Wolfgang Schwarz

Gallow on causal counterfactuals without miracles and backtracking

Gallow (2023) spells out an interventionist theory of counterfactuals that promises to preserve two apparently incompatible intuitions.

Suppose the laws of nature are deterministic. What would have happened if you had chosen some act that you didn't actually choose? The two apparently incompatible intuitions are:

(A1) Had you chosen differently, no law of nature would have been violated.

(A2) Had you chosen differently, the initial conditions of the universe would not have been changed.

Rejecting one of these intuitions is widely thought to spell trouble for Causal Decision Theory. Gallow argues that they can both be respected. I'll explain how. Then I'll explain why I'm not convinced.

We start with some familiar ideas from causal modelling. To begin, we assume that the world contains relations of causal influence between variables. Here, a variable isn't a syntactic object, but a "contrastive generalisation of an event". For our topic, we can focus on binary variables that take only two values, 0 and 1. For example, we'll talk about a variable B that represents whether or not you take a certain bet. If you do, we say that B has value 1, if you don't, it has value 0.

Relations of causal influence can be represented by structural equations that specify how the value of one variable is determined by the value of other variables. For example, if B is a bet that a certain coin will land heads, H is a variable for the outcome of the coin flip (1 = Heads, 0 = Tails), and W is a variable for getting the relevant amount of money ("winning"), then the equation

(1) W := B * H

represents that whether you get the money is determined by whether you bet and by how the coin lands, in multiplicative fashion (so that W = 1 iff B and H are both 1).

What does it take for an equation like this to be a correct representation of the causal structure in the world? Gallow lists three conditions.

First, all the variables must be mereologically distinct, so that all combinations of value assignments are possible.

Second, for any possible values of the variables on the right-hand side of the equation (B and H in my example), the equality (W = B * H) holds at the closest worlds at which the variables on the right-hand side have these values. Closeness is measured by intuitive overall similarity at the time of B and H.

To see what this means, let's continue the example. Suppose you haven't actually bet (B=0) and the coin has landed heads (H=1). We then need to check, for example, whether you win (W=1) in the closest worlds at which you bet (B=1) and the coin lands heads (H=1). The answer is yes. In general, no matter how we set W and B, the equation W = B * H holds (in the closest worlds where W and B have these values).

A third condition pertains to entire systems of structural equations. For systems that contain only a single equation, it says that there is no non-trivial dependence (of the sort that would meet the second condition) between the variables on the right-hand side.

In the coin example, there is no non-trivial dependence between B and H. So (1) is a correct representation of the coin flip scenario.

We need one more piece of machinery to connect all this to conditionals.

Let a causal model be a system of structural equations together with an assignment of values to those ("exogenous") variables that only occur on the right-hand side of the equations. A model is correct if the system of equations is correct (as per the three criteria above) and the values assigned to the exogenous variables are their true values. Finally, if M is a model and V=v is an assignment of value v to some variable V, then M revised by V=v is a model much like M except that V has value v and all equations in which V occurs on the left-hand side are removed.

Now, according to Gallow, a counterfactual conditional A=a > C=c is true (on its "causal" reading) whenever there is a correct causal model that determines C=c when revised by A=a.

In the coin example, the equation W := B * H together with the assignment { B=0, H=1 } is a correct causal model. To check whether B=1 > W=1 ("If you had bet on heads, you would have won"), we consider the revised model that has the same equation but assignment { B=1, H=1 }. Applying the equation, this determines W=1. So the conditional is true.

All this is nice and interesting and useful, and spelled out with more care than at many other places where the same ideas have been discussed. But let's see how it helps with (A1) and (A2).

Imagine a situation in which you have the option of betting that the actual laws of nature are violated. You're not going to choose this option. We wonder what would happen if you did. Would you win the bet? Intuitively, the answer is no. Your choosing otherwise would not have brought about a violation of the laws, as per (A1).

Gallow models the relevant causal relations as follows. We have three variables. B for whether you bet, W for whether you get the payoff, and M ("miracle") for whether the actual laws are violated. These variables are related by the equation

(2) W := B * M.

We can check that this is a correct system of equations by Gallow's three conditions. First, the three variables are distinct, at least insofar as any combination of values is possible. Second, the closest worlds at which B=1 or M=1 (or both) are still worlds at which you get the money iff B=1 and M=1. Finally, there is no dependence between B and M.

Wait. Why isn't there a dependence between B and M? In the actual world, we have B=0 and M=0. According to Lewis (1979), the closest B=1 worlds are M=1 worlds. (The closest worlds where you bet have a "miracle".) This suggests that B and M are related by equation (3), which would make (2) an incorrect system.

(3) M := B.

We could, of course, deny that the closest B=1 worlds are M=1 worlds, but then we'd run into the analogous problem with the past and (A2). So let's assume that the closest B=1 worlds are M=1 worlds, if only for the sake of the argument.

