Complete Works of Lewis Carroll (141 page)

[Thus, we know that
every
banker is a Member of the Genus “men”.
Hence,
every
banker is either in the Class “rich men”, or else in the Class “poor men”.]

Also we have been told that, in the case we are discussing,
some
Members of the Subject are in Class (1).
What
else
do we need to be told, in order to know that
all
of them are there?
Evidently we need to be told that
none
of them are in Class (2); i.e.
that
none
of them are Members of the Class whose Differentia is
contradictory
to that of the Predicate.

[Thus, we may suppose we have been told that
some
bankers are in the Class “rich men”.
What
else
do we need to be told, in order to know that
all
of them are there?
Evidently we need to be told that
none
of them are in the Class “
poor
men”.]

Hence a Proposition of Relation, beginning with “All”, is a
Double
Proposition, and is ‘
equivalent
’ to (i.e.
gives the same information as) the
two
Propositions

(1) “
Some
Members of the Subject are Members of the Predicate”;

(2) “
No
Members of the Subject are Members of the Class whose Differentia is
contradictory
to that of the Predicate”.

[Thus, the Proposition “
All
bankers are rich men” is a
Double
Proposition, and is equivalent to the
two
Propositions

(1) “
Some
bankers are rich men”;

(2) “
No
bankers are
poor
men”.]

§ 4.

What is implied, in a Proposition of Relation, as to the Reality of its Terms?

Note that the rules, here laid down, are
arbitrary
, and only apply to Part I of my “Symbolic Logic.”

A Proposition of Relation, beginning with “Some”, is henceforward to be understood as asserting that there are
some existing Things
, which, being Members of the Subject, are also Members of the Predicate; i.e.
that
some existing Things
are Members of
both
Terms at once.
Hence it is to be understood as implying that
each
Term, taken by itself, is
Real
.

[Thus, the Proposition “Some rich men are invalids” is to be understood as asserting that
some existing Things
are “rich invalids”.
Hence it implies that
each
of the two Classes, “rich men” and “invalids”, taken by itself, is
Real
.]

A Proposition of Relation, beginning with “No”, is henceforward to be understood as asserting that there are
no existing Things
which, being Members of the Subject, are also Members of the Predicate; i.e.
that
no existing Things
are Members of
both
Terms at once.
But this implies nothing as to the
Reality
of either Term taken by itself.

[Thus, the Proposition “No mermaids are milliners” is to be understood as asserting that
no existing Things
are “mermaid-milliners”.
But this implies nothing as to the
Reality
, or the
Unreality
, of either of the two Classes, “mermaids” and “milliners”, taken by itself.
In this case as it happens, the Subject is
Imaginary
, and the Predicate
Real
.]

A Proposition of Relation, beginning with “All”, contains (see § 3) a similar Proposition beginning with “Some”.
Hence it is to be understood as implying that
each
Term, taken by itself, is
Real
.

[Thus, the Proposition “All hyænas are savage animals” contains the Proposition “Some hyænas are savage animals”.
Hence it implies that
each
of the two Classes, “hyænas” and “savage animals”, taken by itself, is
Real
.]

§ 5.

Translation of a Proposition of Relation into one or more Propositions of Existence.

We have seen that a Proposition of Relation, beginning with “Some,” asserts that
some existing Things
, being Members of its Subject, are
also
Members of its Predicate.
Hence, it asserts that some existing Things are Members of
both
; i.e.
it asserts that some existing Things are Members of the Class of Things which have
all
the Attributes of the Subject and the Predicate.

Hence, to translate it into a Proposition of Existence, we take “existing Things” as the new
Subject
, and Things, which have
all
the Attributes of the Subject and the Predicate, as the new Predicate.

Similarly for a Proposition of Relation beginning with “No”.

A Proposition of Relation, beginning with “All”, is (as shown in § 3) equivalent to
two
Propositions, one beginning with “Some” and the other with “No”, each of which we now know how to translate.

[Let us work a few Examples, to illustrate these Rules.

(1)

“Some apples are not ripe.”

Here we arrange thus:—

“Some”

Sign of Quantity
.

“existing Things”

Subject
.

“are”

Copula
.

“not-ripe apples”

Predicate
.

or thus:—

“Some | existing Things | are | not-ripe apples.”

(2)

“Some farmers always grumble at the weather, whatever it may be.”

Here we arrange thus:—

“Some | existing Things | are | farmers who always grumble at the weather, whatever it may be.”

(3)

“No lambs are accustomed to smoke cigars.”

Here we arrange thus:—

“No | existing Things |are | lambs accustomed to smoke cigars.”

(4)

“None of my speculations have brought me as much as 5 per cent.”

Here we arrange thus:—

“No | existing Things | are | speculations of mine, which have brought me as much as 5 per cent.”

(5)

“None but the brave deserve the fair.”

Here we note, to begin with, that the phrase “none but the brave” is equivalent to “no not-brave men.”
We then arrange thus:—

“No | existing Things | are | not-brave men deserving of the fair.”

(6)

“All bankers are rich men.”

