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.Now, hisstatement is either true or false.If his statement is true, thenhe really is human, but the only humans who make true statements are sane humans so in this case he is sane.If, on theother hand, his statement is false, then he is really a vampire,but the only vampires who make false statements are sanevampires (insane vampires make true statements, just likesane humans), so again he is sane.This proves that when aTransylvanian claims to be human, he must be sane, regardless of whether he is really human or not.Suppose a Transylvanian claims to be a vampire; what follows? Well, if his claim is true, then he really is a vampire,but the only vampires who make true claims are insane vampires.If his claim is false, then he is in fact human, but theonly humans who make false claims are insane humans; so inthis case he is also insane.Thus, any Transylvanian whoclaims to be a vampire is insane.We trust that the reader can verify for himself the fact thatany Transylvanian who claims to be sane must in fact behuman, and any Transylvanian who claims to be insane mustin fact be a vampire.Now let us turn to the solutions of the problems.1 Lucy's statement is either true or false.If it is true, thenboth sisters are really insane; hence Lucy is insane, and the53 THE LADY OR THE TI GER?only insane Transylvanian who can make a true statement isan insane vampire.So, if Lucy's statement is true, then Lucyis a vampire.Suppose Lucy's statement is false.Then at least one of thesisters is sane.If Lucy is sane, then, since she has made a falsestatement, she must be a vampire (because sane humansmake only true statements).Suppose Lucy is insane.Then itmust be Minna who is sane.Also, Minna, by contradictingLucy's false statement, has made a true statement.Therefore,Minna is sane and has made a true statement; so Minna ishuman, and again Lucy must be the vampire.This proves that regardless of whether Lucy's statement istrue or false, Lucy is the vampire.2 " We have already established the principle that anyTransylvanian who says he is human must be sane and anyTransylvanian who says he is a vampire must be insane (seeprefacing the solutions).Now, both the Lugosibrothers claim to be human; therefore, they are both sane.Therefore, Bela the Elder makes a true statement when hesays that his brother is sane.So Bela the Elder is both saneand makes true statements; hence he is human.Therefore, itis Bela the Younger who is the vampire.3 " Since Michael claims to be a vampire, he is insane, andsince Peter claims to be human, he is sane.So Michael is insane and Peter is sane; thus the two brothers are not alike asfar as their sanity goes.Therefore, Michael's second statement is false, and since Michael is insane, he must be human(insane vampires don't make false statements!).Therefore,Peter is the vampire.4 " Father and son agree in answering the question abouttheir sanity.This means that they either both make true54 INSPECTOR CRAI G VI S ITS TRANSYL VANIAstatements or both make false statements.But, since only oneof them is human and the other is a vampire, they must necessarily be different as regards their sanity: If they are bothsane, the one who is human would make true statements andthe vampire would make false statements, and they couldnever agree; if they are both insane, the human would makefalse statements and the vampire would make true stateents, and again they could not agree.Therefore, it is reallymtrue that at least one of them is insane.This proves that bothof them make true statements.Then, since the father says heis not a vampire, he really isn't.So it is the son who is thevampire.5 " Suppose Martha is the vampire.Then Karl is human, andalso Karl has made a true statement; hence Karl in this casehas to be a sane human.This would make Martha an insanevampire, since, as we have been told, Karl and Martha aredifferent as regards their sanity.But then Martha, an insanevampire, would have made a false statement-that Karl isinsane-which insane vampires cannot do.Therefore, the assumption that Martha is a vampire leads to a contradiction.So it is Karl who is the vampire.We can also determine their sanity or lack of it: Karl hasmade a false statement; hence, being a vampire, he is sane.But then Martha has also made a false statement; hence,being human, she is insane.So the complete answer is thatKarl is a sane vampire and Martha is an insane human; Karl islying when he says that his sister is a vampire, and Martha isdeluded when she says that her brother is insane.(Quite apair, even for Transylvania!)6 " Now we are in the situation where either both are vampires or both are human.Therefore, the first two statementstcannot bo h be right, nor can they both be wrong (for if they55 T HE LADY OR T HE T IGER?are both wrong, Sylvan would be a vampire and Sylvia wouldbe human).So one of the two statements is right and one istwrong.This means that one of the two people is sane and heother insane (because if.they were both sane, their statementswould both be right if they were human, and both wrong ifthey were vampires).Therefore, Sylvia is right when she saysthat one of the two is sane and the other insane.This meansthat Sylvia makes true statements.Therefore her statementthat her husband is human is true.This means that they areboth human (and, incidentally, SylVia is sane and Sylvan insane).7 " Gloria, in saying that whatever her husband says is true,is assenting to his claim that she is insane; in other words,Gloria is indirectly claiming to be insane.Only vampires canmake such a claim (as we proved in the discussion precedingthe solutions); hence Gloria must be a vampire.Therefore,they are both vampires.8 " Suppose.Uust as inFergusson's system of the last chapter).We shall call a number n an index of a nameable set A if A = An- (Thus, for example, if the sets Az, AT and A13 all happen to be the same,then 2, 7, and 13 are all indices of this set.) As with Fergusson's system, we have associated with every number x andevery number y a certain sentence-written x E Ay-which iscalled t e if x belongs to Ay and false if x doesn't belong toAy- We no longer assume, however, that the sentences x E Ayare the only sentences of the system; there may be others.Butt Composite translation
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