the other
affirmed of the same thing, they can be denied particularly one of the
other (l’un de l’autre)(199)
While no further explication is given, it is easy to see how the principle grounds
the affirmative moods, for in Darapti, “All M is P ” and “All M is S”, both P
and S are affirmed of the entire extension of M, so while there may be further
instances of P and of S, there will at least be instances in common, namely the
M’s. Both conclusions, “Some S is P ” and “Some P is S”, are able to be drawn
from these premises, given the rule.
In the case of the negative moods, it is easy to read the principle as suggesting
that S can be denied of P and P can be denied of S. This, of course, is not
the case. Jean-Claude Pariente pointed out that Arnauld and Nicole’s language
inadvertently suggests that they have forgotten that the O propositions are not
convertible [Pariente, 1985, 343]. However, if we understand the line “l’un de
l’autre” not as the usual “of each other” but instead as “the one of the other” we
can read the principle as licensing only that the term denied of the middle term
can be denied of the term affirmed of the middle term, and not the other way
around. In this way the principle would justify Bocardo, that is that from “Some
M is not P ” and “All M is S”, I can infer “Some S is not P ,” but would not
license the inference from these premises to “Some P is not S”.
Pariente points out that behind the accounts of the third figure syllogisms is
a method similar to that of the ekthesis (setting out) proofs given by Aristotle
in Prior Analytics for the third figure syllogisms. The ekthesis proofs of Bocardo
involves choosing the M which is not P as an S which is not P (Prior Analytics
28b 21-22). While the discussion appears to incorporate the reduction so much
scorned by the Logic, it is intended as giving simple rules to help the mind make
the correct inferences.