Causation and counterfactuals

Suppose that an extremely reliable cannon is loaded with a rock, and pointed at a window, and the extremely reliable timer on the cannon is set for two minutes. Two minutes later, the cannon shoots out the rock causing the window to break.

The Lewisian counterfactual account of causation accounts for the causation by the counterfactual:

1. Were the cannon not to have fired the rock, the window wouldn’t have broken.

But imagine that a risk-taking undersupervised kid was walking by towards the end of the the two minutes, and on a whim considered swapping the rock in the cannon for their steel water bottle. The decision whether to do the swap was an extremely conflicted one, and a single neuron’s made the difference, and resulted in the swap not happening.

We can set up the story in such a way that on Lewis’s way of measuring the closeness of worlds, a world where the kid swapped the rock for the water bottle is closer than any worlds where the timer wasn’t set or where the cannon misfired or where the cannon wasn’t loaded or anything like that. In that case on a Lewisian analysis of counterfactuals:

2. Were the cannon not to have fired the rock, the window would still have broken.

But surely whether the kid walks by or not, the cannon’s firing the rock caused the window to break.

An impartiality premise

In an argument that David Lewis’s account of possible worlds leads to inductive skepticism, I used this premise:

1. If knowing that x is F (where F is purely non-indexical and x is a definite description or proper name) does not epistemically justify inferring that x is G (where G is purely non-indexical), then neither does knowing x is F and that x is I (now, here, etc.: any pure indexical will do) justify inferring that x is G.

This is less clear to me now than it was then. Self-locating evidence might be a counterexample to this principle. I know that the tallest person in the world is the tallest person in the world. But suppose I now learn that I am the tallest person in the world. It doesn’t seem entirely implausible to think that at this point it becomes reasonable (or at least more reasonable) to infer that the number of people in the world is small. For on the hypothesis that the number of people is small, it seems more likely that I am the tallest than on the hypothesis that the number of people is large. (Compare: That I won some competition is evidence that the number of competitors was small.)

But I think I can fix my argument by using this premise:

2. If knowing that x is F (where F is purely non-indexical and x is a definite description or proper name) and that a uniformly randomly chosen person (or other occupied location) is x would not epistemically justify inferring that x is G (where G is purely non-indexical), then neither does knowing x is F and that x is I (now, here, etc.: any pure indexical will do) justify inferring that x is G.

There are multiple versions of (b) depending on how the random choice works, e.g., whether it is a random choice from among actual persons or from among possible persons (cf. self-sampling vs. self-indication).

It takes a bit of work to convince oneself that the rest of the argument still works.

On the plurality of bestnesses

According to the best-systems account of laws (BSA), the fundamental laws of nature are the axioms of the system that are true and optimize a balance of informativeness and brevity in a perfectly natural language (i.e., the language cuts reality perfectly at the joints). There are some complications in probabilistic cases, but those will only make my argument below more compelling.

Here is the issue I want to think about: There are many reasonable ways of defining the “balance of informativeness and brevity”.

First, in the case of theories that rule out all but a finite number of worlds, we can say that a theory is more informative if it is compatible with fewer worlds. In such a case, there may be some natural information-theoretic way of measuring informativeness. But in fact, we do not expect the laws of nature to rule out all but a finite number of worlds. We expect them to be compatible with an infinite number of worlds.

Perhaps, though, we get lucky and the laws place restrictions on the determinables in such a way that provides for a natural state space. Then we can try to measure what proportion of that state space is compatible with the laws. This is going to be technically quite difficult. The state space may well turn out to be unbounded and/or infinite dimensional, without a natural volume measure. But even if there is a natural volume measure, it is quite likely that the restrictions placed by the laws make the permitted subset of the state space have zero volume (e.g., if the state space includes inertial and gravitational mass, then laws that say that inertial mass equals gravitational mass will reduce the number of dimensions of the state space, and the reduced space is apt to have zero volume relative to the full space). So we need some way of comparing subsets with zero volume. And mathematically there are many, many tools for this.

Second, brevity is always measured relative to a language. And while the requirement that the language be perfectly natural, i.e., that it cut nature at the joints, rules out some languages, there will be a lot of options remaining. Minimally, we will have a choice point about grouping, Polish notation, dot notation, parentheses, and a slew of other options we haven’t thought of yet, and we will have choice points about the primitive logical operators.

