Complete Works of Lewis Carroll (200 page)

Thus, the Diagram, here given, exhibits the two Classes, whose respective Attributes are
x
and
y
, as so related to each other that the following Propositions are all simultaneously true:—“All
x
are
y
”, “No
x
are not-
y
”, “Some
x
are
y
”, “Some
y
are not-
x
”, “Some not-
y
are not-
x
”, and, of course, the Converses of the last four.

Similarly, with this Diagram, the following Propositions are true:—“All
y
are
x
”, “No
y
are not-
x
”, “Some
y
are
x
”, “Some
x
are not-
y
”, “Some not-
x
are not-
y
”, and, of course, the Converses of the last four.

Similarly, with this Diagram, the following are true:—“All
x
are not-
y
”, “All
y
are not-
x
”, “No
x
are
y
”, “Some
x
are not-
y
”, “Some
y
are not-
x
”, “Some not-
x
are not-
y
”, and the Converses of the last four.

Similarly, with this Diagram, the following are true:—“Some
x
are
y
”, “Some
x
are not-
y
”, “Some not-
x
are
y
”, “Some not-
x
are not-
y
”, and of course, their four Converses.

Note that
all
Euler’s Diagrams assert “Some not-
x
are not-
y
.”
Apparently it never occured to him that it might
sometimes
fail to be true!

Now, to represent “All
x
are
y
”, the
first
of these Diagrams would suffice.
Similarly, to represent “No
x
are
y
”, the
third
would suffice.
But to represent any
Particular
Proposition, at least
three
Diagrams would be needed (in order to include all the possible cases), and, for “Some not-
x
are not-
y
”, all the
four
.

§ 6.

Venn’s Method of Diagrams.

Let us represent “not-
x
” by “
x
′
”.

Mr.
Venn’s Method of Diagrams is a great advance on the above Method.

He uses the last of the above Diagrams to represent
any
desired relation between
x
and
y
, by simply shading a Compartment known to be
empty
, and placing a + in one known to be
occupied
.

Thus, he would represent the three Propositions “Some
x
are
y
”, “No
x
are
y
”, and “All
x
are
y
”, as follows:—

It will be seen that, of the
four
Classes, whose peculiar Sets of Attributes are
xy
,
xy
′
,
x
′y
, and
x
′y′
, only
three
are here provided with closed Compartments, while the
fourth
is allowed the rest of the Infinite Plane to range about in!

This arrangement would involve us in very serious trouble, if we ever attempted to represent “No
x
′
are
y
′
.”
Mr.
Venn
once
(at p.
281) encounters this awful task; but evades it, in a quite masterly fashion, by the simple foot-note “We have not troubled to shade the outside of this diagram”!

To represent
two
Propositions (containing a common Term)
together
, a
three
-letter Diagram is needed.
This is the one used by Mr.
Venn.