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
“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.