Complete Works of Lewis Carroll (158 page)

CHAPTER III.

SYLLOGISMS.

§ 1.

Representation of Syllogisms.

We already know how to represent each of the three Propositions of a Syllogism in subscript form.
When that is done, all we need, besides, is to write the three expressions in a row, with “†” between the Premisses, and “¶” before the Conclusion.

[Thus the Syllogism

“No
x
are
m
′
;

  All
m
are
y
.

       
∴
No
x
are
y
′
.”

may be represented thus:—

xm
′
0
†
m
1
y
′
0
¶
xy
′
0

When a Proposition has to be translated from concrete form into subscript form, the Reader will find it convenient, just at first, to translate it into
abstract
form, and
thence
into subscript form.
But, after a little practice, he will find it quite easy to go straight from concrete form to subscript form.]

§ 2.

Formulæ for solving Problems in Syllogisms.

When once we have found, by Diagrams, the Conclusion to a given Pair of Premisses, and have represented the Syllogism in subscript form, we have a
Formula
, by which we can at once find, without having to use Diagrams again, the Conclusion to any
other
Pair of Premisses having the
same
subscript forms.

[Thus, the expression

xm
0
†
ym
′
0
¶
xy
0

is a Formula, by which we can find the Conclusion to any Pair of Premisses whose subscript forms are

xm
0
†
ym
′
0

For example, suppose we had the Pair of Propositions

“No gluttons are healthy;

  No unhealthy men are strong”.

proposed as Premisses.
Taking “men” as our ‘Universe’, and making
m
 = healthy;
x
 = gluttons;
y
 = strong; we might translate the Pair into abstract form, thus:—

“No
x
are
m
;

  No
m
′
are
y
”.

These, in subscript form, would be

xm
0
†
m
′y
0

which are identical with those in our
Formula
.
Hence we at once know the Conclusion to be

xy
0

that is, in abstract form,

“No
x
are
y
”;

that is, in concrete form,

“No gluttons are strong”.]

I shall now take three different forms of Pairs of Premisses, and work out their Conclusions, once for all, by Diagrams; and thus obtain some useful Formulæ.
I shall call them “Fig.
I”, “Fig.
II”, and “Fig.
III”.

Fig.
I.

This includes any Pair of Premisses which are both of them Nullities, and which contain Unlike Eliminands.

The simplest case is

xm
0
†
ym
′
0

∴
xy
0

In this case we see that the Conclusion is a Nullity, and that the Retinends have kept 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

m
1
x
0
†
ym
′
0
(which ¶
xy
0
)

xm
′
0
†
m
1
y
0
(which ¶
xy
0
)

x
′m
0
†
ym
′
0
(which ¶
x
′y
0
)

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

If either Retinend is asserted in the
Premisses
to exist, of course it may be so asserted in the
Conclusion
.

Hence we get two
Variants
of Fig.
I, viz.

(α) where
one
Retinend is so asserted;

(β) where
both
are so asserted.

[The Reader had better work out, on Diagrams, examples of these two Variants, such as

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

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

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

The Formula, to be remembered, is

xm
0
†
ym
′
0
¶
xy
0

with the following two Rules:—

(1)
Two Nullities, with Unlike Eliminands, yield a Nullity, in which both Retinends keep their Signs.

(2)
A Retinend, asserted in the Premisses to exist, may be so asserted in the Conclusion.

[Note that Rule (1) is merely the Formula expressed in words.]

Fig.
II.

This includes any Pair of Premisses, of which one is a Nullity and the other an Entity, and which contain Like Eliminands.

The simplest case is

xm
0
†
ym
1

∴
x
′y
1