Venn's experience illustrates another characteristic of mathematical logic. It is necessary, in order to make our arguments conclusive, to devote great care to the elimination of difficulties which rarely occur. The White Knight--who was like Boole in being a pioneer of mathematical logic in this way, and yet seems to have held, like Boole, those philosophical opinions which would base logic on psychology--recognized the necessity of taking precautions against any unusual appearance of mice on a horse's back.[71] FOOTNOTES: [67] _S. L._, 1881, p. 339, note. [68] See Appendix N. [69] _Md._, N. S., vol. xiv., October 1905, pp. 479-93. [70] See Appendix N. [71] See Appendix O. CHAPTER XXVI IS _Is_ has four perfectly distinct meanings in English, besides misuses of the word. Among the misuses, perhaps the most important are those referred to by De Morgan:[72] "... We say 'murder _is_ death to the perpetrator' where the copula is _brings_; 'two and two _are_ four,' the copula being 'have the value of,' etc." Schröder[73] quite satisfactorily pointed out the well-known distinction between an _is_ where subject and predicate can be interchanged (such as: "the class whose members are Shem, Ham and Japhet is the class of the sons of Noah") and an _is_ or _are_ where they cannot (such as: Englishmen are Britons), but failed to see[74] the more important distinction (made by Peano) of is in the sense of "is a member of." If Englishmen are Britons, and Britons are civilized people, it follows that Englishmen are civilized people; but, though the _Harmsworth Encyclopædia_ is a member of the class Book (of one or more volumes), and this class is the member of a class A of which it is the only member, yet the _Harmsworth Encyclopædia_ is not a member of A, for it is not true that it is the whole class of books; and such a statement would not even be made except possibly in the form of an advertisement. The fourth meaning of _is_ is _exists_; it is in certain rare moods a matter for regret that there are difficulties in the way of using one word to denote four different things. For, if there were not, we might prove the existence of any thing we please by making it the subject of a proposition, and thereby earn the gratitude of theologians. FOOTNOTES: [72] _F. L._, p. 268. [73] _A. d. L._, i. pp. 127 sqq. [74] _Ibid._, vol. ii. pp. 461, 597. CHAPTER XXVII _AND_ AND _OR_