Complete Works of Lewis Carroll
Page 101
“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.
CHAPTER II.
COUNTERS.
Let us agree that a Red Counter, placed within a Cell, shall mean “This Cell is occupied” (i.e. “There is at least one Thing in it”).
Let us also agree that a Red Counter, placed on the partition between two Cells, shall mean “The Compartment, made up of these two Cells, is occupied; but it is not known whereabouts, in it, its occupants are.” Hence it may be understood to mean “At least one of these two Cells is occupied: possibly both are.”
Our ingenious American cousins have invented a phrase to describe the condition of a man who has not yet made up his mind which of two political parties he will join: such a man is said to be “sitting on the fence.” This phrase exactly describes the condition of the Red Counter.
Let us also agree that a Grey Counter, placed within a Cell, shall mean “This Cell is empty” (i.e. “There is nothing in it”).
[The Reader had better provide himself with 4 Red Counters and 5 Grey ones.]
CHAPTER III.
REPRESENTATION OF PROPOSITIONS.
§ 1.
Introductory.
Henceforwards, in stating such Propositions as “Some x-Things exist” or “No x-Things are y-Things”, I shall omit the word “Things”, which the Reader can supply for himself, and shall write them as “Some x exist” or “No x are y”.
[Note that the word “Things” is here used with a special meaning, as explained at p. 23.]
A Proposition, containing only one of the Letters used as Symbols for Attributes, is said to be ‘Uniliteral’.
[For example, “Some x exist”, “No y′ exist”, &c.]
A Proposition, containing two Letters, is said to be ‘Biliteral’.
[For example, “Some xy′ exist”, “No x′ are
y”, &c.]
A Proposition is said to be ‘in terms of’ the Letters it contains, whether with or without accents.
[Thus, “Some xy′ exist”, “No x′ are y”, &c., are said to be in terms of x and y.]
§ 2.
Representation of Propositions of Existence.
Let us take, first, the Proposition “Some x exist”.
[Note that this Proposition is (as explained at p. 12) equivalent to “Some existing Things are x-Things.”]
This tells us that there is at least one Thing in the North Half; that is, that the North Half is occupied. And this we can evidently represent by placing a Red Counter (here represented by a dotted circle) on the partition which divides the North Half.
[In the “books” example, this Proposition would be “Some old books exist”.]
Similarly we may represent the three similar Propositions “Some x′ exist”, “Some y exist”, and “Some y′ exist”.
[The Reader should make out all these for himself. In the “books” example, these Propositions would be “Some new books exist”, &c.]
Let us take, next, the Proposition “No x exist”.
This tells us that there is nothing in the North Half; that is, that the North Half is empty; that is, that the North-West Cell and the North-East Cell are both of them empty. And this we can represent by placing two Grey Counters in the North Half, one in each Cell.
[The Reader may perhaps think that it would be enough to place a Grey Counter on the partition in the North Half, and that, just as a Red Counter, so placed, would mean “This Half is occupied”, so a Grey one would mean “This Half is empty”.
This, however, would be a mistake. We have seen that a Red Counter, so placed, would mean “At least one of these two Cells is occupied: possibly both are.” Hence a Grey one would merely mean “At least one of these two Cells is empty: possibly both are”. But what we have to represent is, that both Cells are certainly empty: and this can only be done by placing a Grey Counter in each of them.
In the “books” example, this Proposition would be “No old books exist”.]
Similarly we may represent the three similar Propositions “No x′ exist”, “No y exist”, and “No y′ exist”.
[The Reader should make out all these for himself. In the “books” example, these three Propositions would be “No new books exist”, &c.]
Let us take, next, the Proposition “Some xy exist”.
This tells us that there is at least one Thing in the North-West Cell; that is, that the North-West Cell is occupied. And this we can represent by placing a Red Counter in it.
[In the “books” example, this Proposition would be “Some old English books exist”.]
Similarly we may represent the three similar Propositions “Some xy′ exist”, “Some x′y exist”, and “Some x′y′ exist”.
[The Reader should make out all these for himself. In the “books” example, these three Propositions would be “Some old foreign books exist”, &c.]
Let us take, next, the Proposition “No xy exist”.
This tells us that there is nothing in the North-West Cell; that is, that the North-West Cell is empty. And this we can represent by placing a Grey Counter in it.
[In the “books” example, this Proposition would be “No old English books exist”.]
Similarly we may represent the three similar Propositions “No xy′ exist”, “No x′y exist”, and “No x′y′ exist”.
[The Reader should make out all these for himself. In the “books” example, these three Propositions would be “No old foreign books exist”, &c.]
We have seen that the Proposition “No x exist” may be represented by placing two Grey Counters in the North Half, one in each Cell.
