CHAPTER II.

REPRESENTATION OF PROPOSITIONS OF RELATION.

Let us take, first, the Proposition “Some x are y”.

This, we know, is equivalent to the Proposition of Existence “Some xy exist”.  Hence it may be represented by the expression “xy1”.

The Converse Proposition “Some y are x” may of course be represented by the same expression, viz. xy1”.

Similarly we may represent the three similar Pairs of Converse Propositions, viz.—

“Some x are y”  = “Some y are x”,

“Some x are y”  = “Some y are x”,

“Some x are y” = “Some y are x”.

Let us take, next, the Proposition “No x are y”.

This, we know, is equivalent to the Proposition of Existence “No xy exist”.  Hence it may be represented by the expression “xy0”.

The Converse Proposition “No y are x” may of course be represented by the same expression, viz. xy0”.

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”.

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 “x1xy0” 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. x1y0”, 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.]

It will be convenient to remember that, in translating a Proposition, beginning with “All”, from abstract form into subscript form, or vice versâ, the Predicate changes sign (that is, changes from positive to negative, or else from negative to positive).

[Thus, the Proposition “All y are x” becomes “y1x0”, where the Predicate changes from x to x.

Again, the expression “x1y0” becomes “All x are y”, where the Predicate changes for y to y.]