A logician on holiday in the South Seas finds himself on an island inhabited by the two proverbial tribes of liars and truth-tellers. Members of one tribe always tell the truth, members of the other always lie. The logician comes to a fork in the road and has to ask a native bystander which road he should take to reach the village. He has no way of telling whether the native is a truth-teller or a liar. The logician thinks a moment, then asks one question only. From the reply he knows which road to take. What question does he ask.
If we require that the question be answerable by 'Yes' or 'No', there are several solutions, all exploiting the same basic gimmick. For example, the logician points to one of the roads and say's to the native, 'If I were to ask you if this road leads to the village, would you say "Yes"?' The native is forced to give the right answer, even if he is a liar! If the road does lead to the village, the liar would say 'No' to the direct question, but as the question is put, he lies and says he would respond 'Yes'. Thus the logician can be certain that the road does lead to the village, whether the respondant is a truth teller or a liar. On the other hand if the road does not lead to the village, the liar is forced in the same way to reply 'No' to the inquirer's question.
It is a sad commentary on the rise of logic that it leads to the decay of the art of lying. Even among liars, the life of reason seems to be gaining ground over the better life. We refer to puzzle no. 4 in the February issue, and it's solution. If we accept the proposed solution, we must believe that liars can always be made the dupes of their own principles, a situation, indeed, which is bound to arise whenever lying takes the form of slavish adherance to arbitrary rules.
For the anthropologist to say to the native, 'If I were to ask you if this road leads to the village, would you say "Yes"?' expecting him to interpret this question as counter-factual conditional in meaning as well as form, presupposes a certain preciosity on the part of the native. If the anthropologist asks the question casually, the native is almost certain to mistake the odd phraseology for some civility of manner taught in Western democracies, and answer as if the question were simply, 'Does this road lead to the village?' On the other hand, if he fixes him with a glittering eye in order to emphasize the logical intent of the question, he also reveals it's purpose, arousing the natives suspicion that he is being tricked. The native, if he is worthy the name of liar, will pursue a method of counter-trickery, leaving the antropologist misinformed. On this latter view, the proposed solution is inadequate, but even in terms of strictly formal lying, it is faulty because of its ambiguity.
The investigation of unambiguous solutions leads us to a more detailed analysis of the nature of lying. The traditional definition employed by logicians is that a liar is one who always says what is false. The ambiguity of this definition appears when we try to predict what a liar will answer to a compound truth functional question, such as 'Is it true that if this is the way to town, you are a liar?' Will he evaluate the two components correctly in order to evaluate the function and reverse his evaluation in the telling, or will he follow the impartial policy of lying to himself as well as to others, reversing the evaluation of each component before computing the value of the function, and then reversing the computed value of the function? Here we distinguish the simple liar who always utters what is simply false from the honest liar who always utters the logical dual of the truth.
The question, 'Is it true that if this is the way to town, you are a liar?' is a solution if our liars are honest liars. The honest liar and the truth-teller both answer 'Yes' if the indicated road is not the way to town, and 'No' if it is. The simple liar however, will answer 'No' redardless of where the village is. By substituting equivelance for implication we obtain a solution which works for both simple and honest liars. The question becomes, 'Is it true that this is the way to town if and only if you are a liar?' The answer is unformly 'No' if it is the way, and 'Yes' if it not.
But no lying primitive savage could be expected to display the scrupulous consistency required by these conceptions, nor would any liar capable of such acumen be so easily outwitted. We must therefore consider the case of the artistic liar whose principle is always to deceive. against such an opponent the anthropologist can only hope to maximize the probability of a favourable outcome. No logical question can be an infallible solution, for if the liar's principle is to deceive, he will counter with a strategy of deception which circumvents logic. Clearly the essential feature of the anthropologist's strategy must be it's psychological soundness. Such a strategy is admissible since it is even more effective against the honest and the simple liar than against the more fractory artistic liar.
We therefore propose as the most general solution the following question or it's moral equivelent, 'Did you know that they are serving free beer in the village?' The truth-teller answers 'No' and immediately sets off for the village, the anthropologist following. The simple or honest liar answers 'Yes' and sets off for the village. The artistic liar making the polite assumption that the anthropologist is also devoted to trickery, chooses his strategy accordingly. Confronted with two contrary motives, he may persue the chance of satisfying both of them by answering, 'Ugh! I hate beer!' and starting for the village. This will not confuse a good anthropologist. But if the liar sees through the ruse, he will recognise the inadequacy of the response. He may then make the supreme sacrafice for the sake of art and start down the wrong road. He achieves a technical victory, but even so, the anthropologist may claim a moral victory, for the liar is punished by the gnawing suspicion that he has missed some free beer.
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Taken from the Book:
"Mathematical Puzzles and Diversions"
by Martin Gardner.
Penguin Books - 1965.
Previously appeared in Scientific American.
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Richard Dalton.
Web: http://www.iol.ie/~dalton
Email: dalton@iol.ie