We have also seen that these two Grey Counters, taken separately, represent the two Propositions “No xy exist” and “No xy′ exist”.
Hence we see that the Proposition “No x exist” is a Double Proposition, and is equivalent to the two Propositions “No xy exist” and “No xy′ exist”.
[In the “books” example, this Proposition would be “No old books exist”.
Hence this is a Double Proposition, and is equivalent to the two Propositions “No old English books exist” and “No old foreign books exist”.]
§ 3.
Representation of Propositions of Relation.
Let us take, first, the Proposition “Some x are y”.
This tells us that at least one Thing, in the North Half, is also in the West Half. Hence it must be in the space common to them, that is, in the North-West Cell. Hence the North-West Cell is occupied. And this we can represent by placing a Red Counter in it.
[Note that the Subject of the Proposition settles which Half we are to use; and that the Predicate settles in which portion of it we are to place the Red Counter.
In the “books” example, this Proposition would be “Some old books are English”.]
Similarly we may represent the three similar Propositions “Some x are y′”, “Some x′ are y”, and “Some x′ are y′”.
[The Reader should make out all these for himself. In the “books” example, these three Propositions would be “Some old books are foreign”, &c.]
Let us take, next, the Proposition “Some y are x”.
This tells us that at least one Thing, in the West Half, is also in the North Half. Hence it must be in the space common to them, that is, in the North-West Cell. Hence the North-West Cell is occupied. And this we can represent by placing a Red Counter in it.
[In the “books” example, this Proposition would be “Some English books are old”.]
Similarly we may represent the three similar Propositions “Some y are x′”, “Some y′ are x”, and “Some y′ are x′”.
[The Reader should make out all these for himself. In the “books” example, these three Propositions would be “Some English books are new”, &c.]
We see that this one Diagram has now served to represent no less than three Propositions, viz.
(1) “Some xy exist;
(2) Some x are y;
(3) Some y are x”.
Hence these three Propositions are equivalent.
[In the “books” example, these Propositions would be
(1) “Some old English books exist;
(2) Some old books are English;
(3) Some English books are old”.]
The two equivalent Propositions, “Some x are y” and “Some y are x”, are said to be ‘Converse’ to each other; and the Process, of changing one into the other, is called ‘Converting’, or ‘Conversion’.
[For example, if we were told to convert the Proposition
“Some apples are not ripe,”
we should first choose our Univ. (say “fruit”), and then complete the Proposition, by supplying the Substantive “fruit” in the Predicate, so that it would be
“Some apples are not-ripe fruit”;
and we should then convert it by interchanging its Terms, so that it would be
“Some not-ripe fruit are apples”.]
Similarly we may represent the three similar Trios of equivalent Propositions; the whole Set of four Trios being as follows:—
(1) “Some xy exist” = “Some x are y” = “Some y are x”.
(2) “Some xy′ exist” = “Some x are y′” = “Some y′ are x”.
(3) “Some x′y exist” = “Some x′ are y” = “Some y are x′”.
(4) “Some x′y′ exist” = “Some x′ are y′” = “Some y′ are x′”.
Let us take, next, the Proposition “No x are y”.
This tell us that no Thing, in the North Half, is also in the West Half. Hence there is nothing in the space common to them, that is, in the North-West Cell. Hence the North-West Cell is empty. And this we can represent by placing a Grey Counter in it.
[In the “books” example, this Proposition would be “No old books are English”.]
Similarly we may represent
the three similar Propositions “No x are y′”, and “No x′ are y”, and “No x′ are y′”.
[The Reader should make out all these for himself. In the “books” example, these three Propositions would be “No old books are foreign”, &c.]
Let us take, next, the Proposition “No y are x”.
This tells us that no Thing, in the West Half, is also in the North Half. Hence there is nothing in the space common to them, that is, in the North-West Cell. That is, the North-West Cell is empty. And this we can represent by placing a Grey Counter in it.
[In the “books” example, this Proposition would be “No English books are old”.]
Similarly we may represent the three similar Propositions “No y are x′”, “No y′ are x”, and “No y′ are x′”.
[The Reader should make out all these for himself. In the “books” example, these three Propositions would be “No English books are new”, &c.]
We see that this one Diagram has now served to present no less than three Propositions, viz.
(1) “No xy exist;
(2) No x are y;
(3) No y are x.”
Hence these three Propositions are equivalent.
[In the “books” example, these Propositions would be
(1) “No old English books exist;
(2) No old books are English;
(3) No English books are old”.]
The two equivalent Propositions, “No x are y” and “No y are x”, are said to be ‘Converse’ to each other.
[For example, if we were told to convert the Proposition
“No porcupines are talkative”,
we should first choose our Univ. (say “animals”), and then complete the Proposition, by supplying the Substantive “animals” in the Predicate, so that it would be