Complete Works of Lewis Carroll
Page 102
“No porcupines are talkative animals”, and we should then convert it, by interchanging its Terms, so that it would be
“No talkative animals are porcupines”.]
Similarly we may represent the three similar Trios of equivalent Propositions; the whole Set of four Trios being as follows:—
(1) “No xy exist” = “No x are y” = “No y are x”.
(2) “No xy′ exist” = “No x are y′” = “No y′ are x”.
(3) “No x′y exist” = “No x′ are y” = “No y are x′”.
(4) “No x′y′ exist” = “No x′ are y′” = “No y′ are x′”.
Let us take, next, the Proposition “All x are y”.
We know (see p. 17) that this is a Double Proposition, and equivalent to the two Propositions “Some x are y” and “No x are y′”, each of which we already know how to represent.
[Note that the Subject of the given Proposition settles which Half we are to use; and that its Predicate settles in which portion of that Half we are to place the Red Counter.]
TABLE II.
Some x exist
No x exist
Some x′ exist
No x′ exist
Some y exist
No y exist
Some y′ exist
No y′ exist
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′”.
Let us take, lastly, the Double Proposition “Some x are y and some are y′”, each part of which we already know how to represent.
Similarly we may represent the three similar Propositions, “Some x′ are y and some are y′”, “Some y are x and some are x′”, “Some y′ are x and some are x′”.
The Reader should now get his genial friend to question him, severely, on these two Tables. The Inquisitor should have the Tables before him: but the Victim should have nothing but a blank Diagram, and the Counters with which he is to represent the various Propositions named by his friend, e.g. “Some y exist”, “No y′ are x”, “All x are y”, &c. &c.
TABLE III.
Some xy exist
= Some x are y
= Some y are x
All x are y
Some xy′ exist
= Some x are y′
= Some y′ are x
All x are y′
Some x′y exist
= Some x′ are y
= Some y are x′
All x′ are y
Some x′y′ exist
= Some x′ are y′
= Some y′ are x′
All x′ are y′
No xy exist
= No x are y
= No y are x
All y are x
No xy′ exist
= No x are y′
= No y′ are x
All y are x′
No x′y exist
= No x′ are y
= No y are x′
All y′ are x
No x′y′ exist
= No x′ are y′
= No y′ are x′
All y′ are x′
Some x are y,
and some are y′
Some y are x
and some are x′
Some x′ are y,
and some are y′
Some y′ are x
and some are x′
CHAPTER IV.
INTERPRETATION OF BILITERAL DIAGRAM WHEN MARKED WITH COUNTERS.
The Diagram is supposed to be set before us, with certain Counters placed upon it; and the problem is to find out what Proposition, or Propositions, the Counters represent.
As the process is simply the reverse of that discussed in the previous Chapter, we can avail ourselves of the results there obtained, as far as they go.
First, let us suppose that we find a Red Counter placed in the North-West Cell.
We know that this represents each of the Trio of equivalent Propositions
“Some xy exist” = “Some x are y” = “Some y are x”.
Similarly we may interpret a Red Counter, when placed in the North-East, or South-West, or South-East Cell.
Next, let us suppose that we find a Grey Counter placed in the North-West Cell.
We know that this represents each of the Trio of equivalent Propositions
“No xy exist” = “No x are y” = “No y are x”.
Similarly we may interpret a Grey Counter, when placed in the North-East, or South-West, or South-East Cell.
Next, let us suppose that we find a Red Counter placed on the partition which divides the North Half.
We know that this represents the Proposition “Some x exist.”
Similarly we may interpret a Red Counter, when placed on the partition which divides the South, or West, or East Half.
Next, let us suppose that we find two Red Counters placed in the North Half, one in each Cell.
We know that this represents the Double Proposition “Some x are y and some are y′”.
Similarly we may interpret two Red Counters, when placed in the South, or West, or East Half.
Next, let us suppose that we find two Grey Counters placed in the North Half, one in each Cell.
We know that this represents the Proposition “No x exist”.
Similarly we may interpret two Grey Counters, when placed in the South, or West, or East Half.
