The relation in question is one which logicians have sometimes expressed by "_p_ implies _q_." It is, for instance, the one which Mr. Russell in the _'Principles of Mathematics_ calls "material implication," and which he and Dr. Whitehead in _Principia Mathematica_ call simply "implication." And if we do use "implication" to stand for this relation, we, of course, got the apparently paradoxical results that every false proposition implies every other proposition, both true and false, and that every true proposition implies every other true proposition: since it is quite clear that if _p_ is false then, whatever _q_ may be, "it is not the case that _p_ is true and _q_ false," and quite clear also, that if _p_ and _q_ are both true, then also "it is not the case that _p_ is true and _q_ false." And these results, it seems to me, appear to be paradoxical, solely because, if we use "implies" in any ordinary sense, they are quite certainly false. Why logicians should have thus chosen to use the word "implies" as a name for a relation, for which it never is used by any one else, I do not know. It is partly, no doubt, because the relation for which they do use it--that expressed by saying "It is not the case that _p_ is true and _q_ false"--is one for which it is very important that they should have a short name, because it is a relation which is very fundamental and about which they need constantly to talk, while (so far as I can discover) it simply has no short name in ordinary life. And it is partly, perhaps, for a reason which leads us back to our present reason for giving some name to this relation. It is, in fact, natural to use "_p_ implies _q_" to mean the same as "If _p,_ then _q."_ And though "If _p_ then _q_" is hardly ever, if ever, used to mean the same as "It is not the case that _p_ is true and _q_ false"; yet the expression "If _anything_ has Q, _it_ has R" may, I think, be naturally used to express the proposition that, in the case of _every_ pair of propositions which resembles the pair A Q and A R in respect of the fact that the first of the pair asserts of some particular thing that it has Q and the second, of the same thing, that it has R, it is not the case that the first is true and the second false. That is to say, if (as I propose to do) we express "It is not the case both that AQ is true and AR false" by AQ * AR, and if, further (on the analogy of the similar case with regard to "entails)," we express the proposition that of _every_ pair of propositions which resemble A Q and A R in the respect just mentioned, it is true that the first has the relation * to the second by _x_Q * _x_R