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l. Suppose there is a small town that consists only of men. There are two kinds of men in this town--those who shave themselves and those who are shaved by the barber. Who shaves the barber? If he shaves himself, then he is shaved by the barber. But if he is shaved by the barber, then he shaves himself. If the barber is assumed to be in one set, he appears in the other. This situation occurs because the barber both appears in the set and is used to define the set.

2. A person from the island of Crete asserts, "All Cretans are liars." We can conclude that if he is telling the truth, then he is lying. But if he is lying, then he is telling the truth. Once again an element of the set is referring to the set.

3. Consider a businessman accused of accepting a bribe. He claims, "I did not take the bribe." There are two possible interpretations of this statement. Either he is a knowledgeable observer making a correct statement, or he is a knowledgeable observer lying to avoid going to jail. The businessman is both the observer and the person being observed. We have no way of knowing which role he is playing.

As the third example indicates, paradox leads to "undecidability". When two equally correct interpretations are possible, in the absence of further information, no decision other than a random choice is possible. (Umpleby)

From the Greek para + dokein, "to think more", conventionally, an argument that apparently derives self-contradictory conclusions by valid deductions from acceptable premises (Webster's). More generally, any description or situation that is compelling enough to lead an observer into a vicious cycle involving mutually exclusive interpretations, indications or acts, force him to step out of or transcend the cycle and to construct a logically more powerful (

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