Complete Works of Lewis Carroll (201 page)

Here, again, we have only
seven
closed Compartments, to accommodate the
eight
Classes whose peculiar Sets of Attributes are
xym
,
xym
′
, &c.

“With four terms in request,” Mr.
Venn says, “the most simple and symmetrical diagram seems to me that produced by making four ellipses intersect one another in the desired manner”.
This, however, provides only
fifteen
closed compartments.

For
five
letters, “the simplest diagram I can suggest,” Mr.
Venn says, “is one like this (the small ellipse in the centre is to be regarded as a portion of the
outside
of
c
; i.e.
its four component portions are inside
b
and
d
but are no part of
c
).
It must be admitted that such a diagram is not quite so simple to draw as one might wish it to be; but then consider what the alternative is of one undertakes to deal with five terms and all their combinations—nothing short of the disagreeable task of writing out, or in some way putting before us, all the 32 combinations involved.”

This Diagram gives us 31 closed compartments.

For
six
letters, Mr.
Venn suggests that we might use
two
Diagrams, like the above, one for the
f
-part, and the other for the not-
f
-part, of all the other combinations.
“This”, he says, “would give the desired 64 subdivisions.”
This, however, would only give 62 closed Compartments, and
one
infinite area, which the two Classes,
a
′b′c′d′e′f
and
a
′b′c′d′e′f′
, would have to share between them.

Beyond
six
letters Mr.
Venn does not go.

§ 7.

My Method of Diagrams.

My Method of Diagrams
resembles
Mr.
Venn’s, in having separate Compartments assigned to the various Classes, and in marking these Compartments as
occupied
or as
empty
; but it
differs
from his Method, in assigning a
closed
area to the
Universe of Discourse
, so that the Class which, under Mr.
Venn’s liberal sway, has been ranging at will through Infinite Space, is suddenly dismayed to find itself “cabin’d, cribb’d, confined”, in a limited Cell like any other Class!
Also I use
rectilinear
, instead of
curvilinear
, Figures; and I mark an
occupied
Cell with a ‘I’ (meaning that there is at least
one
Thing in it), and an
empty
Cell with a ‘O’ (meaning that there is
no
Thing in it).

For
two
letters, I use this Diagram, in which the North Half is assigned to ‘
x
’, the South to ‘not-
x
’ (or ‘
x
′
’), the West to
y
, and the East to
y
′
.
Thus the N.W.
Cell contains the
xy
-Class, the N.E.
Cell the
xy
′
-Class, and so on.

For
three
letters, I subdivide these four Cells, by drawing an
Inner
Square, which I assign to
m
, the
Outer
Border being assigned to
m
′
.
I thus get
eight
Cells that are needed to accommodate the eight Classes, whose peculiar Sets of Attributes are
xym
,
xym
′
, &c.

This last Diagram is the most complex that I use in the
Elementary
Part of my ‘Symbolic Logic.’
But I may as well take this opportunity of describing the more complex ones which will appear in Part II.

For
four
letters (which I call
a
,
b
,
c
,
d
) I use this Diagram; assigning the North Half to
a
(and of course the
rest
of the Diagram to
a
′
), the West Half to
b
, the Horizontal Oblong to
c
, and the Upright Oblong to
d
.
We have now got 16 Cells.

For
five
letters (adding e) I subdivide the 16 Cells of the previous Diagram by
oblique
partitions, assigning all the
upper
portions to
e
, and all the
lower
portions to
e
′
.
Here, I admit, we lose the advantage of having the
e
-Class all
together
, “in a ring-fence”, like the other 4 Classes.
Still, it is very easy to find; and the operation, of erasing it, is nearly as easy as that of erasing any other Class.
We have now got 32 Cells.

For
six
letters (adding
h
, as I avoid
tailed
letters) I substitute upright crosses for the oblique partitions, assigning the 4 portions, into which each of the 16 Cells is thus divided, to the four Classes
eh
,
eh
′
,
e
′h
,
e
′h′
.
We have now got 64 Cells.