Philosophers and mathematicians often have a special knack for complicating things. A prime example is the Sleeping Beauty Problem, a twist that turns a children's tale into an adult nightmare. Keep reading to find out why witches shouldn't have coins, why princes no longer kiss as they used to, and—above all—how a couple of brainiacs can find a paradox inside a children's story. Not to be missed.
Everyone knows the classic story of Sleeping Beauty, who pricks her finger on an enchanted spindle and falls into a deep sleep from which only a prince's kiss can awaken her. Surely the plot sounds familiar, because Disney has made a fortune with it. But a philosopher named Adam Elga, famous for creating several difficult puzzles, based on the work of Arnold Zuboff (published as "One Self: The Logic of Experience"), has given it a twist to turn it into a logical problem that is hard to solve.
The Sleeping Beauty Problem
Suppose it is Sunday, and Sleeping Beauty pricks her finger with the spindle. At that moment, the witch appears and—before the girl falls asleep—tosses a coin into the air. If it lands heads, Sleeping Beauty will wake from the curse on Monday and that will be the end of the story, without savior princes and without paradoxes of any kind. But if it lands tails, she will also wake on Monday, only to fall asleep again until Tuesday. When she wakes on Tuesday, she will be free from the curse but will have a small side effect: thanks to the witch's dark arts, she will not remember whether she woke on Monday or not.
Things being what they are, and with the Prince absent from the tale, our Sleeping Beauty wakes up without knowing if it is Monday or Tuesday. Given that if she woke on Monday, that event was erased from her mind by the witch, she has no way of knowing what day she is in. Adam Elga assumes that Sleeping Beauty is perfectly rational and that on Sunday, before falling asleep, she learned of the witch's plan. With this data, the girl can assign probabilities to the fact that it is Monday and to the fact that it is Tuesday. Or, in other words, she can assign probabilities to the coin having landed heads or tails.
The question to resolve is: What subjective probability should she assign to the hypothesis that the coin landed heads? There are two ways to approach the problem. The first assumes that since on Sunday Sleeping Beauty knew the coin was fair, it had a 50% chance of landing heads. Given that when she wakes on Monday she receives no additional data to help her deduce her situation (since her memory is erased), the probability is 50% that it is Monday and 50% that it is Tuesday. Seems logical, right? But before celebrating, let's look at the second way to approach this problem.
Suppose the witch's experiment is carried out a large number of times. As before, since the coin is not rigged, half the time it will land heads and the other half tails. We know that Sleeping Beauty wakes once after "heads" and twice after "tails." Since both sides of the coin have the same probability of occurring, all of the girl's awakenings also have the same probability. If the witch performs her trick 100 times, Sleeping Beauty would wake 50 times on Monday (after heads), 50 times on Monday (after tails), and 50 times on Tuesday after tails. If each time the girl is asked "What day is it?", she knows she has twice the probability of being awakened after tails than after heads. The probabilities she will assign to "tails" will be 66.66% and to "heads" only 33.33%.
Confused? Don't worry, you're not the only one. What is the correct answer? Adam Elga maintains that the second situation is correct, and that there is a 33.33% chance that Sleeping Beauty wakes on Tuesday after heads. David Lewis, another expert in logic, thinks the first answer is correct, and that when tossing a coin in the air, there is no alternative but to assign 50% probability to each possible result. And you, who do you agree with?
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