Complete Works of Lewis Carroll (162 page)

Now, if we agree to
mark
m
and
m
′
as eliminated, and to read the two expressions together, as if they were written in one, the two Premisses will then exactly represent the
Conclusion
, and we need not write it out separately.

Let us agree to mark the eliminated letters by
underscoring
them, putting a
single
score under the
first
, and a
double
one under the
second
.

The two Premisses now become

xm
0
†
ym
′
0

which we read as “
xy
0
”.

In copying out the Premisses for underscoring, it will be convenient to
omit all subscripts
.
As to the “0’s” we may always
suppose
them written, and, as to the “1’s”, we are not concerned to know
which
Terms are asserted to
exist
, except those which appear in the
Complete
Conclusion; and for
them
it will be easy enough to refer to the original list.

[I will now go through the process of solving, by this method, the example worked in § 2.

The Data are

1
k
1
l
′
0
† 2
dh
′
0
† 3
a
1
c
0
† 4
b
1
e
′
0
† 5
k
′h
0
† 6
b
′l
0
† 7
d
′
1
c
′
0

The Reader should take a piece of paper, and write out this solution for himself.
The first line will consist of the above Data; the second must be composed, bit by bit, according to the following directions.

We begin by writing down the first Premiss, with its numeral over it, but omitting the subscripts.

We have now to find a Premiss which can be combined with this,
i.e.
, a Premiss containing either
k
′
or
l
.
The first we find is No.
5; and this we tack on, with a †.

To get the
Conclusion
from these,
k
and
k
′
must be eliminated, and what remains must be taken as one expression.
So we
underscore
them, putting a
single
score under
k
, and a
double
one under
k
′
.
The result we read as
l
′h
.

We must now find a Premiss containing either
l
or
h
′
.
Looking along the row, we fix on No.
2, and tack it on.

Now these 3 Nullities are really equivalent to (
l
′h
†
dh
′
), in which
h
and
h
′
must be eliminated, and what remains taken as one expression.
So we
underscore
them.
The result reads as
l
′d
.

We now want a Premiss containing
l
or
d
′
.
No.
6 will do.

These 4 Nullities are really equivalent to (
l
′d
†
b
′l
).
So we underscore
l
′
and
l
.
The result reads as
db
′
.

We now want a Premiss containing
d
′
or
b
.
No.
4 will do.

Here we underscore
b
′
and
b
.
The result reads as
de
′
.

We now want a Premiss containing
d
′
or
e
.
No.
7 will do.

Here we underscore
d
and
d
′
.
The result reads as
c
′e′
.

We now want a Premiss containing
c
or
e
.
No.
3 will do—in fact
must
do, as it is the only one left.

Here we underscore
c
′
and
c
; and, as the whole thing now reads as
e
′a
, we tack on
e
′a
0
as the
Conclusion
, with a ¶.

We now look along the row of Data, to see whether
e
′
or
a
has been given as
existent
.
We find that
a
has been so given in No.
3.
So we add this fact to the Conclusion, which now stands as ¶
e
′a
0
†
a
1
,
i.e.
¶
a
1
e
′
0
; i.e.
“All
a
are
e
.”

If the Reader has faithfully obeyed the above directions, his written solution will now stand as follows:—

1
k
1
l
′
0
† 2
dh
′
0
† 3
a
1
c
0
† 4
b
1
e
′
0
† 5
k
′h
0
† 6
b
′l
0
† 7
d
′
1
c
′
0

1
kl
′
† 5
k
′h
† 2
dh
′
† 6
b
′l
† 4
be
′
† 7
d
′c′
† 3
ac
    ¶
e
′a
0
†
a
1
   
i.e.
¶
a
1
e
′
0
;

i.e.
“All
a
are
e
.”

The Reader should now take a second piece of paper, and copy the Data only, and try to work out the solution for himself, beginning with some other Premiss.

If he fails to bring out the Conclusion
a
1
e
′
0
, I would advise him to take a third piece of paper, and
begin again
!]

I will now work out, in its briefest form, a Sorites of 5 Premisses, to serve as a model for the Reader to imitate in working examples.

(1) ”I greatly value everything that John gives me;

(2)   Nothing but this bone will satisfy my dog;

(3)   I take particular care of everything that I greatly value;

(4)   This bone was a present from John;

(5)   The things, of which I take particular care, are things I do
not
give to my dog”.

Univ.
“things”;
a
 = given by John to me;
b
 = given by me to my dog;
c
 = greatly valued by me;
d
 = satisfactory to my dog;
e
 = taken particular care of by me;
h
 = this bone.

1
a
1
c
′
0
† 2
h
′d
0
† 3
c
1
e
′
0
† 4
h
1
a
′
0
† 5
e
1
b
0

1
ac
′ † 3
ce
′
† 4
ha
′
† 2
h
′d
† 5
eb
    ¶
db
0

i.e.
“Nothing, that I give my dog, satisfies him,” or, “My dog is not satisfied with
anything
that I give him!”

[Note that, in working a Sorites by this process, we may begin with
any
Premiss we choose.
For instance, we might begin with No.
5, and the result would then be

5
eb
† 3
ce
′
† 1
ac
′
† 4
ha
′
† 2
h
′d
    ¶
bd
0
]

[Work Examples §
4
, 25–30 (p.
100); §
5
, 25–30 (p.
102); §
6
, 13–15 (p.
106); §
7
, 13–15 (p.
108); §
8
, 1–4, 13, 14, 19, 24 (pp.
110, 111); §
9
, 1–4, 26, 27, 40, 48 (pp.
112, 116, 119, 121).]