Complete Works of Lewis Carroll (206 page)
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Authors: Lewis Carroll
(7) John declines to go into Society, but never gives himself airs;
(8) Brothers, who are apt to be self-conscious, though not
both
of them handsome, usually dislike Society;
(9) Men of the same height, who do not give themselves airs, are free from self-consciousness;
(10) Men, who agree on questions of Art, though they differ in Politics, and who are not both of them ugly, are always admired;
(11) Men, who hold opposite views about Art and are not admired, always give themselves airs;
(12) Brothers of the same height always differ in Politics;
(13) Two handsome men, who are neither both of them admired nor both of them self-conscious, are no doubt of different heights;
(14) Brothers, who are self-conscious, and do not both of them like Society, never look well when walking together.
[N.B.
See Note at end of Problem 2.]
8.
(1) A man can always master his father;
(2) An inferior of a man’s uncle owes that man money;
(3) The father of an enemy of a friend of a man owes that man nothing;
(4) A man is always persecuted by his son’s creditors;
(5) An inferior of the master of a man’s son is senior to that man;
(6) A grandson of a man’s junior is not his nephew;
(7) A servant of an inferior of a friend of a man’s enemy is never persecuted by that man;
(8) A friend of a superior of the master of a man’s victim is that man’s enemy;
(9) An enemy of a persecutor of a servant of a man’s father is that man’s friend.
The Problem is to deduce some fact about great-grandsons.
[N.B.
In this Problem, it is assumed that all the men, here referred to, live in the same town, and that every pair of them are either “friends” or “enemies,” that every pair are related as “senior and junior”, “superior and inferior”, and that certain pairs are related as “creditor and debtor”, “father and son”, “master and servant”, “persecutor and victim”, “uncle and nephew”.]
9.
“Jack Sprat could eat no fat:
His wife could eat no lean:
And so, between them both,
They licked the platter clean.”
Solve this as a Sorites-Problem, taking lines 3 and 4 as the Conclusion to be proved.
It is permitted to use, as Premisses, not only all that is here
asserted
, but also all that we may reasonably understand to be
implied
.
NOTES TO APPENDIX.
(A)
It may, perhaps, occur to the Reader, who has studied Formal Logic that the argument, here applied to the Propositions
I
and
E
, will apply equally well to the Propositions
I
and
A
(since, in the ordinary text-books, the Propositions “All
xy
are
z
” and “Some
xy
are not
z
” are regarded as Contradictories).
Hence it may appear to him that the argument might have been put as follows:—
“We now have
I
and
A
‘asserting.’
Hence, if the Proposition ‘All
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 not-
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.
Hence I
cannot
assert.”
This matter will be discussed in Part II; but I may as well give here what seems to me to be an irresistable proof that this view (that
A
and
I
are Contradictories), though adopted in the ordinary text-books, is untenable.
The proof is as follows:—
With regard to the relationship existing between the Class ‘
xy
’ and the two Classes ‘
z
’ and ‘not-
z
’, there are
four
conceivable states of things, viz.
(1)
Some
xy
are
z
,
and
some
are not-
z
;
(2)
〃
〃
none
〃
(3)
No
xy
〃
some
〃
(4)
〃
〃
none
〃
Of these four, No.
(2) is equivalent to “All
xy
are
z
”, No.
(3) is equivalent to “All
xy
are not-
z
”, and No.
(4) is equivalent to “No
xy
exist.”
Now it is quite undeniable that, of these
four
states of things, each is,
a priori
,
possible
, some
one must
be true, and the other three
must
be false.
Hence the Contradictory to (2) is “Either (1) or (3) or (4) is true.”
Now the assertion “Either (1) or (3) is true” is equivalent to “Some
xy
are not-
z
”; and the assertion “(4) is true” is equivalent to “No
xy
exist.”
Hence the Contradictory to “All
xy
are
z
” may be expressed as the Alternative Proposition “Either some
xy
are not-
z
, or no
xy
exist,” but
not
as the Categorical Proposition “Some
y
are not-
z
.”
(B)
There are yet
other
views current among “The Logicians”, as to the “Existential Import” of Propositions, which have not been mentioned in this Section.
One is, that the Proposition “some
x
are
y
” is to be interpreted, neither as “Some
x
exist
and are
y
”, nor yet as “If there
were
any
x
in existence, some of them
would
be
y
”, but merely as “Some
x
can be
y
; i.e.
the Attributes
x
and
y
are
compatible
”.
On
this
theory, there would be nothing offensive in my telling my friend Jones “Some of your brothers are swindlers”; since, if he indignantly retorted “What do you
mean
by such insulting language, you scoundrel?”, I should calmly reply “I merely mean that the thing is
conceivable
——that some of your brothers
might possibly
be swindlers”.
But it may well be doubted whether such an explanation would
entirely
appease the wrath of Jones!
Another view is, that the Proposition “All
x
are
y
”
sometimes
implies the actual
existence
of
x
, and
sometimes
does
not
imply it; and that we cannot tell, without having it in
concrete
form,
which
interpretation we are to give to it.
This
view is, I think, strongly supported by common usage; and it will be fully discussed in Part II: but the difficulties, which it introduces, seem to me too formidable to be even alluded to in Part I, which I am trying to make, as far as possible, easily intelligible to mere
beginners
.
(C)
The three Conclusions are
“No conceited child of mine is greedy”;
“None of my boys could solve this problem”;
“Some unlearned boys are not choristers.”
THE GAME OF LOGIC
To my Child-friend
.
I charm in vain; for never again,
All keenly as my glance I bend,
Will Memory, goddess coy,
Embody for my joy
Departed days, nor let me gaze
On thee, my fairy friend!
Yet could thy face, in mystic grace,
A moment smile on me, 'twould send
Far-darting rays of light
From Heaven athwart the night,
By which to read in very deed
Thy spirit, sweetest friend!
So may the stream of Life's long dream
Flow gently onward to its end,
With many a floweret gay,
Adown its willowy way:
May no sigh vex, no care perplex,
My loving little friend!
NOTA BENE.
With each copy of this Book is given an Envelope, containing a Diagram (similar to the frontispiece) on card, and nine Counters, four red and five grey.
The Envelope, &c.
can be had separately, at 3d.
each.
The Author will be very grateful for suggestions, especially from beginners in Logic, of any alterations, or further explanations, that may seem desirable.
Letters should be addressed to him at "29, Bedford Street, Covent Garden, London."
PREFACE
"There foam'd rebellious Logic, gagg'd and bound."
This Game requires nine Counters—four of one colour and five of another: say four red and five grey.
Besides the nine Counters, it also requires one Player, AT LEAST.
I am not aware of any Game that can be played with LESS than this number: while there are several that require MORE: take Cricket, for instance, which requires twenty-two.
How much easier it is, when you want to play a Game, to find ONE Player than twenty-two.
At the same time, though one Player is enough, a good deal more amusement may be got by two working at it together, and correcting each other's mistakes.
A second advantage, possessed by this Game, is that, besides being an endless source of amusement (the number of arguments, that may be worked by it, being infinite), it will give the Players a little instruction as well.
But is there any great harm in THAT, so long as you get plenty of amusement?