Complete Works of Lewis Carroll (159 page)

In this case we see that the Conclusion is an Entity, and that the Nullity-Retinend has changed its Sign.

And we should find this Rule to hold good with
any
Pair of Premisses which fulfil the given conditions.

[The Reader had better satisfy himself of this, by working out, on Diagrams, several varieties, such as

x
′m
0
†
ym
1
(which ¶
xy
1
)

x
1
m
′
0
†
y
′m′
1
(which ¶
x
′y′
1
)

m
1
x
0
†
y
′m
1
(which ¶
x
′y′
1
).]

The Formula, to be remembered, is,

xm
0
†
ym
1
¶
x
′y
1

with the following Rule:—

A Nullity and an Entity, with Like Eliminands, yield an Entity, in which the Nullity-Retinend changes its Sign.

[Note that this Rule is merely the Formula expressed in words.]

Fig.
III.

This includes any Pair of Premisses which are both of them Nullities, and which contain Like Eliminands asserted to exist.

The simplest case is

xm
0
†
ym
0
†
m
1

[Note that “
m
1
” is here stated
separately
, because it does not matter in which of the two Premisses it occurs: so that this includes the
three
forms “
m
1
x
0
†
ym
0
”, “
xm
0
†
m
1
y
0
”, and “
m
1
x
0
†
m
1
y
0
”.]

∴
x
′y′
1

In this case we see that the Conclusion is an Entity, and that
both
Retinends have changed their Signs.

And we should find this Rule to hold good with
any
Pair of Premisses which fulfil the given conditions.

[The Reader had better satisfy himself of this, by working out, on Diagrams, several varieties, such as

x
′m
0
†
m
1
y
0
(which ¶
xy
′
1
)

m
′
1
x
0
†
m
′y′
0
(which ¶
x
′y
1
)

m
1
x
′
0
†
m
1
y
′
0
(which ¶
xy
1
).]

The Formula, to be remembered, is

xm
0
†
ym
0
†
m
1
¶
x
′y′
1

with the following Rule (which is merely the Formula expressed in words):—

Two Nullities, with Like Eliminands asserted to exist, yield an Entity, in which both Retinends change their Signs.

In order to help the Reader to remember the peculiarities and Formulæ of these three Figures, I will put them all together in one Table.

TABLE IX.

Fig.
I.

xm
0
†
ym
′
0
¶
xy
0

Two Nullities, with Unlike Eliminands, yield a Nullity, in which both Retinends keep their Signs.

A Retinend, asserted in the Premisses to exist, may be so asserted in the Conclusion.

Fig.
II.

xm
0
†
ym
1
¶
x
′y
1

A Nullity and an Entity, with Like Eliminands, yield an Entity, in which the Nullity-Retinend changes its Sign.

Fig.
III.

xm
0
†
ym
0
†
m
1
¶
x
′y′
1

Two Nullities, with Like Eliminands asserted to exist, yield an Entity, in which both Retinends change their Signs.

I will now work out, by these Formulæ, as models for the Reader to imitate, some Problems in Syllogisms which have been already worked, by Diagrams, in Book V., Chap.
II.

(1)

“No son of mine is dishonest;

  People always treat an honest man with respect.”

Univ.
“men”;
m
 = honest;
x
 = my sons;
y
 = treated with respect.

xm
′
0
†
m
1
y
′
0
¶
xy
′
0
        [Fig.
I.

i.e.
“No son of mine ever fails to be treated with respect.”

(2)

“All cats understand French;

  Some chickens are cats.”

Univ.
“creatures”;
m
 = cats;
x
 = understanding French;
y
 = chickens.

m
1
x
′
0
†
ym
1
¶
xy
1
        [Fig.
II.

i.e.
“Some chickens understand French.”

(3)

“All diligent students are successful;

  All ignorant students are unsuccessful.”

Univ.
“students”;
m
 = successful;
x
 = diligent;
y
 = ignorant.

x
1
m
′
0
†
y
1
m
0
¶
x
1
y
0
†
y
1
x
0
        [Fig.
I (β).

i.e.
“All diligent students are learned; and all ignorant students are idle.”

(4)

“All soldiers are strong;

  All soldiers are brave.

          Some strong men are brave.”

Univ.
“men”;
m
 = soldiers;
x
 = strong;
y
 = brave.

m
1
x
′
0
†
m
1
y
′
0
¶
xy
1
        [Fig.
III.

Hence proposed Conclusion is right.

(5)

“I admire these pictures;

  When I admire anything, I wish to examine it thoroughly.

          I wish to examine some of these pictures thoroughly.”

Univ.
“things”;
m
 = admired by me;
x
 = these;
y
 = things which I wish to examine thoroughly.

x
1
m
′
0
†
m
1
y
′
0
¶
x
1
y
′
0
        [Fig.
I (α).