Complete Works of Lewis Carroll (204 page)

The Major declares that all
xm
must be destroyed; erase it.

Then, as some
my
′
is to be saved, it must clearly be
my
′x′
.
That is, there must exist
my
′x′
; or eliminating
m
,
y
′x′
.
In common phraseology,

‘Some
y
′
are
x
′
,’ or, ‘Some not-gamblers are not-philosophers.’”

(5)
Solution by my Method of Diagrams.

The first Premiss asserts that no
xm
exist: so we mark the
xm
-Compartment as empty, by placing a ‘O’ in each of its Cells.

The second asserts that some
my
′
exist: so we mark the
my
′
-Compartment as occupied, by placing a ‘I’ in its only available Cell.

The only information, that this gives us as to
x
and
y
, is that the
x
′y′
-Compartment is
occupied
, i.e.
that some
x
′y′
exist.

Hence “Some
x
′
are
y
′
”: i.e.
“Some persons, who are not philosophers, are not gamblers”.

(6)
Solution by my Method of Subscripts.

xm
0
†
my
′
1
¶
x
′y′
1

i.e.
“Some persons, who are not philosophers, are not gamblers.”

§ 9.

My Method of treating Syllogisms and Sorites.

Of all the strange things, that are to be met with in the ordinary text-books of Formal Logic, perhaps the strangest is the violent contrast one finds to exist between their ways of dealing with these two subjects.
While they have elaborately discussed no less than
nineteen
different forms of
Syllogisms
——each with its own special and exasperating Rules, while the whole constitute an almost useless machine, for practical purposes, many of the Conclusions being incomplete, and many quite legitimate forms being ignored——they have limited
Sorites
to
two
forms only, of childish simplicity; and these they have dignified with special
names
, apparently under the impression that no other possible forms existed!

As to
Syllogisms
, I find that their nineteen forms, with about a score of others which they have ignored, can all be arranged under
three
forms, each with a very simple Rule of its own; and the only question the Reader has to settle, in working any one of the 101 Examples given at p.
101 of this book, is “Does it belong to Fig.
I., II., or III.?”

As to
Sorites
, the only two forms, recognised by the text-books, are the
Aristotelian
, whose Premisses are a series of Propositions in
A
, so arranged that the Predicate of each is the Subject of the next, and the
Goclenian
, whose Premisses are the very same series, written backwards.
Goclenius, it seems, was the first who noticed the startling fact that it does not affect the force of a Syllogism to invert the order of its Premisses, and who applied this discovery to a Sorites.
If we assume (as surely we may?) that he is the
same
man as that transcendent genius who first noticed that 4 times 5 is the same thing as 5 times 4, we may apply to him what somebody (Edmund Yates, I think it was) has said of Tupper, viz., “here is a man who, beyond all others of his generation, has been favoured with Glimpses of the Obvious!”

These puerile——not to say infantine——forms of a Sorites I have, in this book, ignored from the very first, and have not only admitted freely Propositions in
E
, but have purposely stated the Premisses in random order, leaving to the Reader the useful task of arranging them, for himself, in an order which can be worked as a series of regular Syllogisms.
In doing this, he can begin with
any one
of them he likes.

I have tabulated, for curiosity, the various orders in which the Premisses of the Aristotelian Sorites

1.
All
a
are
b
;

2.
All
b
are
c
;

3.
All
c
are
d
;

4.
All
d
are
e
;

5.
All
e
are
h
.

 
∴
All
a
are
h
.

may be syllogistically arranged, and I find there are no less than
sixteen
such orders, viz., 12345, 21345, 23145, 23415, 23451, 32145, 32415, 32451, 34215, 34251, 34521, 43215, 43251, 43521, 45321, 54321.
Of these the
first
and the
last
have been dignified with names; but the other
fourteen
——first enumerated by an obscure Writer on Logic, towards the end of the Nineteenth Century——remain without a name!

§ 10.

Some account of Parts II, III.

