An introductory text-book of logic - Mellone, Sydney - Rare Books at ...

IMPORT OF PROPOSITIONS AND JUDGMENTS. 109
intents and purposes U, wherever there is an unmis
takable affirmation that the subject and predicate
of a proposition are co-extensive. Thus, all Definitions
are practically U propositions [when regarded on the
side of extension] ; so are all affirmative propositions
of which both the subject and the predicate are singular
terms." We have already given instances of such pro
positions, describing them as "A propositions which
can be converted simply." In ordinary logical form
they must be expressed in two propositions : thus,
"all S is all P," is equivalent to (a) "all S is P," (b)
"all P is S." As examples of the Y form, exclusive
and exceptive propositions are usually given. "The
virtuous alone are happy" might be expressed "some
of the virtuous are all of the happy,"
"some S is all
P." Here again we have a compound proposition
which is equivalent to (a)
"some S is P," (b)
"no
not-S is P" (ch. III. 3).
In the case of U propositions in geometry, we have
really two separate forms, propositions which have to be
independently proved : neither of them can be proved
from the other.
The student should bear in mind that Hamilton s
scheme of Quantification is open to the objection (see
2 ad fineni) which applies to every
attempt to read the predicate as a ..-
precise quantity. It can give no /
account of the large class of A pro- .;
positions where we do not yet know
whether P is wider than S or merely
coextensive with it. The accompany-
ing diagram (fig. 1 7) might be adopted
to represent such propositions.
" %
\
Fij. 17.
5. An interpretation of propositions which is ser-