In Part II. will be found some of the matters mentioned in this Appendix, viz., the "Existential Import" of Propositions, the use of a _negative_ Copula, and the theory that "two negative Premisses prove nothing." I shall also extend the range of Syllogisms, by introducing Propositions containing alternatives (such as "Not-all x are y"), Propositions containing 3 or more Terms (such as "All ab are c", which, taken along with "Some bc' are d", would prove "Some d are a'"), &c. I shall also discuss Sorites containing Entities, and the _very_ puzzling subjects of Hypotheticals and Dilemmas. I hope, in the course of Part II., to go over all the ground usually traversed in the text-books used in our Schools and Universities, and to enable my Readers to solve Problems of the same kind as, and far harder than, those that are at present set in their Examinations. In Part III. I hope to deal with many curious and out-of-the-way subjects, some of which are not even alluded to in any of the treatises I have met with. In this Part will be found such matters as the Analysis of Propositions into their Elements (let the Reader, who has never gone into this branch of the subject, try to make out for himself what _additional_ Proposition would be needed to convert "Some a are b" into "Some a are bc"), the treatment of Numerical and Geometrical Problems, the construction of Problems, and the solution of Syllogisms and Sorites containing Propositions more complex than any that I have used in Part II. I will conclude with eight Problems, as a taste of what is coming in Part II. I shall be very glad to receive, from any Reader, who thinks he has solved any one of them (more especially if he has done so _without_ using any Method of Symbols), what he conceives to be its complete Conclusion. It may be well to explain what I mean by the _complete_ Conclusion of a Syllogism or a Sorites. I distinguish their Terms as being of two kinds----those which _can_ be eliminated (e.g. the Middle Term of a Syllogism), which I call the "Eliminands," and those which _cannot_, which I call the "Retinends"; and I do not call the Conclusion _complete_, unless it states _all_ the relations among the Retinends only, which can be deduced from the Premisses. pg186 1. All the boys, in a certain School, sit together in one large room every evening. They are of no less than _five_ nationalities----English, Scotch, Welsh, Irish, and German. One of the Monitors (who is a great reader of Wilkie Collins' novels) is very observant, and takes MS. notes of almost everything that happens, with the view of being a good sensational witness, in case any conspiracy to commit a murder should be on foot. The following are some of his notes:-- (1) Whenever some of the English boys are singing "Rule Britannia", and some not, some of the Monitors are wide-awake;