And _you_ will reply "But suppose there was a mad bull behind you?" And then your innocent friend will say "Hum! Ha! I must think that over a bit!" You may then explain to him, as a convenient _test_ of the soundness of a Syllogism, that, if circumstances can be invented which, without interfering with the truth of the _Premisses_, would make the _Conclusion_ false, the Syllogism _must_ be unsound.] [Review Tables V-VIII (pp. 46-49). Work Examples § =4=, 7-12 (p. 100); § =5=, 7-12 (p. 101); § =6=, 1-10 (p. 106); § =7=, 1-6 (pp. 107, 108).] pg070 BOOK VI. THE METHOD OF SUBSCRIPTS. CHAPTER I. _INTRODUCTORY._ Let us agree that "x_{1}" shall mean "Some existing Things have the Attribute x", i.e. (more briefly) "Some x exist"; also that "xy_{1}" shall mean "Some xy exist", and so on. Such a Proposition may be called an '=Entity=.' [Note that, when there are _two_ letters in the expression, it does not in the least matter which stands _first_: "xy_{1}" and "yx_{1}" mean exactly the same.] Also that "x_{0}" shall mean "No existing Things have the Attribute x", i.e. (more briefly) "No x exist"; also that "xy_{0}" shall mean "No xy exist", and so on. Such a Proposition may be called a '=Nullity='. Also that "+" shall mean "and". [Thus "ab_{1} + cd_{0}" means "Some ab exist and no cd exist".] Also that "¶" shall mean "would, if true, prove". [Thus, "x_{0} ¶ xy_{0}" means "The Proposition 'No x exist' would, if true, prove the Proposition 'No xy exist'".] When two Letters are both of them accented, or both _not_ accented, they are said to have '=Like Signs=', or to be '=Like=': when one is accented, and the other not, they are said to have '=Unlike Signs=', or to be '=Unlike='. pg071 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.--