Finally, we have a lot of freedom in how we combine the informativeness and brevity measures. This is especially true since it is unlikely that the informativeness measure is a simple numerical measure, given the zero-volume issue.

We could suppose that there is some objective fact, unknowable to humans, as to what is the right way to define the informativeness and brevity balance, a fact that yields the truth about the laws of nature. This seems implausible. Absent such a fact, what the laws are will be relative to the choice of informativeness and brevity
measure ρ. We might have gotten lucky, and in our world all the measures yield the same laws, but we have little reason to hope for that, and even if this is correct, that’s just our world.

Thus, the story implies that for any reasonable informativeness and brevity measure ρ, we have a concept of a law_ρ. This in itself sounds a bit wrong. It makes the concept of a law not sound objective enough. Moreover, corresponding to each reasonable choice of ρ, it seems we will have a potentially different way to give a scientific explanation, and so the objectivity of scientific explanations is also endangered.

But perhaps worst of all, what BSA had going for it was simplicity: we don’t need any fundamental law or causal concepts, just a Humean mosaic of the distribution of powerless properties. However, the above shows that there is enormous complexity in the account of laws. This is not ideological complexity, but it is great complexity nonetheless. If I am right in my preceding post that at least on probabilistic BSA the fact that something is a law actually enters into explanation, and if I am right in this post that the BSA concept of law has great complexity, then this will end up greatly complicating not just philosophy of science, but scientific explanations.

On probabilistic best-systems accounts, laws aren't propositions

According to the probabilistic best-systems account of laws (PBSA), the fundamental laws of nature are the axioms of the system that optimizes a balance of probabilistic fit to reality, informativeness, and brevity in a perfectly natural language.

But here is a tricky little thing. Probabilistic laws include statements about chances, such as that an event of a certain type E has a chance of 1/3. But on PBSA, chances are themselves defined by PBSA. What it means to say “E has a chance of 1/3” seems to be that the best system entails that E has a chance of 1/3. On its face, this is circular: chance is defined in terms of entailment of chance.

I think there may be a way out of this, but it is to make the fundamental laws be sentences that need not express propositions. Here’s the idea. The fundamental laws are sentences in an formal language (with terms having perfectly natural meanings) and an additional uninterpreted chance operator. There are a bunch of choice-points here: is the chance operator unary (unconditional) or binary (conditional)? is it a function? does it apply to formulas, sentences, event tokens, event types or propositions? For simplicity, I will suppose it’s unary function applying to event types, even though that’s likely not the best solution in the final analysis. We now say that the laws are the sentences provable from the axioms of our best system. These sentences include the uninterpreted chance(x) function. We then say stuff like this:

1. When a sentence that does not use the chance operator is provable from the axioms, that sentence contributes to informativeness, but when that sentence is in fact false, the fit of the whole system becomes  − ∞.

2. When a sentence of the form chance(E) = p is provable from the axioms, then the closeness of the frequency of event type E to p contributes to fit (unless the fit is  − ∞ because of the previous rule), and the statement as such contributes to informativeness.

I have no idea how fit is to be measured when instead of being able to prove things like chance(E) = p, we can prove less precise statements like chance(E) = chance(F) or chance(E) ≥ p. Perhaps we need clauses to cover cases like that, or maybe we can hope that we don’t need to deal with this.

An immediate problem with this approach is that the laws are no longer propositions. We can no longer say that the laws explain, because sentences in a language that is not fully interpreted do not explain. But we can form propositions from the sentences: instead of invoking a law s as itself an explanation, we can invoke as our explanation the second order fact that s is a law, i.e., that s is provable from the axioms of the best system.

This is counterintuitive. The explanation of the evolution of amoebae should not include meta-linguistic facts about a formal language!

Humeans laws and constants

On Mill-Ramsey-Lewis accounts of laws of nature, the laws are the propositions that best balance informativeness and brevity (in a language that cuts nature precisely at the joints).

Now, the laws of nature include constants, such as the fine-structure constant whose current best measured value is 1/137.035999206. Now, we might be lucky, and it might turn out that the fine-structure constant will have some neat and elegant precise value. There is a history of speculation that it has such a value—for a while, there was hope it was exactly 1/137, and then other guesses took over. But suppose we don’t get so lucky. Suppose it just is some messy number with no simple expression. That should, after all, be a serious possibility.

