Handbook of the History of Logic: - Fordham University Faculty

682 Russell Wahl
idea in the case of a universal, and an indeterminate part in the case of a
particular]. (173-4)
The fifth axiom states that if I assert the proposition “No F are G”, and the idea G
contains in its comprehension the ideas H and K, I am not denying that there are
F ’s that are H or F ’s that are K, but only that there are no F ’s which have both
attributes. The sixth axiom, which corresponds to the claim that the predicate
term of negative propositions is distributed, is argued for somewhat awkwardly by
the statement that “if triangle is denied of squares, everything that is a triangle
will be denied of squares”. “Everything that is a triangle” clearly refers to the
extension of “triangle” here. Presumably in “some figures are not triangles” the
indeterminate part of the extension of “figures” would not be any of the species
of triangles, and no member of the indeterminate part would be a member of the
extension of “triangle”. From these axioms, Arnauld and Nicole derive a third
rule, that universal negative propositions can be converted. While they don’t
give a further rule about particular negatives, they argue that given these axioms,
they cannot be converted. The argument for the third rule is that since, given
the sixth and seventh axioms, and the fact that in the universal propositions the
entire extension of the subject is in question, a proposition such as “No F are G”
separates the entire extension of F from the entire extension of G, so this separation
will be “mutual and reciprocal.” This is not the case with the particular negatives,
since in those cases the entire extension of the subject term is not in question, but
only an indeterminate part. So “some person is not a physician” makes an assertion
concerning only an indeterminate part of the extension of “person” while “some
physician is not a person” makes a claim concerning its entire extension.
In a brief section on the square of opposition the Port-Royal Logic gives the traditional
relations of contradictories, subcontraries, contraries and subaltern (117).
Unfortunately, there is no discussion of existential presupposition. It is clear that
Arnauld and Nicole hold that the affirmative propositions carry existential presupposition,
but they are not explicit with respect to negative propositions, which
were held by many mediaeval logicians not to have any existential presupposition,
thus preserving all the relations of the square. 24 On this view the immediate
inference of obversion doesn’t hold, and there is no mention of this inference in
the Port-Royal Logic. Jean-Claude Pariente has argued that since ideas which
are species of the general ideas are counted in the extension of terms, there is a
sense in which all terms, even those which do not have any existing particulars in
their extensions, do have things in their extension, and he uses this to justify a
claim of existential presupposition for all terms. 25 As far as I can tell, the text of
the Logic does not really lend support to this interpretation. There is, though, a
24 See [Moody, 1953, 50–51], for a discussion of Buriden on this point, and Freddoso, “Ockham’s
Theory of Truth Conditions”, [Freddoso and Schurrman, 1980, 18], for a discussion of Ockham.
25 See [Pariente, 1985, 246]: “These observations explain at least in part the indifference of Port-
Royal to problems posed for logic by empty terms. When the logic is presented as a calculus
applying to ideas, and more precisely to relations between extensions of ideas, the possibility or
impossibility of applying these ideas to concrete realities has no effect on the calculus itself.”