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". (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'".
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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".

[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. "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'".

[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 "y_{1}x_{0}",
where the Predicate changes from x' to x.

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

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