Lastly, let us suppose that we find a Red and a Grey Counter placed in the North Half, the Red in the North-West Cell, and the Grey in the North-East Cell.
We know that this represents the Proposition, “All x are y”.
[Note that the Half, occupied by the two Counters, settles what is to be the Subject of the Proposition, and that the Cell, occupied by the Red Counter, settles what is to be its Predicate.]
Similarly we may interpret a Red and a Grey counter, when placed in any one of the seven similar positions
Red in North-East, Grey in North-West;
Red in South-West, Grey in South-East;
Red in South-East, Grey in South-West;
Red in North-West, Grey in South-West;
Red in South-West, Grey in North-West;
Red in North-East, Grey in South-East;
Red in South-East, Grey in North-East.
Once more the genial friend must be appealed to, and requested to examine the Reader on Tables II and III, and to make him not only represent Propositions, but also interpret Diagrams when marked with Counters.
The Questions and Answers should be like this:—
Q. Represent “No x′ are y′.”
A. Grey Counter in S.E. Cell.
Q. Interpret Red Counter on E. partition.
A. “Some y′ exist.”
Q. Represent “All y′ are x.”
A. Red in N.E. Cell; Grey in S.E.
Q. Interpret Grey Counter in S.W. Cell.
A. “No x′y exist” = “No x′ are y” = “No y are x′”.
&c., &c.
At first the Examinee will need to have the Board and Counters before him; but he will soon learn to dispense with these, and to answer with his eyes shut or gazing into vacancy.
[Work Examples § 1, 5–8 (p. 97).]
BOOK IV.
THE TRILITERAL DIAGRAM.
CHAPTER I.
SYMBOLS AND CELLS.
First, let us suppose that the above left-hand Diagram is the Biliteral Diagram that we have been using in Book III., and that we change it into a Triliteral Diagram by drawing an Inner Square, so as to divide each of its 4 Cells into 2 portions, thus making 8 Cells altogether. The right-hand Diagram shows the result.
[The Reader is strongly advised, in reading this Chapter, not to refer to the above Diagrams, but to make a large copy of the right-hand 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.]
&nbs
p; Secondly, let us suppose that we have selected a certain Adjunct, which we may call “m”, and have subdivided the xy-Class into the two Classes whose Differentiæ are m and m′, and that we have assigned the N.W. Inner Cell to the one (which we may call “the Class of xym-Things”, or “the xym-Class”), and the N.W. Outer Cell to the other (which we may call “the Class of xym′-Things”, or “the xym′-Class”).
[Thus, in the “books” example, we might say “Let m mean ‘bound’, so that m′ will mean ‘unbound’”, and we might suppose that we had subdivided the Class “old English books” into the two Classes, “old English bound books” and “old English unbound books”, and had assigned the N.W. Inner Cell to the one, and the N.W. Outer Cell to the other.]
Thirdly, let us suppose that we have subdivided the xy′-Class, the x′y-Class, and the x′y′-Class in the same manner, and have, in each case, assigned the Inner Cell to the Class possessing the Attribute m, and the Outer Cell to the Class possessing the Attribute m′.
[Thus, in the “books” example, we might suppose that we had subdivided the “new English books” into the two Classes, “new English bound books” and “new English unbound books”, and had assigned the S.W. Inner Cell to the one, and the S.W. Outer Cell to the other.]
It is evident that we have now assigned the Inner Square to the m-Class, and the Outer Border to the m′-Class.
[Thus, in the “books” example, we have assigned the Inner Square to “bound books” and the Outer Border to “unbound books”.]
When the Reader has made himself familiar with this Diagram, he ought to be able to find, in a moment, the Compartment assigned to a particular pair of Attributes, or the Cell assigned to a particular trio of Attributes. The following Rules will help him in doing this:—
(1) Arrange the Attributes in the order x, y, m.
(2) Take the first of them and find the Compartment assigned to it.
(3) Then take the second, and find what portion of that compartment is assigned to it.
(4) Treat the third, if there is one, in the same way.
[For example, suppose we have to find the Compartment assigned to ym. We say to ourselves “y has the West Half; and m has the Inner portion of that West Half.”
