## Thursday, May 8, 2014

### No event can irreducibly depend on infinitely many things

I am now thinking the following principle is likely to be true:

• (NoInfDep) No event can irreducibly depend on infinitely many things.
This might be true even if we drop "irreducibly", but the "irreducibly" makes me more confident. When A causes B in such a way that B is determined to cause C, then I say that C reducibly depends on B. No additional information is given about how C came about by describing B than is already implicit in a sufficiently rich description of A. In other words, reducible dependence is dependence on a merely and totally instrumental cause, one that doesn't make any contribution truly of its own. We can then see the debate between compatibilists and incompatibilists as a debate on whether an agent's free actions have to irreducibly depend on something in the agent.

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:

1. Thomson's Lamp
2. Grim Reapers
3. Coin sequence guessing
4. Infinite fair lotteries resulting from infinitely many fair coin tosses (see the discussion in one of my comments of the paradoxicality)
5. 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)
6. Realizations of the Banach-Tarski Paradox and maybe even things relating to nonmeasurable sets.
There are other solutions for some of these. (I've never been that impressed by Thomson's lamp, but it's a freebie here.) But NoInfDep provides an elegantly uniform solution. Moreover, NoInfDep expresses the intuition that there cannot be a "completed infinity" without committing one to a dubious presentist or growing block ontology. Indeed, NoInfDep shows that the "no completed infinity" intuition goes beyond considerations of time: it is about dependency.

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.

Alexander R Pruss said...

For Thomson's lamp, we need the stronger principle without "irreducibly".

Unknown said...
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Unknown said...

I see the beginnings of a Cosmological Argument here, but I'm wondering how you defend a beginning of time (a premise in the Kalaam argument as you well know) from what you've said here. Suppose one held (what I take to be the crazy thesis) that there have been an infinite series of events with no dependency whatsoever between the events (all events popping into existence out of the blue). There's a successive addition, so to speak, but it's just one brute event after another. Wouldn't you need a stronger principle than NoInfDep to rule that out (or at least an additional principle)? Perhaps that's what you were getting at in the last couple paragraphs. I'm just wondering what that would be such that we end up with the Kalaam and not (say) Leibniz's argument.

Alexander R Pruss said...

Fair enough. I was thinking of the kalaam argument's distinguishing feature as denying a backwards infinite chain. But you're right: normally it's stated in terms of time.