Complete Works of Lewis Carroll (157 page)

∴
“Some
y
are
x
′
.”

Hence proposed Conclusion is right.

(4)

“All soldiers can march;

  Some babies are not soldiers.

      Some babies cannot march”.

Univ.
“persons”;
m
 = soldiers;
x
 = able to march;
y
 = babies.

“All
m
are
x
;

 Some
y
are
m
′
.

   Some
y
are
x
′
.”

There is no Conclusion.

(5)

“All selfish men are unpopular;

  All obliging men are popular.

      All obliging men are unselfish”.

Univ.
“men”;
m
 = popular;
x
 = selfish;
y
 = obliging.

“All
x
are
m
′
;

 All
y
are
m
.

   All
y
are
x
′
.”

∴
“All
x
are
y
′
;

   All
y
are
x
′
.”

Hence proposed Conclusion is
incomplete
, the
complete
one containing, in addition, “All selfish men are disobliging”.

(6)

”No one, who means to go by the train and cannot get a conveyance, and has not enough time to walk to the station, can do without running;

  This party of tourists mean to go by the train and cannot get a conveyance, but they have plenty of time to walk to the station.

  This party of tourists need not run.”

Univ.
“persons meaning to go by the train, and unable to get a conveyance”;
m
 = having enough time to walk to the station;
x
 = needing to run;
y
 = these tourists.

“No
m
′
are
x
′
;

 All
y
are
m
.

   All
y
are
x
′
.”

There is no Conclusion.

[Here is
another
opportunity, gentle Reader, for playing a trick on your innocent friend.
Put the proposed Syllogism before him, and ask him what he thinks of the Conclusion.

He will reply “Why, it’s perfectly correct, of course!
And if your precious Logic-book tells you it
isn’t
, don’t believe it!
You don’t mean to tell me those tourists
need
to run?
If
I
were one of them, and knew the
Premisses
to be true, I should be
quite
clear that I
needn’t
run—and I
should walk!
”

And
you
will reply “But suppose there was a mad bull behind you?”

And then your innocent friend will say “Hum!
Ha!
I must think that over a bit!”

You may then explain to him, as a convenient
test
of the soundness of a Syllogism, that, if circumstances can be invented which, without interfering with the truth of the
Premisses
, would make the
Conclusion
false, the Syllogism
must
be unsound.]

[Review Tables V–VIII (pp.
46–49).
Work Examples §
4
, 7–12 (p.
100); §
5
, 7–12 (p.
101); §
6
, 1–10 (p.
106); §
7
, 1–6 (pp.
107, 108).]

BOOK VI.

THE METHOD OF SUBSCRIPTS.

CHAPTER I.

INTRODUCTORY.

Let us agree that “
x
1
” shall mean “Some existing Things have the Attribute
x
”, i.e.
(more briefly) “Some
x
exist”; also that “
xy
1
” shall mean “Some
xy
exist”, and so on.
Such a Proposition may be called an ‘
Entity
.’

[Note that, when there are
two
letters in the expression, it does not in the least matter which stands
first
: “
xy
1
” and “
yx
1
” mean exactly the same.]

Also that “
x
0
” shall mean “No existing Things have the Attribute
x
”, i.e.
(more briefly) “No
x
exist”; also that “
xy
0
” shall mean “No
xy
exist”, and so on.
Such a Proposition may be called a ‘
Nullity
’.

Also that “†” shall mean “and”.

[Thus “
ab
1
†
cd
0
” means “Some
ab
exist and no
cd
exist”.]

Also that “¶” shall mean “would, if true, prove”.

[Thus, “
x
0
¶
xy
0
” means “The Proposition ‘No
x
exist’ would, if true, prove the Proposition ‘No
xy
exist’”.]

When two Letters are both of them accented, or both
not
accented, they are said to have ‘
Like Signs
’, or to be ‘
Like
’: when one is accented, and the other not, they are said to have ‘
Unlike Signs
’, or to be ‘
Unlike
’.

CHAPTER II.

REPRESENTATION OF PROPOSITIONS OF RELATION.

Let us take, first, the Proposition “Some
x
are
y
”.

This, we know, is equivalent to the Proposition of Existence “Some
xy
exist”.  Hence it may be represented by the expression “
xy
1
”.

The Converse Proposition “Some
y
are
x
” may of course be represented by the
same
expression, viz.
“
xy
1
”.

Similarly we may represent the three similar Pairs of Converse Propositions, viz.—

“Some
x
are
y
′
”  = “Some
y
′
are
x
”,

“Some
x
′
are
y
”  = “Some
y
are
x
′
”,

“Some
x
′
are
y
′
” = “Some
y
′
are
x
′
”.

Let us take, next, the Proposition “No
x
are
y
”.

This, we know, is equivalent to the Proposition of Existence “No
xy
exist”.  Hence it may be represented by the expression “
xy
0
”.

The Converse Proposition “No
y
are
x
” may of course be represented by the
same
expression, viz.
“
xy
0
”.

Similarly we may represent the three similar Pairs of Converse Propositions, viz.—

“No
x
are
y
′
”  = “No
y
′
are
x
”,

“No
x
′
are
y
”  = “No
y
are
x
′
”,

“No
x
′
are
y
′
” = “No
y
′
are
x
′
”.

Let us take, next, the Proposition “All
x
are
y
”.

Now it is evident that the Double Proposition of Existence “Some
x
exist and no
xy
′
exist” tells us that
some
x
-Things exist, but that
none
of them have the Attribute
y
′
: that is, it tells us that
all
of them have the Attribute
y
: that is, it tells us that “All
x
are
y
”.

Also it is evident that the expression “
x
1
†
xy
′
0
” represents this Double Proposition.

Hence it also represents the Proposition “All
x
are
y
”.

[The Reader will perhaps be puzzled by the statement that the Proposition “All
x
are
y
” is equivalent to the Double Proposition “Some
x
exist and no
xy
′
exist,” remembering that it was stated, at p.
33, to be equivalent to the Double Proposition “Some
x
are
y
and no
x
are
y
′
” (i.e.
“Some
xy
exist and no
xy
′
exist”).
The explanation is that the Proposition “Some
xy
exist” contains
superfluous information
.
“Some
x
exist” is enough for our purpose.]

This expression may be written in a shorter form, viz.
“
x
1
y
′
0
”, since
each
Subscript takes effect back to the
beginning
of the expression.

Similarly we may represent the seven similar Propositions “All
x
are
y
′
”, “All
x
′
are
y
”, “All
x
′
are
y
′
”, “All
y
are
x
”, “All
y
are
x
′
”, “All
y
′
are
x
”, and “All
y
′
are
x
′
”.

[The Reader should make out all these for himself.]

It will be convenient to remember that, in translating a Proposition, beginning with “All”, from abstract form into subscript form, or
vice versâ
, the Predicate
changes sign
(that is, changes from positive to negative, or else from negative to positive).

[Thus, the Proposition “All
y
are
x
′
” becomes “
y
1
x
0
”, where the Predicate changes from
x
′
to
x
.

Again, the expression “
x
′
1
y
′
0
” becomes “All
x
′
are
y
”, where the Predicate changes for
y
′
to
y
.]