## Wednesday, August 03, 2005

### The Idle Argument

Thanks to the Stanford encyclopedia, I've discovered that my favourite puzzle in philosophical logic has a name: "The Idle Argument". It has many variations, but here's a fun one that I've come up with, based on a scenario in which I go sky-diving:

1) Either I will survive the fall, or I will not.
2) If I will survive, then I will survive no matter whether I bother to open my parachute.
3) If I will die, then I will die no matter whether I bother to open my parachute.
Therefore:
4) It doesn't matter whether I bother to open my parachute (it will make no difference).

As I've previously argued, the problem lies in equivocating between two interpretations of the conditional premises (2) and (3). In one sense, the premises are true but fail to establish the conclusion. In the other, the logic is valid but the premises are false. It's only by equivocation that the argument gets its plausibility: the premises all seem true, and the logic seems valid. But in actual fact you can't get both at once. Let me elaborate.

Intuitively, the problem with the Idle Argument is that our actions might affect which outcome results, whereas the argument (wrongly) takes the outcomes as given, and fixed independently of our actions. For suppose that I won't bother to open my parachute while skydiving, and so will die. But further suppose that I could open the parachute if I wanted, even though I actually don't. And if I did do so, then I would survive. Here I have described a scenario where the conclusion is clearly false: it really would make a (life-or-death!) difference whether I open the parachute or not. The difficulty lies in tracing this falsehood back into the premises.

Let's look at premise (3). In the scenario I described above, the antecedent is true: I will die. But what about the consequent? It claims that "I will die no matter whether I bother to open my parachute." But in my scenario, this is false. If I were to open my parachute, then I would survive. It just so happens that I actually won't open my parachute. But I could have. So premise (3) would be false in my imagined possible scenario. Thus (3) is not necessarily true, and the fatalistic argument fails.

That's one interpretation of the conditionals. There are others. For clarity and rigour, let's explicate these rival interpretations in terms of 'possible worlds'. Clearly the antecedent "If I will die" is to be assessed at the actual world. But the consequent is not so clear. My previous interpretation was something like as follows:

(3c-1) "I will die" is true at all close possible worlds w, no matter whether I bother to open my parachute at w.

Note that the actual truth of "I will die" does not guarantee the truth of "I will die" at other close possible worlds. So (3c-1) is clearly false. But perhaps the fatalist isn't interested in what happens at close possible worlds. They might just be talking about the actual world, yielding the interpretation:

(3c-2) "I will die" is actually true, no matter whether it is actually true that I try to open my parachute.

This interpretation of the consequent would make (3) a logical truth. Assuming that "I will die" is actually true, of course (3c-2) follows, since it basically just repeats this assumption. But this doesn't have any fatalistic implications. All it says is that I will actually die, which we've already stipulated. It doesn't make the further claim that I would still die even if I opened my parachute. In other words, it renders the argument logically invalid: the truth of the premises (on this interpretation) fail to establish the truth of the conclusion. The fatalist needs to appeal to counterfactual possibilities after all. Perhaps what they want is instead the following:

(3c-3) "I will die" is true at the actual world, no matter whether I bother to open my parachute at close possible worlds w.

But, as David Buller points out, this sort of claim is similarly inconsequential. Causal relations only hold within a world, not between possible worlds. So if w is distinct from the actual world, then of course opening my parachute in that world will not affect what happens in this world! This trivial truth is entirely consistent with the fact that if I did open my parachute (even though I actually won't) then I would survive rather than die.

My counterexample to the argument is any system of worlds satisfying the following two conditions:
(i) in the actual world, I don't open my parachute, and so I die.
(ii) in those close possible worlds where I do open my parachute, I instead survive.

Conditions (i) and (ii) together establish the falsity of (4). How we trace this falsity back into the argument will depend, as I have argued, upon how we interpret the premises (2) and (3). If those premises are interpreted as making counterfactual claims of the sort found in (3c-1), then they contradict condition (ii) and so are false. Alternatively, if interpreted in the weaker senses of (3c-2) or even (3c-3), then the premises are consistent with both (i) and (ii), and thus consistent with the falsity of the conclusion (4). In other words, the argument is invalid.

Any persisting unease regarding this argument might derive from concerns about the import of other possible worlds. After all, if it is true in the actual world that I will die, and there is nothing that can (categorically) be done to make this not actually true, then isn't that grounds for fatalism? What good is it if I survive in some other possible world, if that other possible world isn't -- and cannot be -- this one? (I have raised such concerns before, here.) But these more general fatalistic concerns are independent of the Idle Argument, so I will discuss them further in a separate post.

1. Hilarious, Richard. Wonderful satire. ;^)

I've given it a 'plug' here.

2. Premises 2 and 3 are false.

3. Sure, that's what I argued too. But the trick is to show why they are false.

Indeed, there is an interpretation where they turn out true. Assuming that I will survive, it follows trivially that I will survive (since we have stipulated this already!), whatever else might also happen to be the case. On this interpretation, then, premise 2 is undeniably true. But I also showed that, on this interpretation, the conclusion no longer follows validly from the premises.

Needless to say, merely asserting "Premises 2 and 3 are false", without any further argumentation, is insufficient to establish that your assertion is correct.

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