I am now thinking the following principle is likely to be true:
- (NoInfDep) No event can irreducibly depend on infinitely many things.
Why think NoInfDep is true? The general line of argument is this. There are a number of paradoxes that NoInfDep rules out. Now in the case of each paradox, there is a narrower modal principle that could rule out the paradox, but the narrower principle is ad hoc in a way that NoInfDep isn't, and so our best explanation as to why the paradoxes are ruled out is (1).
Here are the paradoxes I currently have in mind:
- Thomson's Lamp
- Grim Reapers
- Coin sequence guessing
- Infinite fair lotteries resulting from infinitely many fair coin tosses (see the discussion in one of my comments of the paradoxicality)
- Satan's Apple and some other decision-theoretic paradoxes (e.g., the game where we have dollar bills numbered 1,2,3,... and you start with dollar bill #1, and in each round you give me your lowest numbered bill, and I give you two bills with higher numbers; at the end you have nothing)
- Realizations of the Banach-Tarski Paradox and maybe even things relating to nonmeasurable sets.
I want to say something about the Banach-Tarski case. The paradox there is purely mathematical. But to realize this paradox in real life--to actually decompose a solid ball (if there were such a thing) into two of equal size--you would need to make something like a choice function, which would require infinitely many data points, and those would require, I suspect, irreducibly infinitely many events to generate.
And now we have the Kalaam argument.