Complete Works of Lewis Carroll (199 page)

”If there were any
m
in existence, all of them would be
x
;

  If there were any
m
in existence, all of them would be
y
.

         
∴
If there were any
y
in existence, some of them would be
x
”.

That this Conclusion does
not
follow has been so briefly and clearly explained by Mr.
Keynes (in his “Formal Logic”, dated 1894, pp.
356, 357), that I prefer to quote his words:—

“
Let no proposition imply the existence either of its subject or of its predicate.

“Take, as an example, a syllogism in
Darapti
:—

‘
All M is P
,

 
All M is S
,

         
∴
Some S is P
.’

“Taking
S
,
M
,
P
, as the minor, middle, and major terms respectively, the conclusion will imply that, if there is an
S
, there is some
P
.
Will the premisses also imply this?
If so, then the syllogism is valid; but not otherwise.

“The conclusion implies that if
S
exists
P
exists; but, consistently with the premisses,
S
may be existent while
M
and
P
are both non-existent.
An implication is, therefore, contained in the conclusion, which is not justified by the premisses.”

This seems to
me
entirely clear and convincing.
Still, “to make sicker”, I may as well throw the above (
soi-disant
) Syllogism into a concrete form, which will be within the grasp of even a
non
-logical Reader.

Let us suppose that a Boys’ School has been set up, with the following system of Rules:—

“All boys in the First (the highest) Class are to do French, Greek, and Latin.
All in the Second Class are to do Greek only.
All in the Third Class are to do Latin only.”

Suppose also that there
are
boys in the Third Class, and in the Second; but that no boy has yet risen into the First.

It is evident that there are no boys in the School doing French: still we know, by the Rules, what would happen if there
were
any.

We are authorised, then, by the
Data
, to assert the following two Propositions:—

“If there were any boys doing French, all of them would be doing Greek;

  If there were any boys doing French, all of them would be doing Latin.”

And the Conclusion, according to “The Logicians” would be

“If there were any boys doing Latin, some of them would be doing Greek.”

Here, then, we have two
true
Premisses and a
false
Conclusion (since we know that there
are
boys doing Latin, and that
none
of them are doing Greek).
Hence the argument is
invalid
.

Similarly it may be shown that this “non-existential” interpretation destroys the validity of
Disamis
,
Datisi
,
Felapton
, and
Fresison
.

Some of “The Logicians” will, no doubt, be ready to reply “But we are not
Aldrichians
!
Why should
we
be responsible for the validity of the Syllogisms of so antiquated an author as Aldrich?”

Very good.
Then, for the
special
benefit of these “friends” of mine (with what ominous emphasis that name is sometimes used!
“I must have a private interview with
you
, my young
friend
,” says the bland Dr.
Birch, “in my library, at 9 a.m.
tomorrow.
And you will please to be
punctual
!”), for their
special
benefit, I say, I will produce
another
charge against this “non-existential” interpretation.

It actually invalidates the ordinary Process of “Conversion”, as applied to Proposition in ‘
I
’.

Every
logician, Aldrichian or otherwise, accepts it as an established fact that “Some
x
are
y
” may be legitimately converted into “Some
y
are
x
.”

But is it equally clear that the Proposition “If there
were
any
x
, some of them
would
be
y
” may be legitimately converted into “If there
were
any
y
, some of them would be
x
”?
I trow not.

The example I have already used——of a Boys’ School with a non-existent First Class——will serve admirably to illustrate this new flaw in the theory of “The Logicians.”

Let us suppose that there is yet
another
Rule in this School, viz.
“In each Class, at the end of the Term, the head boy and the second boy shall receive prizes.”

This Rule entirely authorises us to assert (in the sense in which “The Logicians” would use the words) “Some boys in the First Class will receive prizes”, for this simply means (according to them) “If there
were
any boys in the First Class, some of them
would
receive prizes.”

Now the Converse of this Proposition is, of course, “Some boys, who will receive prizes, are in the First Class”, which means (according to “The Logicians”) “If there
were
any boys about to receive prizes, some of them
would
be in the First Class” (which Class we know to be
empty
).