In that case, the exact value of the fine-structure constant cannot be a part of the Mill-Ramsey-Lewis “world in a nutshell” system of laws, since the system would then be infinitely long, and we lose our hope of defining laws in terms of brevity.

So we have two options. First, the system of laws might not include any specific information on the value of the fine-structure constant, but might instead be of the form ∃αF(α) where F(α) says nothing about what α is, except maybe that it’s real-valued and positive. If we go for this option, then we have to say that all the things that depend on the actual value of the fine-structure constant—and that apparently includes all of chemistry—are not in fact laws of nature. This will likely fail to yield some counterfactuals that we want, and while the laws will be briefer, they will be far less informative than if they had something to say about the value of α.

So that moves us to the second option, which is that the laws are of the form ∃αF(α) and F(α) includes some constraints on α, such as that it lies between 1/137.04 and 1/137.03. These constraints are sufficiently tight to generate the nomic implications we need for chemistry and biology. But while this result seems a better fit for science, it is metaphysically very strange. For it is very strange to think that the laws allow the fine-structure constant to have any of an infinite number of values, but these values must lie in a narrow range.

Furthermore, the exact narrow range for α would be determined by fine details (I am not sure if the pun is intended) of exactly how informativeness and brevity are balanced in the definition of the laws.

The same issue comes up for other constants in the laws of nature. Either Mill-Ramsey-Lewis laws do not include anything about the values of constants or else they include oddly specific, but not completely specific, ranges.

Failures of supervenience on Lewis's system

Start with the concept of “narrowly physical” for facts about the arrangement of physical entities and first-order physical properties such as “charge” and “mass”.

Here are two observations I have not seen made:

1. On Lewis-Ramsey accounts of laws, laws of nature concerning narrowly physical facts do not supervene on narrowly physical facts.

2. On Lewis’s account of causation, causal facts about narrowly physical events do not supervene on narrowly physical facts.

This means that in a Lewisian system we have at least four things we could mean by “physical”:

3. narrowly physical

4. grounded in the laws of narrowly physical facts and/or the narrowly physical facts themselves

5. grounded in the causal facts about narrowly physical events and/or the narrowly physical facts themeselves

6. grounded in the causal facts about narrowly physical events, the laws concerning narrowly physical facts and/or the narrowly physical facts themselves.

Here’s a corollary for the philosophy of mind:

7. On a Lewisian system, we should not even expect the mental properties of purely narrowly physical beings to supervene on narrowly physical facts.

Argument for (1): The laws are the optimal systematization of particular facts. But now imagine a possible world where there is just a coin that is tossed a trillion times, and with no discernible pattern lands heads about half the time. In the best systematization, we attribute a chance of 1/2 to the coin landing heads. But now imagine a possible world with the same narrowly physical facts, but where there is an angel that thought about ℵ₃ about a million times—each time, with a good prior mental explanation of the train of thought—and each of these times was a time just before the coin landed heads. Then the best systematization of the coin tosses will no longer make them simply have a chance of 1/2 of landing heads. Rather, they will have a chance 1/2 of landing heads when the angel didn’t just think about ℵ₃.

Argument for (2): Add to the world in the above argument some cats and suppose that on any day when the fattest cat in the world eats n mice, that leads the angel to think about ℵ_n, though there are other things that can get the angel to think about ℵ_n. We can set things up so that the fattest cat’s eating three mice in a day causes the coin to land heads on the Lewisian counterfactual account of causation, but if we subtract the angel from the story, this will no longer be the case.

Is Lewis's identity theory a type-type identity theory?

David Lewis’s 1983 identity theory of mind holds that:

1. For each mental state type M there is a causal role R_M such that to be a state of type M is to fulfill R_M.

2. For each actually occurring mental state type M, the causal role R_M is fulfilled by physical states and only physical states.

It is normal to take Lewis’s identity theory to be a type-type identity theory.

But a type-type identity theory identifies being a state of type M with some physical state type. So whether Lewis’s identity theory is a type-type identity theory depends then on whether fulfilling R_M counts as a physical state type.

Here are two accounts of what makes a type T be a physical type:

3. Everything falling under T is physical.

4. Necessarily everything falling under T is physical.

If (3) is the right account of the physicality of a state type, then Lewis’s theory is a type-type identity theory, because everything that fulfills R_M is physical according to (2).

