Newcomb's Paradox

Newcomb's paradox, named after its creator, physicist William Newcomb, is one of the most widely debated paradoxes of recent times. It was first made popular by Harvard philosopher Robert Nozick. The paradox goes like this:

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. Of course there's a catch.

The being claims that she is able to predict what any human being will decide to do. If she predicted you would take only the closed box, then she placed a million dollars in it. But if she predicted you would take both boxes, she left the closed box empty. Furthermore, she has run this experiment with 9,999 people before, and has been right every time.

It seems that those who believe in free will decide to take both boxes, whereas determinists take one. More on this later.

Here is Franz Kiekeben's solution:
If the being predicts in the manner of a scientist, that means that there is a certain state of affairs, A, which holds at some point in time prior to your decision and the prediction, and which causes both. This connection between the prediction and the decision is what prevents your actions from being independent of the states of the box. And it is realizing this that makes it rational to take the closed box only.

Preamble

Firstly, we are told that the "superior being" claims to be able to predict human behaviour. We are also told that she predicted correctly in all 9,999 cases so far. The first task for the problem solver is to decide how much credence or importance should be given to this claim of clairvoyance. Statistically, the probability of her being right is greater than 99.99%.

Probability arguments are tricky. For instance I might observe that a certain man runs past my house every morning for three years. Based on this I would say that the probability he would appear tomorrow would be better than 99.9%, a near certainty. Yet tomorrow the man does not appear and I learn that the Olympic team has just left the country.

Part of the background of the problem is that logic is taken to be a universal and infallible method of reasoning, producing unique and reliable results whenever applied correctly by anyone anywhere. The catch is that logic is not applicable to all problems. So is Newcomb's paradox one to which logic, so defined, may be validly applied?

If it is, then we must be able to present an argument for our choice that is so sound that it cannot be reasonably disputed (as in the Monty Hall puzzle). We must also be able to shoot down any argument in favour of the contrary choice.

If, on the other hand, we are able to present water-tight arguments in favour of both decisions then we can conclude that this puzzle does not admit a rational solution, cf "I am lying".

My solution

Firstly, assume that the problem has a logical solution.

There is a 99.99% probability that the superior being will have predicted our choice correctly, so the one box solution is correct.

On the other hand, taking both boxes is the logical solution as I must get $1,000 more by so doing. The act of taking both boxes cannot remove anything from the closed box. There is no backward causality. Ergo we have a contradiction, so our original assumption is wrong. Hence the solution is not "one box" or "two", but that "there is no rational solution".

The puzzle is insoluble, like the liar paradox, due to self-reference. Self-reference comes into it because the choice I am to make is indirectly based on itself, ie on what it already was before I even made it. This is because the choice is identical to the prediction (to the extent that the superior being's clairvoyance is to be trusted) and it is the prediction on which my choice is to be based. Of course the prediction is based on my choice...

All forms of the liar's paradox (eg the proof that English is infinite, also on this page) result from treating as meaningful statements that are self-contradictory and hence nonsensical. Newcomb's paradox is a little different in that it posits a situation that is self-referential, rather than a self-referential statement. While the situation in Newcomb's paradox is not self-contradictory, it embodies a vicious circle, as described above.

Note that if we were to randomly generate decision problems involving money and closed boxes, we would find that (except for those that explicitly told us where the money was) the vast majority of puzzles so created would not have a rational solution. Eg "One box contains $1, the other $10. Should you take the green or the red box?" or "If the superior being likes you she puts money in the red box, if not, in the blue one. Which do you take?"

My point is that we should not expect logic to be able to answer any question that we apply it to. On the contrary, only a miniscule subset of all English sentences that express intelligible puzzles have logical solutions. So the conclusion that Newcomb's puzzle lacks a logical solution should not surprise us.

Thus the paradox is yet another case of self-reference tying us in knots. I believe that the solution is that there is no logical answer, just the common-sense one of: grab all the money and run!

Alternately, we can argue that foretelling the future is not possible, so the paradox is putting forward an impossible scenario. No superior being could predict the future and especially not something as volatile as the wayward thinking of human beings.

Postscript

As for free will versus determinism, even if determinism is true, fore-knowledge of the person's choice of boxes may be impossible, so that the two positions (free will and determinism) become operationally identical. In general it is impossible to know what people will do, or what the future will show. To have such knowledge one would need another universe as complex as ours to do the calculations or modelling.

Human beings cannot live as though they believed in total predestination.

What is the meaning of "free will"? It means we have the subjective conviction that we could have acted otherwise if we had wanted to.

My view is that most statements about the world are made within a level of description eg science, chemistry, subjectivity or particle physics. To me the term "free will" applies to the level of subjectivity or consciousness, but possibly not elsewhere.

Tad Boniecki