Again, suppose we have to find the Cell assigned to x′ym′. We say to ourselves “x′ has the South Half; y has the West portion of that South Half, i.e. has the South-West Quarter; and m′ has the Outer portion of that South-West Quarter.”]
The Reader should now get his genial friend to question him on the Table given on the next page, in the style of the following specimen-Dialogue.
Q.
Adjunct for South Half, Inner Portion?
A.
x′m.
Q.
Compartment for m′?
A.
The Outer Border.
Q.
Adjunct for North-East Quarter, Outer Portion?
A.
xy′m′.
Q.
Compartment for ym?
A.
West Half, Inner Portion.
Q.
Adjunct for South Half?
A.
x′.
Q.
Compartment for x′y′m?
A.
South-East Quarter, Inner Portion.
&c. &c.
TABLE IV.
Adjunct
of
Classes.
Compartments,
or Cells,
assigned to them.
x
North
Half.
x′
South
〃
y
West
〃
y′
East
〃
m
Inner
Square.
m′
Outer
Border.
xy
North-
West
Quarter.
xy′
〃
East
〃
x′y
South-
West
〃
x′y′
〃
East
〃
xm
North
Half,
Inner
Portion.
xm′
〃
〃
Outer
〃
x′m
South
〃
Inner
〃
x′m′
〃
〃
Outer
〃
ym
West
〃
Inner
〃
ym′
〃
〃
Outer
〃
y′m
East
〃
Inner
〃
y′m′
〃
〃
Outer
〃
xym
North-
West
Quarter,
Inner
Portion.
xym′
〃
〃
〃
Outer
〃
xy′m
〃
East
〃
Inner
〃
xy′m′
〃
〃
〃
Outer
〃
x′ym
South-
West
〃
Inner
〃
x′ym′
〃
〃
〃
Outer
〃
x′y′m
〃
East
〃
Inner
〃
x′y′m′
〃
〃
〃
Outer
〃
CHAPTER II.
REPRESENTATION OF PROPOSITIONS IN TERMS OF x AND m, OR OF y AND m.
§ 1.
Representation of Propositions of Existence in terms of x and m, or of y and m.
Let us take, first, the Proposition “Some xm exist”.
[Note that the full meaning of this Proposition is (as explained at p. 12) “Some existing Things are xm-Things”.]
This tells us that there is at least one Thing in the Inner portion of the North Half; that is, that this Compartment is occupied. And this we can evidently represent by placing a Red Counter on the partition which divides it.
[In the “books” example, this Proposition would mean “Some old bound books exist” (or “There are some old bound books”).]
Similarly we may represent the seven similar Propositions, “Some xm′ exist”, “Some x′m exist”, “Some x′m′ exist”, “Some ym exist”, “Some ym′ exist”, “Some y′m exist”, and “Some y′m′ exist”.
Let us take, next, the Proposition “No xm exist”.
This tells us that there is nothing in the Inner portion of the North Half; that is, that this Compartment is empty. And this we can represent by placing two Grey Counters in it, one in each Cell.
Similarly we may represent the seven similar Propositions, in terms of x and m, or of y and m, viz. “No xm′ exist”, “No x′m exist”, &c.
These sixteen Propositions of Existence are the only ones that we shall have to represent on this Diagram.
§ 2.
Representation of Propositions of Relation in terms of x and m, or of y and m.
Let us take, first, the Pair of Converse Propositions
“Some x are m” = “Some m are x.”
We know that each of these is equivalent to the Proposition of Existence “Some xm exist”, which we already know h
ow to represent.
Similarly for the seven similar Pairs, in terms of x and m, or of y and m.
Let us take, next, the Pair of Converse Propositions
“No x are m” = “No m are x.”
We know that each of these is equivalent to the Proposition of Existence “No xm exist”, which we already know how to represent.
Similarly for the seven similar Pairs, in terms of x and m, or of y and m.
Let us take, next, the Proposition “All x are m.”
We know (see p. 18) that this is a Double Proposition, and equivalent to the two Propositions “Some x are m” and “No x are m′ ”, each of which we already know how to represent.