However, (3) is an inadequate account of the physicality of a type. Consider the type ghost. That’s paradigmatically not a physical type. But in fact, trivially, everything that is a ghost is physical, simply because there are no ghosts. If one objects that only instantiated types count, then we can note that by (3) the type ghost-or-pig also counts as a physical type, whereas it surely does not.

It seems to me that (4) is a much better account of a physical type. However, on (4) for Lewis’s theory to count as a type-type identity theory, he would need a version of (2) strengthened by deleting “actually” and inserting “Necessarily” in front. And Lewis’s arguments do not establish such a stronger version of (2). Lewis’s arguments are quite compatible with R_M having non-physical realizers in other possible worlds.

That said, perhaps (4) is not the right account of the physicality of a type either. Consider the type believed by God to be an electron. Necessarily, everything falling under this type is an electron, hence physical. But because the definition of the type makes use of supernaturalist vocabulary, the type does not seem to be physical. This criticism points towards an acocunt of physical type like this:

5. The type T is expressible wholly in terms that natural science uses.

It’s essential for this to fit with Lewis’s theory that causation be one of the terms that natural science uses. But now imagine that we live in a world where one being causes spacetime, and it’s a non-physical being. Clearly, the type cause of spacetime is expressible wholly in natural scientific vocabulary, but given that the one and only instance of this type is non-physical, it sure doesn’t sound like a physical type! Indeed, if this (5) is how we understand physical types, then a type-type identity theory does not even imply a token-token identity theory!

We might try to combine (3) with (5):

6. Everything falling under T is physical and the type T is expressible wholly in terms that natural science uses.

But now imagine that there is no being that causes spacetime and all spatiotemporal entities, but that it is possible for there to be such a being, and that any such being would necessarily be non-physical. In that case causes spacetime and all spatiotemporal entities satisfies (6) trivially, but is surely not a physical type, because the only possible instances of it would be non-physical. (If one objects that types need to be instantiated, just disjoin this type with the type pig, as we did in the ghost case.)

So perhaps our best bet is to combine (4) with (5). But any account on which (4) is a necessary condition for the physicality of a type is an account that goes beyond Lewis’s, because it requires the stronger version of (2) with actuality replaced by necessity.

I conclude that Lewis’s account isn’t really a type-type identity theory, except in the inadequate senses of physicality of type given by (3), (5) or (6).

Naturalness and induction

David Lewis’s notion of the naturalness of predicates may seem at first sight like just the thing to solve Goodman’s new puzzle of induction: unlike green, grue is too unnatural for induction with respect to grue to be secure.

But this fails.

Roughly speaking, an object is green provided its emissivity or reflectivity as restricted to the visible range has a sufficiently pronounced peak around 540 nm. But in reality, it’s more complicated than that. An object’s emissivity and reflectivity might well have significantly different spectral profiles (think of a red LED that is reflectively white, as can be seen by turning it off), and one needs to define some sort of “normal conditions” combination of the two features. Describing these normal conditions will be quite complex, thereby making the concept of green be quite far from natural.

Now, it is much easier to define the concepts of emissively black (eblack) and emissively white (ewhite) than of green (or black or white, for that matter) in terms of the fundamental concepts of physics. And emeralds, we think, are eblack (since they don’t emit visible light). Then, just as Goodman defined grue as being observed before a certain date and being green and or being observed after that date and being blue, we can define eblite as existing wholly before 2100 and being eblack or existing wholly after 2100 and being ewhite. And here is the crucial thing: the concept of eblite is actually way more natural, in the Lewis sense of “natural”, than the concept of green. For the definition of eblite does not require the complexities of the normal conditions combination of emissivity and reflectivity.

Thus, if what makes induction with green work better than induction with grue is that greenness is more natural than grueness, then induction with eblite (over short-lived entities like snowflakes, say) should work even better than induction with green, since ebliteness is much more natural than grueness. But we know that we shouldn’t do induction with eblite: even though all the snowflakes we have observed are eblite, we shouldn’t assume that in the next century the snowflaskes will still be eblite (i.e., that they will start to have a white glow). Or, contrapositively, if eblite is insufficiently natural for induction, green is much too unnatural for induction.