Of this Pair of Converse Propositions, the first is undoubtedly
true
: the second,
as
undoubtedly,
false
.

It is always sad to see a batsman knock down his own wicket: one pities him, as a man and a brother, but, as a
cricketer
, one can but pronounce him “Out!”

We see, then, that, among all the conceivable views we have here considered, there are only
two
which can
logically
be held, viz.

I
and
A
“assert”, but
E
does not.

E
and
A
“assert”, but
I
does not.

The
second
of these I have shown to involve great practical inconvenience.

The
first
is the one adopted in this book.

Some further remarks on this subject will be found in Note (B), at p.
196.

§ 3.

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

Is it better to say “John
is-not
in-the-house” or “John
is
not-in-the-house”?
“Some of my acquaintances
are-not
men-I-should-like-to-be-seen-with” or “Some of my acquaintances
are
men-I-should-
not
-like-to-be-seen-with”?
That is the sort of question we have now to discuss.

This is no question of Logical Right and Wrong: it is merely a matter of
taste
, since the two forms mean exactly the same thing.
And here, again, “The Logicians” seem to me to take much too humble a position.
When they are putting the final touches to the grouping of their Proposition, just before the curtain goes up, and when the Copula——always a rather fussy ‘heavy father’, asks them “Am
I
to have the ‘not’, or will you tack it on to the Predicate?”
they are much too ready to answer, like the subtle cab-driver, “Leave it to
you
, Sir!”
The result seems to be, that the grasping Copula constantly gets a “not” that had better have been merged in the Predicate, and that Propositions are differentiated which had better have been recognised as precisely similar.
Surely it is simpler to treat “Some men are Jews” and “Some men are Gentiles” as being both of them,
affirmative
Propositions, instead of translating the latter into “Some men are-not Jews”, and regarding it as a
negative
Propositions?

The fact is, “The Logicians” have somehow acquired a perfectly
morbid
dread of negative Attributes, which makes them shut their eyes, like frightened children, when they come across such terrible Propositions as “All not-x are y”; and thus they exclude from their system many very useful forms of Syllogisms.

Under the influence of this unreasoning terror, they plead that, in Dichotomy by Contradiction, the
negative
part is too large to deal with, so that it is better to regard each Thing as either included in, or excluded from, the
positive
part.
I see no force in this plea: and the facts often go the other way.
As a personal question, dear Reader, if
you
were to group your acquaintances into the two Classes, men that you
would
like to be seen with, and men that you would
not
like to be seen with, do you think the latter group would be so
very
much the larger of the two?

For the purposes of Symbolic Logic, it is so
much
the most convenient plan to regard the two sub-divisions, produced by Dichotomy, on the
same
footing, and to say, of any Thing, either that it “is” in the one, or that it “is” in the other, that I do not think any Reader of this book is likely to demur to my adopting that course.

§ 4.

The theory that “two Negative Premisses prove nothing”.

This I consider to be
another
craze of “The Logicians”, fully as morbid as their dread of a negative Attribute.

It is, perhaps, best refuted by the method of
Instantia Contraria
.

Take the following Pairs of Premisses:—

“None of my boys are conceited;

  None of my girls are greedy”.

“None of my boys are clever;

  None but a clever boy could solve this problem”.

“None of my boys are learned;

  Some of my boys are not choristers”.

(This last Proposition is, in
my
system, an
affirmative
one, since I should read it “are not-choristers”; but, in dealing with “The Logicians,” I may fairly treat it as a
negative
one, since
they
would read it “are-not choristers”.)

If you, dear Reader, declare, after full consideration of these Pairs of Premisses, that you cannot deduce a Conclusion from
any
of them——why, all I can say is that, like the Duke in Patience, you “will have to be contented with our heart-felt sympathy”!

§ 5.

Euler’s Method of Diagrams.

Diagrams seem to have been used, at first, to represent
Propositions
only.
In Euler’s well-known Circles, each was supposed to contain a class, and the Diagram consisted of two circles, which exhibited the relations, as to inclusion and exclusion, existing between the two Classes.