O Paradoxo da Pergunta

Li este texto que vai a seguir há algum tempo e ele sempre me impressionou pela maneira criativa como consegue apresentar um dos paradoxos mais interessantes da lógica. Ei-lo:

The Paradox of the Question
Ned Markosian

Once upon a time, during a large and international conference of the world’s leading philosophers, an angel miraculously appeared and said, “I come to you as a messenger from God. You will be permitted to ask any one question you want – but only one! – and I will answer that question truthfully. What would you like to ask?” The philosophers were understandably excited, and immediately began a discussion of what would be the best question to ask. But it quickly became obvious that they needed more time to discuss the matter, so they asked the angel if he could get back to them. The angel was obliging, and said that he would return at the same time the next day. “But be prepared then,” he warned them, “for you will only get this one chance.”

All of the philosophers gathered at the convention worked at a frenzied pace for the next twenty-four hours, proposing and weighing the merits of various questions. Other philosophers from around the world became involved as well, faxing and emailing their suggestions. Some were in favor of asking the kind of practical question that lots of people might like to know the answer to, such as this one:

Q1 Is it better to check your oil when the car is hot or when it is cold?

But others said they should not squander this rare opportunity, which gave them a chance to learn something about a truly important and intrinsically interesting topic, and after some discussion it was generally agreed that this was right.

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The philosophers were puzzled, however, about which truly important and intrinsically interesting topic they should address in their question. The problem was that they really needed to know in advance what would be the best question to ask, in order to make the most of their marvelous opportunity. One proposal was to try to sneak in two questions, by asking something like this:

Q2 What would be the best question for us to ask, and what is the

answer to that question?

But this proposal was quickly voted down when it was pointed out that the angel had explicitly said that they would get just one question.

Another proposal was simply to ask the first of the questions in Q2, in the hopes that some day they would have another opportunity similar to this one, when they could then ask the question they knew to be the best. This proposal was ruled out, however, on the grounds that if they adopted it then they would probably never get a chance to ask the best question once they knew what it was.

For a while there was a growing consensus that they should ask this question:

Q3 What is the answer to the question that would be the best question

for us to ask?

That way, it was argued, they would at least have the all-important information contained in the relevant answer. But eventually concerns were raised about the possibility of receiving, in response to Q3, an answer such as ’seven’, or ‘yes’, which would mean nothing to them unless they knew which question was being answered.

Finally, just as the philosophers were running out of time, a bright young logician made a proposal that was quickly and overwhelmingly approved. Here was her question:

Q4 What is the ordered pair whose first member is the question that

would be the best one for us to ask you, and whose second member

is the answer to that question?

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Nearly everyone (remember, these are philosophers we’re talking about) agreed that this was the ideal way to solve their little puzzle. By asking Q4 the philosophers could ensure that they would learn both what the best question was, and also what the answer to that question was. There was a great deal of celebrating and back-clapping, and as the minutes ticked down to the time when the angel had promised to return, the mood among philosophers throughout the world was one of nearly feverish anticipation. Everyone was excited about the prospect of learning some wonderful and important truth. They were also more than a little pleased with themselves for hitting upon such a clever way to solve the problem of how to find out what the best question was, and also get the answer to that question, when they had only one question to work with.

Then the angel returned. The philosophers solemnly asked their question – Q4 – and the angel listened carefully. Then he gave this reply:

A4 It is the ordered pair whose first member is the question you just

asked me, and whose second member is this answer I am giving

you.

As soon as he had given his answer, the angel disappeared, leaving the philosophers to pull out their hair in frustration.

The above story leaves us with another little puzzle to solve. At the time the philosophers asked Q4, it seemed like that question was the ideal one for their peculiar situation. But as it turned out, Q4 was obviously not at all the right thing to ask. (They would have been better off asking whether one should check one’s oil when the car is hot or when it is cold.) The puzzle, then, is this: What went wrong?

I am grateful to Mark Aronszajn and Donald Turner for dicussions of this topic, and to Donald Turner for telling me the joke that the puzzle is based on.

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