In a previous post, I talked about something that has been confusing me regarding free will and decision theory. I want to revisit this topic and express a different way to frame the issue.

Here goes:

A decision theory is an algorithm used for calculating the expected utilities of the different possible actions you could take: EU(A). It returns a recommendation for you to take the action that maximizes expected utility: A* = argmax EU(A).

I have underlined the word that is the source of the confusion. The question is: how can we make sense of this notion of possible actions given determinism? If we take determinism very seriously, then the set of possible actions is a set with a single member, which is the action that you end up actually taking. There’s an intuitive sense of possibility at play here that looks benign enough, but upon closer examination becomes problematic.

For instance, we obviously want our set of actions to be restricted to some degree – we don’t want our decision theory telling us to snap our fingers and magically turn the world into a utopia. One seemingly clear line we could draw is to say that possibility here just means physical possibility. Actions that require us to exceed the speed of light, violate conservation of energy, or other such physical impossibilities are not allowed to be included in the set of possible actions.

But this is no solution at all! After all, if the physical laws are the things uniquely generating our actual actions, then all other actions must be violations! Determinism dictates that there can’t be multiple different answers to the question of “what happens next?”. We have an intuitive notion of physical possibility that includes things like “wave my hand through the air” and “take a nap”. But upon close examination, these seem to really just be the product of our ignorance of the true workings of the laws of nature. If we could deeply internalize the way that physics generates behaviors like hand-waving and napping, then we would be able to see why in a particular case hand-waving is possible (and thus happens), and why in other cases it cannot happen.

In other words, the claim I am making is that there is no clear distinction on the level of physics between the claim that I can jump to the moon and the claim that I could have waved my hand around in front of me even though I didn’t. The only difference between these two, it seems to me, is in terms of the intuitive obviousness of the impossibility of the first, and the lack of intuitive obviousness for the second.

Let’s say that eventually physicists reduce all of fundamental physics to a single principle, for example the Principle of Minimum Action. Then for any given action, either it is true that this action minimizes Action, or it is false. (sorry for the two meanings of the word ‘action’, it couldn’t be helped) If it is true, then the action is physically possible, and will in fact happen. And if it is false, then the action is physically impossible, and will not happen. We can explicitly lay out an explanation of why me jumping to the moon does not minimize Action, but it is much much much harder to lay out explicitly why me waving my hand in front of my face right now does not minimize Action. The key point is that the only difference here is an epistemic one – some actions are easier for us to diagnose as non-action-minimizing than others, but in reality, they either are or are not.

If this is all true, then physical possibility is hopeless as a source to ground a choice of the set of possible actions, and any formal decision theory will ultimately rest on an unreal distinction between possible and impossible actions. This distinction will not be represented in any real features of the physical world, and will be vulnerable to future discoveries or increases in computational power that expand our knowledge of the causal determinants of our actions.

Are there other notions of possibility that might be more fruitful for grounding the choice of the set of possible actions? I think not. Here’s a general argument for why not.

Ought implies can

(1) If you should do something, then you can do it.
(2) There is only a single thing that you can do.
(3) Therefore, there is at most a single thing that you should do.

This is an argument that I initially saw in the context of morality. I regarded it as a mere intellectual curiosity, fun to ponder but fairly unimportant (given that I didn’t expect much out of ethics in the first place).

But I think that the exact same argument applies for any theory of normative instrumental rationality. This is much more troubling to me! Unlike morality, I actually feel fairly strongly that there are objective facts about instrumental rationality – that is, facts about how an agent should act in order to optimize their values. (This is no longer an ethical should, but an epistemic one)

But I also feel strongly tempted to endorse both premises (1) and (2) with regard to this epistemic should, and want to reject the conclusion. Let’s lay out our options.

Reject (1): But then this means that there are some actions that it is true that you should do, even though you can’t do them. Do we really want a theory of instrumental rationality that tells us that the most rational course of action is one that we definitely cannot take? This seems obviously undesirable, for the same reason that the decision theory that says that the optimal action is to snap your fingers and turn the world into a utopia is undesirable. If this premise is not true of our decision theory, then we might sometimes have to accept that the action we should take is physically impossible, and what’s the use of a decision theory like that?

Reject (2): But this entails an abandonment of our best understanding of physical reality. Even in standard formulations of quantum mechanics, the wave function that describes the state of the universe evolves completely deterministically. (You might now wonder why quantum mechanics is always thought of as a fundamentally probabilistic theory, but this is definitely too big of a topic to go into here.) So it seems likely that this premise is just empirically correct.

Accept (3): But then our theory of rationality is useless, as it tells us nothing besides “Just do what you are going to do”!

This is the puzzle. Do you see any way out?