Similarly we may represent the three similar Pairs of Converse Propositions, viz.—
“No x are y′” = “No y′ are x”,
“No x′ are y” = “No y are x′”,
“No x′ are y′” = “No y′ are x′”.
[pg072]Let us take, next, the Proposition “All x are y”.
Now it is evident that the Double Proposition of Existence “Some x exist and no xy′ exist” tells us that some x-Things exist, but that none of them have the Attribute y′: that is, it tells us that all of them have the Attribute y: that is, it tells us that “All x are y”.
Also it is evident that the expression “x1 † xy′0” represents this Double Proposition.
Hence it also represents the Proposition “All x are y”.
[The Reader will perhaps be puzzled by the statement that the Proposition “All x are y” is equivalent to the Double Proposition “Some x exist and no xy′ exist,” remembering that it was stated, at [p. 33], to be equivalent to the Double Proposition “Some x are y and no x are y′” (i.e. “Some xy exist and no xy′ exist”). The explanation is that the Proposition “Some xy exist” contains superfluous information. “Some x exist” is enough for our purpose.]
This expression may be written in a shorter form, viz. “x1y′0”, since each Subscript takes effect back to the beginning of the expression.
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′”.
[The Reader should make out all these for himself.]