Gallow argues that (3) is incorrect even if the closest B=1 worlds are M=1 worlds, because the closest B=0 worlds aren't all M=0 worlds: they include worlds at which the (actual) laws of nature are violated at around the time of the bet.

This makes no sense if closeness is a matter of intuitive similarity at the relevant time (as we are told on p.14). The actual world is surely more similar to itself than any world in which the actual laws are violated at the time of the bet.

So closeness doesn't measure intuitive similarity. It's a technical concept that measures something else. I'm not sure what it measures. Gallow doesn't really explain. He describes a different example involving Jesus that he thinks motivates the idea that the actual world is not the most similar world to itself. I don't really follow his intuitions here. The idea seems to be that if J is a variable for whether Jesus was born then an equation like D := J means that D=1 is guaranteed no matter how J=1 is realised. Even if J actually has value 1, we therefore need to check that D=1 is the case at nearby worlds at which J=1 is realised in some other way. OK.

Perhaps B=0 can also be realised in different ways. On Gallow's reading, equation (3) then requires that at nearby worlds at which B=0 is realised in some other way, the actual laws obtain without violation. On a Lewisian conception of closeness, these worlds all involve violations of the laws. So (3) is false. And so (2) is correct. Here it is again.

(2) W := B * M.

Now it's easy to see why 'B=1 > W=0' ("if you were to bet that the laws are violated you wouldn't win") is true. The actual values of B and M are B=0 and M=0. To evaluate the conditional, we set B=1, leaving M at 0, and plug these values into equation (2). We get W=0.

The same reasoning applies for the past. If C describes the initial conditions of the universe and B is a bet that the initial conditions are not C, then (4) is a correct system of equations:

(4) W := B * (1-C).

We get the desired result that if you had bet that the initial conditions are different then you would have lost, because the initial conditions would not have been changed, as per (A2).

If we're only interested in (A1) and (A2), we can actually use simpler models.

Let M1 be a model with variables B and M, an empty system of equations, and the assignment { B=0, M=0 }. This model is correct by Gallow's criteria. Revising M1 by B=1 ("you choose otherwise") yields a model with M=0. So we have 'B=1 > M=0' ("the laws would not be violated").

Similarly, let M2 be a model with B and C, an empty system of equations, and assignment { B=0, C=1 }. Revising this correct model by B=1 ("you choose otherwise") determines C=1 ("the initial conditions would be the same").

But how could the deterministic laws and the initial conditions both be the same, if you had chosen a different act?! Would an impossible situation have obtained?

According to Gallow, we're not licensed to conclude that the laws and the initial conditions would both have been the same. The "agglomeration" rule for counterfactuals fails.

What if we try to build a model to check whether 'B=1 > (M=0 ∧ C=1)' is true? We can't. The three variables B, M, and C are not independent. M=0 and C=1 entail B=0. Any system with these three variables violates the first condition for correct systems: that the variables must be "distinct". Gallow's semantics therefore doesn't tell us whether 'B=1 > (M=0 ∧ C=1)' is true.

The distinctness condition has an independent motivation. We don't want to say that your playing poker causally determines that you play cards. But if we could use one variable P for poker and another C for cards, then the equation C := P would be a correct system. We need the distinctness condition to rule it out.

So much for my summary. I have eleven worries/objections.

One. As Gallow points out, the coin scenario looks like the famous Morgenbesser case. It is widely thought that to vindicate the intuition that B=1 > W=1 ("if you had bet on heads you would have won"), one must put explicitly causal notions into the analysis of the conditional. Gallow gets the intuition right without invoking any casual notions. This looks like an advantage. But I think it's a problem. The Morgenbesser intuition depends on the extent to which the betting is isolated from the coin flip. Take an extreme case of non-isolation: if you bet on heads, the coin is flipped gently with the left hand; if you don't, it is flipped vigorously with the right hand. As before, you don't bet, the coin is flipped (vigorously with the right hand) and lands heads. If you had bet on heads, would you have won? This is far from clear. I'd say no. There would have been an entirely different coin flip. Who knows how it would have landed? But the system composed only of equation (1) is still correct by Gallow's standards. And the exogenous variables still have values { B=0, H=1 }. Revising this model by B=1 still yields W=1. So 'B=1 > W=1' comes out true, in my version of the story with extreme non-isolation. That's the wrong result.

Two. It is crucial to Gallow's account that there is more than one closest B=0 world, even if B=0 is actually true. I already mentioned that I don't really see an independent motivation for this assumption. If the idea is that B=0 can be realised in different ways, and that the equations should be robust across these ways, then OK, but that's not enough. What if B=0 is a very precise description of a bet that can't be realised in relevantly different ways? We don't want to say that if you had taken the bet then there would have been no law violations but if you had taken the bet in such-and-such specific way then there would have been law violations.

