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”.]
[pg032]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.]