(_t_). not (_x_). [Greek: ph] (_x_, _t_); which can be thus read: "If at every instant of his life there was at least one person _x_ to whom he did no wrong (at that instant)." It is difficult to imagine any one so sunk in iniquity that he would not satisfy this hypothesis. We are forced, then, unless our imagination for evil is to be distrusted, to conclude that any one might have been there to have heard that song. Now this conclusion is plainly false, possibly on physical grounds, and certainly on æsthetic grounds. It may be added, by the way, that it is quite possible that De Morgan was mistaken in his interpretation of the above proposition owing to the fact that he was unacquainted with Frege's work. In fact, if he had not noticed the fact that _any_ two of the "not's" cannot be cancelled against one another he would have concluded that the interpretation was: "If he had never done any wrong to anybody." According as the symbol for "not" comes before the (_x_) or between the (_x_) and the [Greek: ph], we have an expression of what Frege called respectively the denial of generality, and the generality of denial. The denial of the generality of a denial is the form of all existential propositions, while the assertion of or denial of generality is the general form of all non-existential or universal propositions. FOOTNOTES: [48] To which De Morgan drew attention in a letter; see (Mrs.) S. E. De Morgan, _Memoir of Augustus De Morgan_, London, 1882, p. 324. [49] _Pa. Ma._, p. 16. [50] However, here, for the printer's convenience, we depart from Mr. Russell's usage so far as to write "not" for a curly minus sign. CHAPTER XIX IMPLICATION A good illustration of the fact that what is called "implication" in logic is such that a false proposition implies any other proposition, true or false, is given by Lewis Carroll's puzzle of the three barbers.[51] Allen, Brown, and Carr keep a barber's shop together; so that one of them must be in during working hours. Allen has lately had an illness of such a nature that, if Allen is out, Brown must be accompanying him. Further, if Carr is out, then, if Allen is out, Brown must be in for obvious business reasons. The problem is, may Carr ever go out? Putting _p_ for "Carr is out," _q_ for "Allen is out" and _r_ for "Brown is out," we have: (1) _q_ implies _r_, (2) _p_ implies that _q_ implies not-_r_.