CHAPTER II.. _representation of Propositions of Relation._

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". (See p. 31.) Hence it may be represented by the expression "xy_{1}".

The Converse Proposition "Some y are x" may of course be represented by the same expression, viz. "xy_{1}".

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". (See p. 33.) Hence it may be represented by the expression "xy_{0}".

The Converse Proposition "No y are x" may of course be represented by the same expression, viz. "xy_{0}".

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 "x{1} + xy'{0}" represents this Double Proposition.

Hence it also represents the Proposition "All x are y".

This expression may be written in a shorter form, viz. "x{1}y'{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'".

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).

Again, the expression "x'{1}y'{0}" becomes "All x' are y", where the Predicate changes for y' to y.]

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