Complete Works of Lewis Carroll (202 page)

For
seven
letters (adding
k
) I add, to each upright cross, a little inner square.
All these 16 little squares are assigned to the
k
-Class, and all outside them to the
k
′
-Class; so that 8 little Cells (into which each of the 16 Cells is divided) are respectively assigned to the 8 Classes
ehk
,
ehk
′
, &c.
We have now got 128 Cells.

For
eight
letters (adding
l
) I place, in each of the 16 Cells, a
lattice
, which is a reduced copy of the whole Diagram; and, just as the 16 large Cells of the whole Diagram are assigned to the 16 Classes abcd, abcd′, &c., so the 16 little Cells of each lattice are assigned to the 16 Classes ehkl, ehkl′, &c.
Thus, the lattice in the N.W.
corner serves to accommodate the 16 Classes
abc
′d′ehkl
,
abc
′d′eh′kl′
, &c.
This Octoliteral Diagram (see next page) contains 256 Cells.

For
nine
letters, I place 2 Octoliteral Diagrams side by side, assigning one of them to
m
, and the other to
m
′
.
We have now got 512 Cells.

Finally, for
ten
letters, I arrange 4 Octoliteral Diagrams, like the above, in a square, assigning them to the 4 Classes
mn
,
mn
′
,
m
′n
,
m
′n′
.
We have now got 1024 Cells.

§ 8.

Solution of a Syllogism by various Methods.

The best way, I think, to exhibit the differences between these various Methods of solving Syllogisms, will be to take a concrete example, and solve it by each Method in turn.
Let us take, as our example, No.
29 (see p.
102).

“No philosophers are conceited;

  Some conceited persons are not gamblers.

 
∴
Some persons, who are not gamblers, are not philosophers.”

(1)
Solution by ordinary Method.

These Premisses, as they stand, will give no Conclusion, as they are both negative.

If by ‘Permutation’ or ‘Obversion’, we write the Minor Premiss thus,

‘Some conceited persons are not-gamblers,’

we can get a Conclusion in
Fresison
, viz.

“No philosophers are conceited;

  Some conceited persons are not-gamblers.

 
∴
Some not-gamblers are not philosophers”

This can be proved by reduction to
Ferio
, thus:—

“No conceited persons are philosophers;

  Some not-gamblers are conceited.

 
∴
Some not-gamblers are not philosophers”.

The validity of
Ferio
follows directly from the Axiom ‘
De Omni et Nullo
’.

(2)
Symbolic Representation.

Before proceeding to discuss other Methods of Solution, it is necessary to translate our Syllogism into an
abstract
form.

Let us take “persons” as our ‘Universe of Discourse’; and let
x
 = “philosophers”,
m
 = “conceited”, and
y
 = “gamblers.”

Then the Syllogism may be written thus:—

“No
x
are
m
;

  Some
m
are
y
′
.

 
∴
Some
y
′
are
x
′
.”

(3)
Solution by Euler’s Method of Diagrams.

The Major Premiss requires only
one
Diagram, viz.

1

The Minor requires
three
, viz.

2

3

4

The combination of Major and Minor, in every possible way requires
nine
, viz.

Figs.
1 and 2 give

5

6

7