Zog runs a casino on the planet of Vega Sals. Like his competitors on the galaxy-famous Landing Strip he's been using probability to estimate his earnings based on the odds that he offers in his games. It needs quite a bit of care - offer odds that are too good and he'll lose the games, offer stingy odds and his competitors undercut him.
Anyway, Zog decided to attend some adult education classes at the local university and learnt about the concept of falsifiable propositions. The effect was disastrous, he found that he couldn't attach a meaning to the statement "this coin has a 50% probability of turning up left-head". (Note that the inhabitants of Vega Sals have two heads.) Whether it turns up left-head or right-head it's consistent with the original statement which can never be falsified. Yet surely it has a meaning of some sort.
Zog wrestled and wrestled with this problem, suffering nights of insomnia, when suddenly in a flash of smoke the Devil appeared in front of him.
"I'll do a deal with you," he said. "You're struggling because you can't get a falsifiable proposition from a probabilistic one. So here's what I'll do for you. Whenever you want, I'll let you convert a probabilistic statement into a falsifiable one with probability one.". "
"Sounds great," replied Zog, "but what's the catch?"
"Hmmm...I see my reputation precedes me. Anyway, it's just a small catch. For every proposition that I convert you'll have to suffer the fires of Hell. If it's an unlikely proposition you might suffer for trillions of years. But if it's 90% likely you'll only have to suffer, oh, let's say a billion years. If you can make it 99.99% likely you might only suffer for a million years but if you make it something like 99.9999999% likely you'll only have to suffer a short while. I have a brochure here that explains the details."
"Sign me up right away," said Zog, and he slept well that night.
But the next day when Zog was doing the weekly profit and loss forecast he found he had a problem. If he had a thousand people play space-roulette that day then it'd cost him trillions of years of sufffering to make the outcomes of these games certain. In the cold light of day he seemed no better off than before. But then he had an idea. Why not batch together the days takings? Instead of predicting the outcome of one game he would predict the outcome for the entire day. By using a suitable variant of the central limit theorem he found that he could predict the day's winnings to within 10 galactic credits with 99.9999% certainty. "Great!" he thought looking this up in the brochure. "99.9999% - 3 days of Hellfire." As the Astro-Priest had told him that even one second in Hell was worse than a lifetime of suffering it didn't seem such a good idea after all.
So Zog decided to consult a mathematician to buy a theorem. "I can construct just the thing you need," the mathematician said, "for only 1000 galactic credits." So the mathematician sold him a theorem a bit like this: "if, over the course of your lifetime, you make your decisions based on probability theory, then you have a 99.999999999% chance that you won't make a total loss at your casino tables and that nobody else will undercut you." The cunning mathematician had taken Zog's batching trick and batched together the entirety of Zog's future transactions to make this prediction. It was a perfectly valid theorem and Zog thought it was money well spent. He also no longer needed to worry about the meaning of probabilistic statements. He knew that if he just used probability theory, instead of thinking about its meaning, he would be guaranteed to succeed in the casino business.
And Zog lived happily ever after. When he died, the hundredth of a second that he suffered in Hell was pretty bad, but the everlasting happiness that came afterwards more than made up for it.
Of course you're probably getting suspicious by now. There isn't really such a place as Vega Sals. And I just made up Zog. This is of course a fable. The truth of the matter is that it was me who made the deal with the Devil, not Zog. And it wasn't exactly this deal. The deal I made is based on the theorem that I got for free from here. The Devil said that any time there was a system whose wavefunction was ψ he could make it φ where the amount of suffering I would have to endure was computed from |ψ-φ|. Using that theorem I was able to get the devil to remove from the universal wavefunction that part in which the probabilities of the Copenhagen interpretation didn't work. It was a pretty small part so I should suffer even less than Zog did. And the Devil offered me a 50% discount if I blogged aobut how good my deal was.
But I'm not the only person who's made such a deal with the Devil. Another person is David Deutsch in this paper. But I feel a little sorry for David. You see he claims "we have proved that a rational decision maker will maximise the expectation value of his utility." Well, yes, this is true. But only because of how he's defined 'rational'. But by using sleight of hand he's made it appear as if this statement is true for people who are rational in the usual sense of the word and of course people who read this paper will be less inclined to deal with the Devil. The Devil really doesn't like it when people deny they have had dealings with him, it makes him feel sordid and shady. So even as we speak the Devil is warming a very special place in Hell for David.
Oh...and maybe you can now see why I've been thinking about "almost certainty" a bit recently. I should also add that S4 doesn't quite do the job I want. I think ◊¬(◊p∧◊¬p) really ought to be a theorem of my logic but it's provably not a theorem of S4. (Where I now use ◊ to mean "almost certainly".) I think I need some more axioms in addition to S4...
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