## Fixing the Past and the Laws (EDC, ch.5)

Chapter 5 of Evidence, Decision and Causality presents a powerful challenge to CDT (drawing on Ahmed (2013) and Ahmed (2014)).

Imagine you have strong evidence that a certain deterministic system S is the true system of laws in our universe. You also believe that questions about what you should do are meaningful even in a deterministic world. Now consider the following two decision problems.

Past: You must choose between raising and lowering your hand. Let DL (mnemonic for "determined lower") be the hypothesis that the state of the world 10000 years ago, together with S, entails that you will lower your hand. For some reason, you strongly prefer DL to ¬DL. You also like raising your hand.

Laws: You must choose between raising and lowering your hand. Raising your hand will be interpreted as affirming S, lowering as denying S. You want to affirm truths and deny falsehoods.

According to Arif, if is obvious that you should lower your hand in Past and raise your hand in Laws. CDT appears to give the opposite advice. Here is one way to derive these verdicts.

Next, Laws. If DL is true and you raise your hand then it is logically guaranteed that S is false (since DL and S together entail that you lower your hand) and therefore that you affirm a falsehood. If DL is true and you lower your hand then it is likely (but not certain) that you deny a truth, since it is likely (but not certain) that S is true. In sum, DL is bad news because it indicates that you'll either affirm a falsehood or deny a truth. It's better to lower your hand (if DL is true), as that gives you at least a small chance of denying a falsehood. What if DL is false? If you then lower you hand it is logically guaranteed that S is false and therefore that you deny a falsehood. If DL is false and you raise your hand then it is likely (but not certain) that you affirm a truth. In sum, the falsity of DL is good news; you have a better chance of satisfying your desire by lowering your hand. Since lowering your hand is the better choice either way, you should lower your hand.

We could block these arguments by denying that the state of the world 10000 years ago is outside your present control. Dorr (2016) makes a strong case that the past in a deterministic world counterfactually depends on the present. If we follow Dorr and use a formulation of CDT in terms of counterfactuals (a la Gibbard and Harper (1978), perhaps) we get Arif's preferred recommendations.

Arif objects that a decision theory that appeals to a non-causal relation of counterfactual dependence does not deserve the name 'Causal Decision Theory'. This reminds us that for Arif, CDT is motivated by a specific view about causation. Many friends of CDT do not share this view. As Dorr points out, one might instead motivate CDT by intuitions about Newcomb cases, and by the intuitive idea that if we want to know whether a choice is right, we need to consider what would happen (or would have happened) if a different choice were made. I, for one, would certainly prefer Dorr's version of CDT to EDT.

But I'm not happy with Dorr's version of CDT. In part, this is because I think Arif is right that this version goes against some ideas that motivate CDT. Suppose we say that the past counterfactually depends on your present choices, even though there is no corresponding causal dependence of the past on the present. We can then say that you should lower your hand in Past on the grounds that the desirable state DL would then obtain. But DL would not obtain because you lower your hand. We couldn't praise you for bringing about the desirable state DL. Nothing good would happen as a result of your choice to lower your hand, nor is the act intrinsically good in itself. In fact, it is intrinsically undesirable. Why then should we think the act is any good?

Relatedly, I'm worried that Dorr's decision theory actually yields the wrong verdict in Past. Arif intuits that you should here lower you hand. But the case really looks like Newcomb's Problem to me. DL is, in effect, a prediction that you will lower your hand. It would be good news if this particular prediction had been made, but you have no control over that. The only thing over which you do have control is whether you perform the intrinsically pleasant act of raising your hand. Shouldn't you raise your hand?

I'm not sure if this is the right way to think about Past. I don't have strong intuitions about the case. But to the extent that I have an intuition, it is the opposite of Arif's: Raising your hand looks like the better choice.

I therefore don't think that Past presents a serious challenge to CDT. The real challenge is Laws. Here I share Arif's intuition. It you have strong evidence for S and you want to tell the truth, then surely the right choice is to affirm S and not to deny it!

What shall we do about that?

I've already mentioned Dorr's response. This would yield the intuitively right result for Laws, but at some costs.

In an earlier post, I suggested a different response that tries to deliver the intuitively right result by invoking impossible worlds. This proposal is also developed in Williamson and Sandgren (2021). It has other costs.

A third possibility, also suggested by Williamson and Sandgren, is that Laws violates some condition on genuine decision problems, wherefore CDT shouldn't give any verdict at all. Joyce (2016) makes a similar move (although he actually ends of defending the counterintuitive verdict that you should lower your hand). In fact, as Arif himself points out, some prominent formulations of CDT (like that of Skyrms (1984)) already do fall silent about Laws, without adding new constraints.

My currently preferred response is a version of the third. I'm tempted to say that there is (surprisingly) no fact of the matter about what you should do if you find yourself in Laws or in Past – but not for the reasons suggested by Williamson and Sandgren or Joyce, who assume a Savage-style formulation of CDT. I prefer a formulation in terms of subjunctive supposition. The reason why CDT might fall silent is that the supposition operation is undefined for the propositions involved in Laws and Past.

I might say more about this another time. In the remainder of this post, I want to say a little more about Arif's discussion of Past.

As I said, I share Arif's intuition about Laws but not about Past. Arif uses a trick to lure his readers into sharing his judgment about Past. He initially presents the case as follows:

You must choose one of two bets. By raising your hand, you will accept a bet on the hypothesis DL* that the past is such that it causes you to lower your hand; the bet will give you $10 if DL* is true and cost$1 if DL* is false. By lowering your hand, you will accept a bet on ¬DL*; this bet pays $1 if DL* is false and costs$10 if DL* is true. What should you do?

