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

IMPORT OF PROPOSITIONS AND JUDGMENTS. 1O7
tains some has a negative proposition as its secondary
implication. For example, take the proposition
"
all
men are some animals," represented in fig. 14. It im
plies that there are other animals than men e.g., lions,
in other words
" no men are some animals
tigers, &c. ;
(i.e., lions, tigers, &c.)." That is, Hamilton s A pro
position implies rj ; they are not independent forms.
In a similar way we may show that Hamilton s Y
"
"
some elements are all metals
proposition e.g.,
(fig.
15) implies O, "some elements are no metals." These
also are not independent forms.
The proposition w is peculiarly useless, for it is
compatible with each of the five diagrams already given ;
equilateral
Tnarujles
ami
equiangular
iriancjks
Fig. 14. Fig. 15- i Fig. 6.
it is thus compatible with U, unless S and P are the
names of an individual (and therefore logically
divisible) object. This seems paradoxical ; hence we
must show it in detail. Let S and P be both
names of classes. The proposition U says that "all S
is all P,"
"
all equilateral triangles are all equiangular
"
triangles (fig. 1 6).
"
"
"
Now some means only a "
part ;
and hence, if we divide the circle which represents the
coincident classes into any two separate portions, or
mark off two separate smaller parts within it by smaller
circles, we may call one part,
and the other, "some equiangular triangles"; and it will
be true that
"
some equilateral triangles are not some
in
"
"
some equilateral triangles