Moreover, this points to a better story. Lewisian unnaturalness measures the complexity of a property relative to the properties that are in themselves perfectly natural. But this is unsatisfactory for epistemological purposes, since the perfectly natural properties are ones that we are far from having discovered as yet. Rather, for epistemological purposes, what we want to do is measure the complexity of a property relative to the properties that are for us perfectly natural. (This, of course, is meant to recall Aristotle’s distinction between what is more understandable in itself and what is more understandable for us.) The properties that are for us perfectly natural are the directly observable ones. And now the in itself messy property of greenness beats not only grue and eblite, but even the much more in itself natural property of eblack.

This can’t be the whole story. In more scientifically developed cases, we will have an interplay of induction with respect to for us natural properties (including ones involved in reading data off lab equipment) and in themselves natural properties.

And there is the deep puzzle of why we should trust induction with respect to what is merely for us natural. My short answer is it that it is our nature to do so, and our nature sets our epistemic norms.

The context problem for Lewisian functionalism

One problem for functionalism is the problem of defect. David Lewis, for instance, talks of a madman for whom pain is triggered by something other than damage and whose pain triggers something other than avoidance. Lewis’s functionalist solution is to define the function of a mental state in terms of the role it normally plays in the species.

Here is a problem with this. Suppose that in mammals pains is realized by C-fiber firing. But now take the C-fibers inside a living mammalian skull, disconnect their outputs and connect external electrodes to their inputs. Make the C-fibers fire. Since the C-fiber outputs are disconnected, causing them to fire does not cause any of the usual pain behaviors, the formation of memories of pain, etc. In fact, it seems very plausible that there is no pain at all. Yet according to Lewisian functionalism, there is pain, because it is the normal connections of the C-fibers that define their functional role.

This thought experiment shows that the physical realizers of mental states need to occur in their proper context. But this bumps up against Lewis’s madman, in whom the pain states, and presumably their physical realizers, do not occur in their proper context.

It seems that what the functionalist needs to say is that in order to realize a mental state, a physical state must occur in a sufficient approximation to its proper context. If it’s too far, as in the case of the C-fibers with severed outputs, there is no mental state. If it’s close enough, as in a moderate version of the madman case (I don’t know what to say about Lewis’s more extreme one), the mental state occurs.

But how is the line to be drawn?

Perhaps there is no problem. Pain is not in fact C-fiber firing. Perhaps enough of the brain needs to be involved in conscious states that one cannot plausibly remove the states from their normal functional context? Still, this is worth thinking about.

Counterparts and singletons

Here is an interesting problem for Lewis. Lewis says that sets are necessary beings, and hence count as existing in all worlds. Very plausibly then:

1. If A is a set and w₁ and w₂ are worlds, then A in w₂ is a counterpart of A in w₁.

After all, if identity isn’t good enough for being a counterpart, nothing is. Note that (1) does not say that A at w₂ is the only counterpart of A in w₁. To handle some identical twin scenarios, Lewis may need to allow a world to have more than one counterpart of an object.

Let α be Aristotle. Let A = {α} be the singleton of α. Lewis is now committed to the truth of:

2. Possibly α is not a member of A.

For Lewis’s criterion for whether F(a, b) is possible is whether there is a world w with counterparts a′ and b′ of a and b respectively such that F(a′,b′) holds at w. Let w be a non-actual world where there is a counterpart β of Aristotle. Since individuals are world-bound, β ≠ α. Moreover set membership is necessary, so:

3. β is not a member of A at w.

Since β is a counterpart of α and A is a counterpart of A by (1), it follows that (2) is true. But (2) seems clearly wrong: it is impossible for Aristotle not to be a member of A.

Here’s what seems to me to be the best way out for Lewis: Require pairwise counterparts rather than individual counterparts (in fact, I vaguely remember that Lewis may do that somewhere) for possibility claims involving two objects. Thus that β is not a member of A and β and A are individually counterparts of α and A isn’t enough to make it be that possibly α is not a member of A. One would need β and A to be pairwise counterparts of α and A. But perhaps they’re not. Perhaps, rather, it is β and B = {β} that are pairwise counterparts of α and A. However, this greatly complicates the counterpart relation as well as Lewis’s identification of properties with sets.

Are there unicorns here?

Multiverse theories like David Lewis’s or Donald Turner’s populate reality with a multitude of universes containing strange things like unicorns and witches riding broomsticks. One might think that positing unicorns and witches makes a theory untenable, but the theorists try to do justice to common sense by saying that the unicorns and witches aren’t here. Each universe occupies its own spacetime, and the different spacetimes have no locations in common.

