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