"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". [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.] pg073 CHAPTER III. _SYLLOGISMS._ § 1. _Representation of Syllogisms._ We already know how to represent each of the three Propositions of a Syllogism in subscript form. When that is done, all we need, besides, is to write the three expressions in a row, with "+" between the Premisses, and "¶" before the Conclusion. [Thus the Syllogism "No x are m'; All m are y. .'. No x are y'." may be represented thus:-- xm'_{0} + m_{1}y'_{0} ¶ xy'_{0} When a Proposition has to be translated from concrete form into subscript form, the Reader will find it convenient, just at first, to translate it into _abstract_ form, and _thence_ into subscript form. But, after a little practice, he will find it quite easy to go straight from concrete form to subscript form.] pg074 § 2.