Little-Known Resolutions of Well-Known Paradoxes

Paradoxides

The following resolutions of paradoxes deserve to be better known. (All of these paradoxes are resolved; you don’t need to worry about them any more.)

The Liar Paradox

Paradox

Consider the sentence “This sentence is false.” If it is true, then it is false, and if it is false, then it is true.

Resolution

The Liar Paradox was resolved in Semantic Paradoxes as Equations by Lan Wen, The Mathematical Intelligencer, vol. 23, issue 1, December 2001, pp. 43–48. Wen showed how to take any system of boolean equations that doesn’t have a solution and create a corresponding semantic paradox. He presents the Three Cards Paradox:

The Three Cards Paradox. Consider three cards with the following three sentences:
  1. The sentence on the second card is true, and the sentence on the third card is false.
  2. Either the sentence on the first card is false, or the sentence on the third card is true.
  3. The sentences on the first and second cards are both true.

The Liar Paradox is the semantic equivalent of assuming that all systems of equations have solutions. See Wen’s article for details. Note that there is no self-reference in the Three Cards Paradox, so the self-reference in the Liar paradox is a red herring.

Prisoner’s Dilemma

Paradox

Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of communicating with the other. The prosecutors lack sufficient evidence to convict the pair on the principal charge. They hope to get both sentenced to a year in prison on a lesser charge. Simultaneously, the prosecutors offer each prisoner a bargain. Each prisoner is given the opportunity either to betray the other by testifying that the other committed the crime or to cooperate with the other by remaining silent. The offer is:

The Nash equilibrium is for both prisoners to betray the other. So far, this isn’t a paradox. However, it is often stated or implied that the Nash equilibrium is the optimal strategy, even though both prisoners do better if they both remain silent.

Resolution

The Prisoner’s Dilemma was resolved by Douglas Hofstadter. If both prisoners are rational, then they should not choose the Nash equilibrium. Because the situation is symmetric and because both prisoners are rational, whichever strategy prisoner A decides is best, prisoner B will also decide is best. Since both remaining silent is better than both betraying, each should decide to remain silent. The Nash equilibrium is not relevant because it considers situations that are impossible if both prisoners are rational (i.e., one prisoner remaining silent and the other betraying).

This is discussed in detail in Metamagical Themas: Questing for the Essence of Mind and Pattern by Douglas R. Hofstadter, Basic Books, March 1996, ISBN 978-0465045662. Also see my Prisoner’s Dilemma, Letter to the Editor, Notices of the American Mathematical Society, vol. 51, no. 7, August 2004, p. 735.

Newcomb’s Paradox

Paradox

A highly superior being from another part of the galaxy presents you with two boxes, one open and one closed. In the open box there is a thousand-dollar bill. In the closed box there is either one million dollars or there is nothing. You are to choose between taking both boxes or taking the closed box only. But there’s a catch.

The being claims that he is able to predict what any human being will decide to do. If he predicted you would take only the closed box, then he placed a million dollars in it. But if he predicted you would take both boxes, he left the closed box empty. Furthermore, he has run this experiment with 999 people before, and has been right every time.

What do you do?

This is a paradox because there seem to be good arguments for both options. Considering the being accuracy in the past, the odds are that he/she will be right again, so you should take only the closed box. On the other hand, the money is already in the box, so you might as well take both.

Resolution

This is basically just the Prisoner’s Dilemma again, but dressed up on one side to try to make the player think it very likely that both he and the being will choose the same strategy and on the other side to try to bring causality into it. Douglas Hofstadter’s resolution of the Prisoner’s Dilemma also resolves Newcomb’s Paradox. (I'm not sure if I read this resolution somewhere or thought of it myself. Let me know if you see it somewhere.)

Is It Rational to Vote?

Paradox

Is it rational to vote? The chance of one vote affecting the election is small, so one could argue that it isn’t worth the trouble to vote. But people do vote.

Resolution

There are counter arguments that say that it is rational to vote. See Voting as a rational choice: why and how people vote to improve the well-being of others by Aaron Edlin, Andrew Gelman, and Noah Kaplan, Rationality and Society, 2007, vol 19(3), pages 293–314. They argue that you should consider the benefit not just to the voter, but to the country. Also see posts on Andrew Gelman’s blog here and here. While some people may use this reasoning to justify voting, I want to suggest a rationale that is rational and probably closer to the way that most people think.

If you believe that there are other rational voters, then Douglas Hofstadter’s resolution of the Prisoner’s Dilemma applies to voting. So, you don’t have just one vote, but rather all you rational voters are voting together. (I have not seen this resolution anywhere else. Let me know if you see it somewhere.)

The Unexpected Hanging

Paradox

On Sunday evening a judge tells a condemned prisoner that he will be awakened and hanged on the morning of one of the following five days. The judge says that it will happen unexpectedly, i.e., the prisoner will be uncertain about when the hanging will occur until the moment the attendants arrive. But the prisoner’s attorney convinces him that no such hanging is possible. The first step in the attorney’s argument is to eliminate Friday as execution day: If the judge sets Friday as the morning of the hanging, the prisoner will know it on Thursday because he is still alive and realizes that tomorrow is the final day of the execution period. Friday is ruled out. But then, by the process of elimination, so are Thursday, Wednesday, Tuesday, and Monday. Of course, on Wednesday, the prisoner is hauled out of bed, much to his surprise, and hanged.

Resolution

The resolution is to realize that there is a difference between being able to guarantee something will happen and it happening because you got lucky. The prisoner is correct that the judge cannot guarantee that the hanging will be unexpected. (A judge's sentence is normally interpreted as a guarantee that something will happen.) But, this does not prevent the judge from getting lucky. An analogy would be if the judge said that the prisoner would be hanged in the morning and it would not rain for the whole day of the hanging. The judge cannot guarantee that it won't rain, but he might get lucky. (I have not seen this resolution anywhere else. Let me know if you see it somewhere.)

Moral

Resolutions of paradoxes should (usually) be short. If you need a whole book to explain your resolution, then it probably is not the correct resolution.

Page published 2016-09-11.