Handbook of the History of Logic: - Fordham University Faculty

684 Russell Wahl
of what propositions constitute the contradictory of the given propositions, and
in some of these discussions Arnauld and Nicole appear to be closer to a truthfunctional
analysis of the propositions. Disjunctions are not given a simple truthfunctional
analysis, as the authors hold that the truth of the disjunction depends
on “the necessary opposition between the parts, which must not permit a middle.”
The contradiction of a disjunction is said simply to be a proposition denying
the disjunction, without further explication. The truth of discretives, such as
“Happiness depends not on wealth but on knowledge”, is said to depend “on the
truth of both parts and the separation between them”. This suggests a connection
stronger than a mere conjunction. Nevertheless, Arnauld and Nicole do not say
that a discretive proposition without a contrast between the terms is false, but
rather that it is ridiculous [ridicule]. When it comes to stating what contradicts
discretive propositions, they say that such propositions can be contradicted in
three ways:
Happiness depends on wealth and not knowledge
Happiness depends neither on wealth nor knowledge
Happiness depends on both wealth and knowledge.
As they started with a proposition of the sort “Not P but Q,” the propositions
denying the discretive are the three possibilities, “P and not Q,” “Not P and not
Q,” “P and Q.” This gives a nice modern truth table for “Not P and Q.” Such
a list was not given for the denial of copulatives (i.e. conjunctions). Instead of
recognizing that the contradiction of the discretive is equivalent to a disjunction,
they remark that the last two of these alternatives are copulatives and so say,
“Thus we see that copulatives are contradictories of discretives” (137).
With respect to conditionals, Arnauld and Nicole say that to determine their
truth “we consider only the truth of the inference,” stating that even if both
propositions are false, “if the inference from the one to the other is valid, the
proposition insofar as it is conditional is true”. This way of putting things clearly
rules out a simple truth-functional analysis. In their analysis of the denial of “if
you eat the forbidden fruit, you will die” they give “although you eat the forbidden
fruit, you will not die”, which is then seen as equivalent to “It is not true that
if you eat the forbidden fruit, you will die” (135). This may suggest an analysis
closer to modern truth functions, but we should understand that “although” is
not treated simply as a truth function. Unfortunately it isn’t clear whether they
accept their own analysis as a complete account of the denial of a conditional,
since by their account for this proposition to be true the inference has to be valid.
They could conceivably hold that in the case that someone dies after eating the
fruit, the proposition may still be false because the inference from “you eat the
fruit” to “you die” is not valid.
peuvent ni chasser la fiévre du corps de celui qui les possede, ni déelivrer son esprit d’inquietude
et de chagrin. (132)