This is equivalent to the two Propositions “Some bankers are rich men” and “No bankers are poor men.”

Here we arrange thus:—

“Some | existing Things | are | rich bankers”; and “No | existing Things | are | poor bankers.”]

[Work Examples §
1
, 1–4 (p.
97).]

BOOK III.

THE BILITERAL DIAGRAM.

CHAPTER I.

SYMBOLS AND CELLS.

First, let us suppose that the above Diagram is an enclosure assigned to a certain Class of Things, which we have selected as our ‘Universe of Discourse.’
or, more briefly, as our ‘Univ’.

[For example, we might say “Let Univ.
be ‘books’”; and we might imagine the Diagram to be a large table, assigned to all “books.”]

[The Reader is strongly advised, in reading this Chapter,
not
to refer to the above Diagram, but to draw a large one for himself,
without any letters
, and to have it by him while he reads, and keep his finger on that particular
part
of it, about which he is reading.]

Secondly, let us suppose that we have selected a certain Adjunct, which we may call “
x
,” and have divided the large Class, to which we have assigned the whole Diagram, into the two smaller Classes whose Differentiæ are “
x
” and “not-
x
” (which we may call “
x
′
”), and that we have assigned the
North
Half of the Diagram to the one (which we may call “the Class of
x
-Things,” or “the
x
-Class”), and the
South
Half to the other (which we may call “the Class of
x
′
-Things,” or “the
x
′
-Class”).

[For example, we might say “Let
x
mean ‘old,’ so that
x
′
will mean ‘new’,” and we might suppose that we had divided books into the two Classes whose Differentiæ are “old” and “new,” and had assigned the
North
Half of the table to “
old
books” and the
South
Half to “
new
books.”]

Thirdly, let us suppose that we have selected another Adjunct, which we may call “
y
”, and have subdivided the
x
-Class into the two Classes whose Differentiæ are “
y
” and “
y
′
”, and that we have assigned the North-
West
Cell to the one (which we may call “the
xy
-Class”), and the North-
East
Cell to the other (which we may call “the
xy
′
-Class”).

[For example, we might say “Let
y
mean ‘English,’ so that
y
′
will mean ‘foreign’”, and we might suppose that we had subdivided “old books” into the two Classes whose Differentiæ are “English” and “foreign”, and had assigned the North-
West
Cell to “old
English
books”, and the North-
East
Cell to “old
foreign
books.”]

Fourthly, let us suppose that we have subdivided the
x
′
-Class in the same manner, and have assigned the South-
West
Cell to the
x
′y
-Class, and the South-
East
Cell to the
x
′y′
-Class.

[For example, we might suppose that we had subdivided “new books” into the two Classes “new
English
books” and “new
foreign
books”, and had assigned the South-
West
Cell to the one, and the South-
East
Cell to the other.]

It is evident that, if we had begun by dividing for
y
and
y
′
, and had then subdivided for
x
and
x
′
, we should have got the
same
four Classes.
Hence we see that we have assigned the
West
Half to the
y
-Class, and the
East
Half to the
y
′
-Class.

[Thus, in the above Example, we should find that we had assigned the
West
Half of the table to “
English
books” and the
East
Half to “
foreign
books.”

We have, in fact, assigned the four Quarters of the table to four different Classes of books, as here shown.]

The Reader should carefully remember that, in such a phrase as “the x-Things,” the word “Things” means that particular
kind
of Things, to which the whole Diagram has been assigned.

[Thus, if we say “Let Univ.
be ‘books’,” we mean that we have assigned the whole Diagram to “books.”
In that case, if we took “
x
” to mean “old”, the phrase “the
x
-Things” would mean “the old books.”]

The Reader should not go on to the next Chapter until he is
quite familiar
with the
blank
Diagram I have advised him to draw.

He ought to be able to name,
instantly
, the
Adjunct
assigned to any Compartment named in the right-hand column of the following Table.

Also he ought to be able to name,
instantly
, the
Compartment
assigned to any Adjunct named in the left-hand column.

To make sure of this, he had better put the book into the hands of some genial friend, while he himself has nothing but the blank Diagram, and get that genial friend to question him on this Table,
dodging
about as much as possible.
The Questions and Answers should be something like this:—

TABLE I.

 Adjuncts 

 of 

 Classes. 

 Compartments, 

 or Cells, 

 assigned to them. 

 
x

 North 

Half.

 
x
′

 South 

〃

 
y

 West 

〃

 
y
′

 East 

〃

 
xy

 North -

West

Cell.

 
xy
′

 
〃
  

East

〃

 
x
′y

 South -

West

〃

 
x
′y′

 
〃
  

East

〃

Q.

“Adjunct for West Half?”

A.

“
y
.”

Q.

“Compartment for
xy
′
?”

A.

“North-East Cell.”

Q.

“Adjunct for South-West Cell?”

A.

“
x
′y
.”

&c., &c.

After a little practice, he will find himself able to do without the blank Diagram, and will be able to see it
mentally
(“in my mind’s eye, Horatio!”) while answering the questions of his genial friend.
When
this
result has been reached, he may safely go on to the next Chapter.