an empty machine, such as "Every
X is Y"; it may be supplied with _matter_, as in "Every _man_ is _animal_."
The logicians will not see that their _formal_ proposition, "Every X is Y,"
is material in three points, the degree of assertion, the quantity of the
proposition, and the copula. The purely formal proposition is "There is the
probability [alpha] that X stands in the relation L to Y." The time will
come when it will be regretted that logic went without paradoxers for two
thousand years: and when much that has been said on the distinction of form
and matter will breed jokes.

I give one instance of one mood of each of the systems, in the order of the
letters first written above.

_Relative._--In this system the formal relation is taken, that is, the
copula may be any whatever. As a material instance, in which the
_relations_ are those of consanguinity (of men understood), take the
following: X is the brother of Y; X is not the uncle of Z; therefore, Z is
not the child of Y. The discussion of relation, and of the objections to
the extension, is in the _Cambridge Transactions_, Vol. X, Part 2; a
crabbed conglomerate.

_Undecided._--In this system one premise, and want of power over another,
infer want of power over a conclusion. {335} As "Some men are not capable
of tracing consequences; we cannot be sure that there are beings
responsible for consequences who are incapable of tracing consequences;
therefore, we cannot be sure that all men are responsible for the
consequences of their actions."

_Exemplar._--This, long after it suggested itself to me as a means of
correcting a defect in Hamilton's system, I saw to be the very system of
Aristotle himself, though his followers have drifted into another. It makes
its subject and predicate examples, thus: Any one man is an animal; any one
animal is a mortal; therefore, any one man is a mortal.

_Numerical._--Suppose 100 Ys to exist: then if 70 Xs be Ys, and 40 Zs be
Ys, it follows that 10 Xs (at least) are Zs. Hamilton, whose mind could not
generalize on symbols, saw that the word _most_ would come under this
system, and admitted, as valid, such a syllogism as "most Ys are Xs; most
Ys are Zs; therefore, some Xs are Zs."

_Onymatic._--This is the ordinary system much enlarged in propositional
forms. It is fully discussed in my _Syllabus of Logic_.

_Transposed._--In this syllogism the quantity in one premise is transposed
into the other. As, some Xs are not Ys; for