This version of the story has two problems. First, it's not clear that the truth of DL* is outside your causal control. What does the hypothesis that the past is such that it causes you to lower your hands amount to? It implies that you will lower your hand, and that this will not happen a product of chance. Arguably that's all DL* says. By raising your hand, you can make DL* false. So it's not outside your control. And if you can influence whether DL* is true by raising or lowering your hand, then CDT agrees that you should lower your hand.

(Assume you go ahead and lower your hand. If you had raised your hand, would the past have been such that it causes you to lower your hand? Surely not.)

Arif doesn't mention this problem. But he seems to recognize it, since he goes on to replace the initial Past scenario (which he calls "approximate") by a different scenario (which he calls "precise"). In the "precise" version, the problematic proposition DL* is replaced by DL.

Or rather, that's what happens in the book. In the article (Ahmed (2014)), DL* is instead replaced by the hypothesis DL** that the true system of laws of the actual world, together with the state of the world 10000 years ago, entails that you lower your hand.

DL** raises another problem. To see the problem, consider the following situation.

You have to choose one of two bets. The first is on the proposition Q, that everything is just as it actually is. The bet pays $1 if Q is true and$-10 if Q is false. The second bet is on ¬Q. This bet pays $10 if Q is false and$-1 if Q is true.

Note that Q is a logical truth. Its negation is a contradiction. It may therefore seem reasonable to bet on Q rather than ¬Q. But suppose you're confident that you'll accept the bet on Q. What would be the case if instead you took the bet on ¬Q? Well, at least one thing would be different from how it actually is: you would take a different bet. So you would win that bet! You would get $10. That's better than the$1 you'll get by betting on Q.

Shocking news: CDT says you should not bet on a logical truth!

What went wrong here? A sentence like 'everything is just as it actually is' really expresses two different propositions. In one sense, it expresses a trivial truth – the set of all worlds. This is the sentence's "1-intension". In another sense, it expresses a highly non-trivial truth – the unit set of the actual world. This is the sentence's "2-intension". Betting on the 1-intension is a good idea: you are guaranteed to win. There is no counterfactual situation in which you would do better by betting on the negation of the 1-intension, which is true at no world. Betting on the 2-intension is usually a terrible idea, unless you have perfect information about which world you inhabit.

When we consider the verbal presentation of the scenario, it's natural to read the case as involving 1-intensions. We interpret a bet on Q as a bet that you are guaranteed to win. But when Q is put in a counterfactual context ('what would happen if you had bet on ¬Q?'), we evaluate it by its 2-intension. To avoid equivocation, we should make clear from the start at which worlds the relevant bets are won and at which they are lost.

The statement DL** – that the true system of laws of the actual world, together with the state of the world 10000 years ago, entails that you lower your hand – also expresses two propositions. As before, the bet on DL** is naturally understood as a bet on the 1-intension. (A bet on the 2-intension would be highly unattractive.) But the 1-intension of DL** is essentially equivalent to DL*. On that reading, you have control over whether DL** is true. The "precise" version of the scenario is no better than the "approximate" version.

Even my presentation of DL is not ideal. It is easy to conflate DL with DL* – especially since we haven't been told what S is. Ideally, we would spell out DL as a disjunction of claims about the intrinsic state of the world 10000 years ago.

The second problem with Arif's initial (and revised) presentation of Past concerns the resolution of the bets. In Arif's presentation, somebody offers the bets to somebody else. Let's say I offer the bets to you. How do I figure out whether DL is true, so that I know whether you won or lost? Arif suggests that I could simply observe your act. If you raise your hand, I can infer that DL is false; if you lower your hand, I can infer that DL is true.

In this scenario, the payoff doesn't actually depend on the truth of DL at all. It rather depends on your act. Unlike DL, this is something over which you have complete control. So of course you should "bet on DL" by lowering your hand.

Arif mentions this problem rather late into the discussion (on p.681 in the article and p.136 in the book). In response, he suggests that one could instead use the Past scenario as I described it.

In this final version, CDT might indeed recommend raising your hand. In the other versions it does not. We should make sure we squarely focus on the final version and don't assume that our intuitive judgments about the irrelevant other versions carries over.

Ahmed, Arif. 2013. “Causal Decision Theory: A Counterexample.” The Philosophical Review 122 (2): 289–306. doi.org/10.1215/00318108-1963725.
Ahmed, Arif. 2014. “Causal Decision Theory and the Fixity of the Past.” The British Journal for the Philosophy of Science 65 (4): 665–85. doi.org/10.1093/bjps/axt021.
Dorr, Cian. 2016. “Against Counterfactual Miracles.” Philosophical Review 125 (2): 241–86. doi.org/10.1215/00318108-3453187.
Gibbard, Allan, and William Harper. 1978. “Counterfactuals and Two Kinds of Expected Utility.” In Foundations and Applications of Decision Theory, edited by C. A. Hooker, J. J. Leach, and E. F. McClennen, 125–62. Dordrecht: D. Reidel.
Joyce, James. 2016. “Review of Arif Ahmed, Evidence, Decision and Causality.” Journal of Philosophy 13 (4).
Skyrms, Brian. 1984. Pragmatics and Empiricism. Yale: Yale University Press.
Williamson, Timothy Luke, and Alexander Sandgren. 2021. “Law-Abiding Causal Decision Theory.” British Journal for the Philosophy of Science.