Three. Interventionist accounts tend to neglect the need for a "ramp". An example from Bennett (2003): At time t, a dam bursts. Several cars on the valley road are swept away. Some drivers and passengers die. What would have happened if there had been no cars on the road at t? Nobody would have died. (Let's say.) Would there have been an inquiry into the mystery of the disappearing cars? Intuitively, no. We can construct a model with variables for E = whether cars enter the road at t-1, R = whether there are cars on the road at t, I = whether there is an inquiry into the mystery of disappearing cars. The equation I := E * (1-R) is plausibly correct. (This is so even if our closeness standards require a ramp, so that the closest R=0 worlds are E=0 worlds.) Intervening with R=0 therefore yields I=1. But 'R=0 > I=1' is intuitively false.

Four. Continuing the valley road example, if our closeness standards require a ramp, so that cars don't magically appear or disappear at nearby worlds, then the equation E := R comes out correct. But it has the wrong direction. (Lewis (1981a) tried to get around this by hoping that ramp "events" like E would be too disjunctive to count as genuine events. But (a) this depends on the details of the case, and (b) Gallow doesn't have a rule that events/variables can't be disjunctive.) If our closeness standards don't have a ramp, even the equation I := 1-R is correct. Intuitively, neither E := R nor I := 1-R correctly represents the causal structure of the world.

Five. "If you had played poker you would have played cards" is true. It may not be a strictly "causal" counterfactual, but it's the kind of counterfactual that could well be relevant to CDT. (Imagine you assign basic value to playing cards and wonder whether to play poker or go for a walk.) We need a theory of counterfactuals that doesn't just cover conditionals between distinct variables.

Six. Is M actually distinct from B, as required for the correctness of equation (2)? Gallow doesn't formally define distinctness. He suggests that we can model variables as classes of Lewisian events, each of which is a class of possible spacetime regions (see Lewis (1986)). On this understanding, M=0 comprises all of spacetime in all worlds where the actual laws aren't violated, and M=1 comprises all other regions in all worlds. Wherever B=0 and B=1 happen, they're part of the M=0 region or the M=1 region. Without further constraints on events, B=0 and M=0 definitely aren't distinct by the somewhat complicated criteria from Lewis (1986, 258–60).

Seven. The distinctness constraint isn't enough. Let 'Xanthippe's widowing' be the class of possible spacetime regions that are fully occupied by Xanthippe and simultaneous to Socrates's death. This "event" is distinct from Socrates's death. And it "occurs" whenever Socrates dies. If we allow a variable X for Xanthippe's widowing and a variable S for Socrates dying, the equation S := X is correct. (As is X := S.) But there's no causal relationship between S and X. (This problem was raised by Kim (1973) against Lewis (1973).) I worry that whatever constraint rules out equations with S and X (Lewis (1981b) suggested that X is too gerrymandered to count as an event) will also rule out equations with B and M.

Eight. This is really just a feeling of uneasiness. M doesn't look like the kind of variable that should figure in a causal model. Causal models are useful to represent relations of causal influence between concrete and local types of events: whether a brake was released, how fast a ball was thrown, etc. I wouldn't expect that one could have, say, a variable for whether the equation W := B * H is correct in a causal model. But M is just like that.

Nine. Suppose the deterministic laws of nature have a certain parametric form L(x). There are only two possible parameter values, 0 and 1. The true value is 1. Let L state that there are unviolated laws of the form L(x). B is a bet that a certain event in the past (say, the holocaust) never happened. C are the initial conditions. You are sure that the event happened, so you don't choose B. This means that L(1) and C together entail ¬B. L(0) and C together entail that the history of the universe takes a different path. Let's say it leads to a history in which the event didn't happen and you now choose B. We can model the structure of this scenario with variables for L, B, C, and W (for winning the bet). All combinations of values are possible, because L doesn't settle whether the laws are L(1) or L(0). The equation W := C * L * B is correct. Since we have C=1 and L=1, setting B=1 yields W=1. On Gallow's account, we have to conclude that if you had bet that the holocaust didn't happen, then the holocaust would not have happened. (In general, both the laws and the past seem to depend on our present actions if we look at models with suitably weakened versions of M and C.)

Ten. I would have thought that if there are two correct but partial models of the world's causal structure, then they can be combined into a larger correct model. According to Gallow, M1 and M2 can't be combined in this way. Is the total causal structure of the world unrepresentable?

Eleven. Suppose I'm confident that there are no violations of the actual laws (M=0) and that the initial conditions are C (C=1). Someone offers me a deal that pays $1 if M=0 and $-1 if M=1. Gallow says that I should accept. Then someone offers me a deal that pays $1 if C=1 and $-1 if C=0. Again Gallow says that I should accept. But if someone offers me deal that pays $2 if M=0 and C=1, $0 if M=1 and C=0, and $1 otherwise, then Gallow says that it's unclear what I should do because the situation can't be modelled. But isn't the last bet equivalent to the first two? How could it be rationally required to accept the first two and not the last?