Lewis Carroll supposed that "_q_ implies _r_" and "_q_ implies not-_r_" are inconsistent, and hence that _p_ must be false. But these propositions are not inconsistent, and are, in fact, both true if _q_ is false. The contradictory of "_q_ implies _r_" is "_q_ does not imply _r_" which is not a consequence of "_q_ implies not-_r_." It seems to be true theoretically that, if Mr. X is a Christian, he is not an Atheist, but we cannot conclude from this alone that his being a Christian does not imply that he is an Atheist, unless we assume that the class of Christians is not null. Thus, if _p_ is true, _q_ is false; or, if Carr is out, Allen is in. The odd part of this conclusion is that it is the one which common-sense would have drawn in that particular case. A distinguished philosopher (M) once thought that the logical use of the word "implication"--any false proposition being said to "imply" any proposition true or false--is absurd, on the grounds that it is ridiculous to suppose that the proposition "2 and 2 make 5" implies the proposition "M is the Pope." This is a most unfortunate instance, because it so happens that the false proposition that 2 and 2 make 5 can rigorously be proved to imply that M, or anybody else other than the Pope, is the Pope. For if 2 and 2 make 5, since they also make 4, we would conclude that 5 is equal to 4. Consequently, subtracting 3 from both sides, we conclude that 2 would be equal to 1. But if this were true, since M and the Pope are two, they would be one, and obviously then M would be the Pope. The principle that the false implies the true has very important applications in political arguments. In fact, it is hard to find a single principle of politics of which false propositions are not the main support. If _p_ and _q_ are two propositions, and _p_ implies _q_; then, if, and only if, _q_ and _p_ are both false or both true, we also have: _q_ implies _p_. The most important applications of this invertibility were made by the late Samuel Butler[52] and Mr. G. B. Shaw. A political application may be made as follows: In a country where only those with middling-sized incomes are taxed, conservative and _bourgeois_ politicians would still maintain that the proposition "the rich are taxed" implies the proposition "the poor are taxed," and this implication, which is true because both premiss and conclusion are false, would be quite unnecessarily supported by many false practical arguments. It is equally true that "the poor are taxed" implies that "the rich are taxed." And this can be proved, in certain cases, on other grounds. For the taxation of the poor would imply, ultimately, that the poor could not afford to pay a little more for the necessities of life than, in strict justice, they ought; and this would mean the cessation of one of the chief means of production of individual wealth.