Handbook of the History of Logic: - Fordham University Faculty

Port Royal: The Stirrings of Modernity 689
With the fourth figure, Arnauld and Nicole give a set of rules, but do not bother
to give any justifying principle, since they hold the fourth figure to be “unnatural”.
7 A GENERAL METHOD FOR EVALUATING SYLLOGISMS
In their account of complex syllogisms, Arnauld and Nicole do two things. First
they give an account, at least for some specific examples, of how to rewrite complex
syllogisms into non-complex ones. The examples generally have the middle term
as only part of a complex term. For example in the argument,
Divine law commands us to honor kings
Louis XIV is [a or the] king.
Therefore, divine law commands us to honor Louis XIV. (206)
“kings” is the middle term, but occurs in a part of the predicate of the first premiss.
The syllogism is rewritten ultimately as the non-complex syllogism:
Kings ought to be honored
Louis XIV is king.
Therefore, Louis XIV ought to be honored. (207)
This rewriting, though, is governed by the observation that the term king is taken
generally (that is, through its entire extension) in the first premiss, along with the
observation that it is not the entire predicate of the first proposition.
The second thing they do is to form a general principle from these considerations.
Since a valid (bon) argument is one where the conclusion is contained in
the premises, they say that in any syllogism, there will be a proposition, called
the containing proposition, which contains the conclusion, but only implicitly, and
another proposition, called the applicative proposition which shows this. Thus in
the above example, they would hold that the first premise, “Divine law commands
us to honor kings,” contains the conclusion, and the second premise shows that it
does. With syllogisms such as Barbara they say that either premise can be taken
to be the containing premise though it is customary to take the major premise as
that one, since it is “more general” (212). In negative syllogisms, though, they
hold that the negative premise is always the containing one. They then argue
that all the rules that governed the syllogisms can be derived from this general
observation. What they argue for, in effect, is that what we call the rules of distribution
follow from this general claim about valid arguments. They argue that
if a syllogism is valid, in order for the conclusion to be contained in the premises
it cannot have a term taken more generally than in the premise, and for the applicative
premise to show the subject of the conclusion can be substituted in the
containing premise, the middle term will have to be taken generally at least once.
It is in the demonstration of this last claim by an example where Arnauld and