But why take the different universes to have no locations in common? Surely, just as a unicorn can have the same charge or color as I, it can have the same location as I. From the fact that a unicorn can have the same charge or color as I, we infer in a Lewisian setting that some unicorn does have the same charge or color as I (and likewise in Turner’s, with some plausible auxiliary assumptions about values). Well, by the same token, from the fact that a unicorn can have the same location as I, we should be able to infer that some unicorn does have the same location as I.

Not so, says Lewis. Counterpart theory holds for locations, but not for charges and colors. What makes it true that a unicorn can have the same charge as I now have is that some unicorn does have the same charge as I. But what makes it true that a unicorn can have the same location as I now have is that some unicorn has a counterpart of my location in a different spacetime.

But what justifies this asymmetry between the properties of charge and location? The asymmetry seems to require clauses in Lewis’s modal semantics that work differently for different properties. It seems there are properties—say, being green—whose possible possession is grounded in something’s having the property, and there are properties—say, being at this location—whose possible possession is grounded in something’s having a a counterpart of the property.

Specifying in a non-ad hoc way which properties are which rather complicates the system. Moreover, it leads to this oddity. Lewis thinks abstract objects exist in all worlds. So, he has to say that being at this location exists in all worlds. And yet the counterpart of being at this location in another world is a different property, even though this exact property does exist at that world.

There is a solution for Lewis. Lewis is committed to counterpart theory holding for objects. It is reasonable for him thus to take counterpart theory also to hold for properties defined de re in terms of particular non-abstract objects. Thus, what makes it true that a unicorn could have had the property of being a mount of Socrates is not that some unicorn in some universe has this property—for no unicorn in our universe has that property, and Socrates according to Lewis only exists in our world—but that some unicorn has a counterpart to this property, which counterpart property is the property of being a mount of S where S is a counterpart of Socrates.

If Lewis can maintain that location properties are defined de re by relation to non-abstract objects, then he has a way out of the objection. Two kinds of theories allow a Lewisian to do this. First, the Lewisian can be a substantivalist who thinks that points or regions of space are non-abstract. Then being here will consist in being locationally related to some point or region L, and Lewis can take counterpart theory to apply to points or regions L. Second, the Lewisian can be a relationalist and say that location is defined by relations to other physical objects, in such a way that if all the objects were numerically different from what they are, nothing could be in the same place it is, and counterpart theory is applied to physical objects by Lewis.

What Lewis cannot do, however, is take a view of location that either takes location to be a relation to abstract objects—say, sets of points in a mathematical manifold—or that takes location to simply be a non-relational determinable like charge or rest mass.

In particular, multiverse theorists like Lewis and Turner are committed to treating location as different from other properties. Anecdotally, most philosophers do treat location like that. But for those of us who are attracted to the idea that location is just another determinable, this is a real cost.

An account of laws

According to the Lewisian best system analysis of laws, a proposition p is a fundamental law if and only if it is an axiom in the best system. There is room for variation in the concept of a best system, but a standard version in deterministic settings is that the best system comprises only truths and optimizes the brevity of its axioms and the informativeness of its theorems about the world. The biggest problem for me with the best system account is that the fact that something is an axiom in the best system simply does not make it be explanatory.

I think this is a better account. A proposition p is a fundamental law of nature provided that:

1. p is an axiom in the best system, and
2. God wills p, as such.

(I am not sure if the will in (2) should be taken to be antecedent or consequent. If miracles are counterinstances to laws, it must be antecedent. But a lot of people think that's a bad account of miracles, and that allows it to be consequent.)

The "as such" in (2) rules out a case where God instead of willing p, wills something that entails p.

This account solves the explanatory problem with Lewis's account by making the fundamental laws be explanatory. They are not explanatory directly because they are axioms in the best system, but rather because God wills them.

Interestingly, I think (1) may imply (2) in the actual world, by divine omnirationality. For that p has the kind of simplicity and fecundity that axioms in the best system are going to have gives God a reason to will p. And since p in fact holds, presumably God willed p. The only exception is going to be if p reports the sort of thing that God has reason to distance himself from. Suppose, for instance, that everyone who is tempted a certain way sins. Then that universal generalization might be a best system axiom, but God has reason not to will it. But in fact it does not seem that any axioms in our world's best system are going to be like that—such regularities don't seem to be far-reaching enough. All the candidates we hear about from physics are propositions that God does not seem to have reason to distance his will from.

