The Mathematical Works of Lewis Carroll. Ð›ÑŒÑŽÐ¸Ñ ÐšÑрролл
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.”