In Part II.
will be found some of the matters mentioned in this Appendix, viz., the “Existential Import” of Propositions, the use of a
negative
Copula, and the theory that “two negative Premisses prove nothing.”
I shall also extend the range of Syllogisms, by introducing Propositions containing alternatives (such as “Not-all
x
are
y
”), Propositions containing 3 or more Terms (such as “All
ab
are
c
”, which, taken along with “Some
bc
′
are
d
”, would prove “Some
d
are
a
′
”), &c.
I shall also discuss Sorites containing Entities, and the
very
puzzling subjects of Hypotheticals and Dilemmas.
I hope, in the course of Part II., to go over all the ground usually traversed in the text-books used in our Schools and Universities, and to enable my Readers to solve Problems of the same kind as, and far harder than, those that are at present set in their Examinations.

In Part III.
I hope to deal with many curious and out-of-the-way subjects, some of which are not even alluded to in any of the treatises I have met with.
In this Part will be found such matters as the Analysis of Propositions into their Elements (let the Reader, who has never gone into this branch of the subject, try to make out for himself what
additional
Proposition would be needed to convert “Some
a
are
b
” into “Some
a
are
bc
”), the treatment of Numerical and Geometrical Problems, the construction of Problems, and the solution of Syllogisms and Sorites containing Propositions more complex than any that I have used in Part II.

I will conclude with eight Problems, as a taste of what is coming in Part II.
I shall be very glad to receive, from any Reader, who thinks he has solved any one of them (more especially if he has done so
without
using any Method of Symbols), what he conceives to be its complete Conclusion.

It may be well to explain what I mean by the
complete
Conclusion of a Syllogism or a Sorites.
I distinguish their Terms as being of two kinds——those which
can
be eliminated (e.g.
the Middle Term of a Syllogism), which I call the “Eliminands,” and those which
cannot
, which I call the “Retinends”; and I do not call the Conclusion
complete
, unless it states
all
the relations among the Retinends only, which can be deduced from the Premisses.

1.

All the boys, in a certain School, sit together in one large room every evening.
They are of no less than
five
nationalities——English, Scotch, Welsh, Irish, and German.
One of the Monitors (who is a great reader of Wilkie Collins’ novels) is very observant, and takes MS.
notes of almost everything that happens, with the view of being a good sensational witness, in case any conspiracy to commit a murder should be on foot.
The following are some of his notes:—

(1) Whenever some of the English boys are singing “Rule Britannia”, and some not, some of the Monitors are wide-awake;

(2) Whenever some of the Scotch are dancing reels, and some of the Irish fighting, some of the Welsh are eating toasted cheese;

(3) Whenever all the Germans are playing chess, some of the Eleven are
not
oiling their bats;

(4) Whenever some of the Monitors are asleep, and some not, some of the Irish are fighting;

(5) Whenever some of the Germans are playing chess, and none of the Scotch are dancing reels, some of the Welsh are
not
eating toasted cheese;

(6) Whenever some of the Scotch are
not
dancing reels, and some of the Irish
not
fighting, some of the Germans are playing chess;

(7) Whenever some of the Monitors are awake, and some of the Welsh are eating toasted cheese, none of the Scotch are dancing reels;

(8) Whenever some of the Germans are
not
playing chess, and some of the Welsh are
not
eating toasted cheese, none of the Irish are fighting;

(9) Whenever all the English are singing “Rule Britannia,” and some of the Scotch are
not
dancing reels, none of the Germans are playing chess;

(10) Whenever some of the English are singing “Rule Britannia”, and some of the Monitors are asleep, some of the Irish are
not
fighting;

(11) Whenever some of the Monitors are awake, and some of the Eleven are
not
oiling their bats, some of the Scotch are dancing reels;

(12) Whenever some of the English are singing “Rule Britannia”, and some of the Scotch are
not
dancing reels, * * * *

Here the MS.
breaks off suddenly.
The Problem is to complete the sentence, if possible.

[N.B.
In solving this Problem, it is necessary to remember that the Proposition “All
x
are
y
” is a
Double
Proposition, and is equivalent to “Some
x
are
y
, and none are
y
′
.”
See p.
17.]