If this is right, then in the actual world, all the axioms in the best system are fundamental laws, and Lewis is contingently right. Moreover, this line of thought shows that the fact that p is an axiom in the best system makes it likely that God wills it. Consequently, as long as we know that God exists, we get to keep the epistemological benefits of Lewis's system.

Of chocolates and laws

According to David Lewis, the fundamental laws of nature are those propositions that collectively optimize a balance of the twin desiderata of informativeness and simplicity. (You can get maximal informativeness by listing all the facts about the world, at the expense of maximal complexity of description, and you can get maximal simplicity by saying nothing, at the expense of minimal informativeness.) And laws are what follows from the fundamental laws.

But laws are explanatory. While Lewis's laws aren't.

Imagine a finite world w₁ which contains a powerful contingent magical being who creates a very large number N of golden boxes, and places a chocolate in each one, to fulfill some aesthetic goal—chocolate goes well with gold. No other explanation of the chocolate content of the boxes exists at w₁.

We can ask at w₁ why the third golden box contains a chocolate. And the answer is that the magical being put a chocolate in every golden box.

Now imagine a world w₂ just like w₁ but where the magical being doesn't exist and the boxes and chocolates come into existence ex nihilo. Nothing gets added to w₂ that wasn't there in w₁. Plainly at w₂ there just is no explanation of why the third golden box contains a chocolate.

If the number of boxes N is large enough, the proposition that every box contains a chocolate will be informative enough that it'll be included in the system that maximizes informativeness and simplicity (as N increases, the informativeness of the proposition increases but its simplicity stays constant).

But then if Lewis's account of laws is correct, then at w₂ it will be a law that all golden boxes contain chocolates. But laws are explanatory. So at w₂ we'll be able to explain why the third golden box contains a chocolate. But we said that at w₂ there is no explanation of this!

So the Lewisian account of laws is wrong.

Options: (1) Deny that laws are explanatory. (2) Abandon the Lewisian account of laws. (3) Deny that it is possible to have uncaused chocolates.

I think that (3) by itself won't do, because we can run the argument on a counterpossible. And (1) is unattractive. That leaves (2).

In summary, I think "Lewisian laws" aren't explanatory, and hence aren't laws.

... because the world is not in the world

Yesterday, I overheard my six-year-old son Dominic saying in another room "... because the world is not in the world."  In later conversation, he told me what the conclusion of the argument was: "The world isn't the biggest thing in the world."

If Dominic is right that the world is not in the world, then on Lewis's semantics the worlds are not possible entities and in particular are not possible worlds.  For on Lewis's semantics, something is possible provided it exists in some world.

A partial fix to Lewis's account of counterfactuals

A serious problem with Lewis's account of counterfactuals is that as Elga and Pruss have shown, Lewis's similarity account licenses errant counterfactuals of the form "Were B to occur at t₁, A would have occurred at t₀", where t₀ < t₁. A quick argument for this conclusion is as follows. We have good reason to think the universe's future is longer than the past. On the grounds of divine revelation this is clear: the future is forever but the world was created a finite amount of time ago. On the grounds of science this is also likely: the universe is expanding, and either it will extend forever, or it will eventually reverse and collapse, but even in the latter case we're not yet at the half-way point. Therefore, it seems, a world that exactly matches our world in all of the future but in very little of the past will always be more like our world than a world that matches our world in all of the past but in very little of the future. Hence, if we look at worlds close to ours in which the antecedent of a subjunctive holds, we will do better to look at worlds whose future matches the future of our world, and then retrodict to the past from the antecedent of the subjunctive.

But here is an interesting partial solution to this problem. Suppose we abandon Lewis's counterpart theory. Suppose, further, that we assume that all of the causal history of an event is essential to the event's identity. (It is plausible that some of the causal history is essential, but the possibility of drawing a line as to what is and what is not essential is implausible.) Now if exact match of spatiotemporal regions of worlds requires the numerical identity of the events in these regions, it is impossible for two worlds in which causation goes orthodoxly from past to future to match in the future but not in the past. The asymmetry in causation together with the essentiality of origins of events thus induces an asymmetry in counterfactuals, ruling out most backtracking counterfactuals. Of course Lewis wouldn't have liked this, since it means that causation is prior to counterfactuals.

Of course, other problems for Lewis remain.