Complete Works of Lewis Carroll (198 page)

APPENDIX,

ADDRESSED TO TEACHERS.

§ 1.

Introductory.

There are several matters, too hard to discuss with
Learners
, which nevertheless need to be explained to any
Teachers
, into whose hands this book may fall, in order that they may thoroughly understand what my Symbolic Method
is
, and in what respects it differs from the many other Methods already published.

These matters are as follows:—

The “Existential Import” of Propositions.

The use of “is-not” (or “are-not”) as a Copula.

The theory “two Negative Premisses prove nothing.”

Euler’s Method of Diagrams.

Venn’s Method of Diagrams.

My Method of Diagrams.

The Solution of a Syllogism by various Methods.

My Method of treating Syllogisms and Sorites.

Some account of Parts II, III.

§ 2.

The “Existential Import” of Propositions.

The writers, and editors, of the Logical text-books which run in the ordinary grooves——to whom I shall hereafter refer by the (I hope inoffensive) title “The Logicians”——take, on this subject, what seems to me to be a more humble position than is at all necessary.
They speak of the Copula of a Proposition “with bated breath”, almost as if it were a living, conscious Entity, capable of declaring for itself what it chose to mean, and that we, poor human creatures, had nothing to do but to ascertain
what
was its sovereign will and pleasure, and submit to it.

In opposition to this view, I maintain that any writer of a book is fully authorised in attaching any meaning he likes to any word or phrase he intends to use.
If I find an author saying, at the beginning of his book, “Let it be understood that by the word ‘
black
’ I shall always mean ‘
white
’, and that by the word ‘
white
’ I shall always mean ‘
black
’,” I meekly accept his ruling, however injudicious I may think it.

And so, with regard to the question whether a Proposition is or is not to be understood as asserting the existence of its Subject, I maintain that every writer may adopt his own rule, provided of course that it is consistent with itself and with the accepted facts of Logic.

Let us consider certain views that may
logically
be held, and thus settle which of them may
conveniently
be held; after which I shall hold myself free to declare which of them
I
intend to hold.

The
kinds
of Propositions, to be considered, are those that begin with “some”, with “no”, and with “all”.
These are usually called Propositions “in
I
”, “in
E
”, and “in
A
”.

First, then, a Proposition in
I
may be understood as asserting, or else as
not
asserting, the existence of its Subject.
(By “existence” I mean of course whatever kind of existence suits its nature.
The two Propositions, “
dreams
exist” and “
drums
exist”, denote two totally different kinds of “existence”.
A
dream
is an aggregate of ideas, and exists only in the
mind of a dreamer
: whereas a
drum
is an aggregate of wood and parchment, and exists in
the hands of a drummer
.)

First, let us suppose that
I
“asserts” (i.e.
“asserts the existence of its Subject”).

Here, of course, we must regard a Proposition in
A
as making the
same
assertion, since it necessarily
contains
a Proposition in
I
.

We now have
I
and
A
“asserting”.
Does this leave us free to make what supposition we choose as to
E
?
My answer is “No.
We are tied down to the supposition that
E
does
not
assert.”
This can be proved as follows:—

If possible, let
E
“assert”.
Then (taking
x
,
y
, and
z
to represent Attributes) we see that, if the Proposition “No
xy
are
z
” be true, some things exist with the Attributes
x
and
y
: i.e.
“Some
x
are
y
.”

Also we know that, if the Proposition “Some
xy
are
z
” be true, the same result follows.

But these two Propositions are Contradictories, so that one or other of them
must
be true.
Hence this result is
always
true: i.e.
the Proposition “Some
x
are
y
” is
always
true!

Quod est absurdum.
).

We see, then, that the supposition “
I
asserts” necessarily leads to “
A
asserts, but
E
does not”.
And this is the
first
of the various views that may conceivably be held.

Next, let us suppose that
I
does
not
“assert.”
And, along with this, let us take the supposition that
E
does
“assert.”

Hence the Proposition “No
x
are
y
” means “Some
x
exist, and none of them are
y
”: i.e.
“
all
of them are
not
-
y
,” which is a Proposition in
A
.
We also know, of course, that the Proposition “All
x
are not-
y
” proves “No
x
are
y
.”
Now two Propositions, each of which proves the other, are
equivalent
.
Hence every Proposition in
A
is equivalent to one in
E
, and therefore “
asserts
”.

Hence our
second
conceivable view is “
E
and
A
assert, but
I
does not.”

This view does not seen to involve any necessary contradiction with itself or with the accepted facts of Logic.
But, when we come to
test
it, as applied to the actual
facts
of life, we shall find I think, that it fits in with them so badly that its adoption would be, to say the least of it, singularly inconvenient for ordinary folk.

Let me record a little dialogue I have just held with my friend Jones, who is trying to form a new Club, to be regulated on strictly
Logical
principles.

Author.
“Well, Jones!
Have you got your new Club started yet?”

Jones
(
rubbing his hands
).
“You’ll be glad to hear that some of the Members (mind, I only say ‘
some
’) are millionaires!
Rolling in gold, my boy!”

Author.
“That sounds well.
And how many Members have entered?”

Jones
(
staring
).
“None at all.
We haven’t got it started yet.
What makes you think we have?”

Author.
“Why, I thought you said that some of the Members——”

Jones
(
contemptuously
).
“You don’t seem to be aware that we’re working on strictly
Logical
principles.
A
Particular
Proposition does
not
assert the existence of its Subject.
I merely meant to say that we’ve made a Rule not to admit
any
Members till we have at least
three
Candidates whose incomes are over ten thousand a year!”

Author.
“Oh,
that’s
what you meant, is it?
Let’s hear some more of your Rules.”

Jones.
“Another is, that no one, who has been convicted seven times of forgery, is admissible.”

Author.
“And here, again, I suppose you don’t mean to assert there
are
any such convicts in existence?”

Jones.
“Why, that’s exactly what I
do
mean to assert!
Don’t you know that a Universal Negative
asserts
the existence of its Subject?
Of course
we didn’t make that Rule till we had satisfied ourselves that there are several such convicts now living.”

The Reader can now decide for himself how far this
second
conceivable view would fit in with the facts of life.
He will, I think, agree with me that Jones’ view, of the ‘Existential Import’ of Propositions, would lead to some inconvenience.

Thirdly, let us suppose that neither
I
nor
E
“asserts”.

Now the supposition that the two Propositions, “Some
x
are
y
” and “No
x
are not-
y
”, do
not
“assert”, necessarily involves the supposition that “All
x
are
y
” does
not
“assert”, since it would be absurd to suppose that they assert, when combined, more than they do when taken separately.

Hence the
third
(and last) of the conceivable views is that neither
I
, nor
E
, nor
A
, “asserts”.

The advocates of this third view would interpret the Proposition “Some
x
are
y
” to mean “If there
were
any
x
in existence, some of them
would
be
y
”; and so with
E
and
A
.

It admits of proof that this view, as regards
A
, conflicts with the accepted facts of Logic.

Let us take the Syllogism
Darapti
, which is universally accepted as valid.
Its form is

“All
m
are
x
;

  All
m
are
y
.

         
∴
Some
y
are
x
”.

This they would interpret as follows:—