Explanatory parsimony versus theoretical parsimony

Like the previous Ptolemaic model, Copernicus’ heliocentric model of the solar system required epicycles to predict the astronomical observations of the positions of the planets in the night sky. (The reason was that Copernicus’ orbits were circular instead of elliptical, and I suppose he needed epicycles to simulate the ellipticality.) Indeed, by Gingerich’s count, the Copernican system had more epicycles than the Ptolemaic.

What advantage did the Copernican theory have over the Ptolemaic, then?

The Copernican theory did not need epicycles to predict the qualitative movements of the planets in the night sky, with the famous retrograde motion where sometimes planets move from night to night in the opposite of their usual direction. Such qualitative features are nicely accounted for by the fact that the planets, including earth, are all orbiting the sun, and thus sometimes the earth overtakes a slower outer planet which thus appears to be moving backwards. On the other hand, Ptolemaic theory is both quantitatively and qualitatively wrong without epicycles: it needs epicycles to predict the existence of retrograde motion.

We think of the Copernican theory as superior due to its greater simplicity. But the theory as a whole is not simpler. However, it provides a simpler explanation of a puzzling or significant phenomenon.

This suggests an interesting theoretical virtue, which we might call not just parsimony, but explanatory parsimony, where the theory offers a particularly parsimonious explanation of one or more specific phenomena against a competitor, even if there happens to be little difference in the theories’ overall parsimony as needed for explaining all the phenomena, and even if there is less overall parsimony, as Gingerich claims was the case for Copernicus.

Take a more human case. We could explain the assassination of JFK in terms S₁, a theory about Oswald working alone, or we could explain it in terms of, S₂, where Oswald is working together with the CIA. The non-conspiracy explanation is more parsimonious: it involves two persons—Oswald and JFK—as opposed to many more. But notice that neither S₁ nor S₂ is capable of explaining all the humanly significant phenomena of 1963. To that end, both theories need to be enriched to include the activities of all of the 3.2 billion people of 1963. Let T₁ and T₂ be such respective enrichments of S₁ and S₂. These are vast theories.

Is T₁ more parsimonious than T₂? I doubt it. After all, the people at the CIA who, according to S₂, were working with Oswald were still doing something according to T₁, and indeed something that explains events of human significance. This activity is included in T₁.

Theory T₁ superior to T₂ not in terms of overall parsimony—or even the overall parsimony of the explanatorily fundamental posits—but in terms of how large a portion of (or approximation to?) the theory is needed to explain the specific events around the assassination of JFK. On T₁, a much smaller theory (with primary focus on two persons) S₁ is sufficient for the assassination, while on T₂, a much larger portion S₂ is needed.

If this is right, then we like focused explanations to be neater and smaller, even if the overall story is equally complex.

Here is an interesting application. Physicists have a strong preference for local theories, in the broad sense of the word according to which the behavior of a system depends only on what is happening near the system. But notice that in practice locality is often an approximation. Each system is affected to some degree by everything in its past light cone. To get a full explanation of the system, we need the whole past light cone. But we still have locality, because we get explanations of how the system works to a high degree of approximation in terms of how things are nearby, because all of the forces of physics drop off with distance. A preference for locality is thus a special case of a preference for explanatory parsimony: we want the approximate behavior of the system to have a more parsimonious explanation, even if the fully precise behavior needs a profligate one.

The general feature thus is this. It counts in favor of a theory if a puzzling or significant coarse-grained feature of our observations is explained in a more parsimonious way according to the theory, even if there is no parsimony advantage with respect to finer grained observations.

From a Bayesian point of view, this translates into a view on which we should have higher priors for theories that explain more of puzzling or significant phenomena with less.

Is the anthropocentrism in the notion of “puzzling or significant phenomena” a problem? Maybe not on my natural law epistemology: our epistemology is human epistemology.

I wonder if there is a philosophy of religion application. For instance, consider that theism provides a parsimonious explanation of the coarse-grained fact that there are rational animals: a perfect being has good reason to create rational animals given the objective value of such animals. But once we look at more fine-grained facts, such as that the rational animals have ten fingers and ten toes, it can be contended that we no longer have a more parsimonious explanation. Indeed, to explain all the details of humans, the theist is likely to need to invoke evolutionary mechanisms just as the naturalist does. Could the theist say that the more parsimonious explanation of the coarse-grained fact still counts for something?

I am also now thinking that this story gives the nominalist a good response to the Platonist in the context of yesterday’s post. The fact that a theory includes lots of fundamental nameable things is not very significant. What is significant is the ideology that needs to be brought to bear on particular explanations. When explaining our reaction to seeing grass, say, the Platonist needs to invoke not just the predicate “instantiates” but also the name “greenness”, while the nominalist only needs the predicate “is green”.

Existing for all times is still puzzling

Let’s assume causal finitism is false, and that the past can be infinite.

If an object pops into existence ex nihilo at a specific time t, that feels objectionably arbitrary, since we want to know why did it pop into existence then rather than earlier or later (or if it couldn’t have done that, for essentiality of origins reasons, why didn’t something just like it do that).

But if the object has existed for infinite time, that does not feel arbitrary, and some people have the intuition that the object does not call out for an explanation even if it is metaphysically contingent.

This failure to be puzzled is I think based on a mistake. For even if the object existed for an infinite time, we can ask why it didn’t exist for a longer infinite time. That question makes sense. After all, if the object existed on days ...,  − 3,  − 2,  − 1, 0, where 0 is today, then we can ask why before all those days it didn’t exist on an earlier day, one which we might denote  − ω, where ω is the first infinite ordinal.

One might answer: “Well, it didn’t exist on day  − ω, because time does not include day  − ω.” But that doesn’t solve the puzzle. After all, imagine that on day  − 18 an object popped into existence, and when we are puzzled by this, we are told: “Well, day  − 18 is the first day of time. The object couldn’t have existed on day  − 19 because there was no such day.” But now we want to know: Why did time itself begin on day  − 18 rather than on day  −19 or day −10⁹⁴? And similarly, we should want to know why time extends only to the dates ...,  − 3,  − 2,  − 1, 0 and not to day  − ω, or day  − ω − 1, or day  − 2^ω₉₄.

In fact, there is a sense in which the first of the following questions might be less puzzling than the second:

• Why did time begin on day  − 18 rather than day  − 10⁹⁴?

• Why did time extend only to days ...,  − 3,  − 2,  − 1, 0 rather than beginning with day  − ω, inclusive?

For if time is relational, then there might be no real metaphysical difference between time beginning on day  − 18 and time beginning on day  − 10⁹⁴, but even so there is a metaphysical difference between time extending only to days ...,  − 3,  − 2,  − 1, 0 and its starting on  − ω. If time extends only to days ...,  − 3,  − 2,  − 1, 0, then it has no first day, but if it starts on day  − ω, then it has a first day, namely  − ω.

In fact, there are infinities of different possible temporal orders, all of them linear (i.e., reflexive, transitive and total). Why is time one of these rather than another? If times are representable as real numbers (as classical physics assumes), why? Why not merely as rational numbers? Or, more expansively, why not as hyperreals? Or why not as an order that looks like the real numbers with zero removed?

Whatever is the time sequence occupied by an object, even if the object fills the time sequence, it calls out for an explanation why the time sequence that it fills is this one rather than another one.

Explanatory infinitism

I wish one could formulate and defend a version of explanatory finitism, roughly saying that nothing has infinitely many explanations or an infinite chain of explanations, even if these explanations are not causal (causal finitism of course rules out the causal case). But it seems hard to do so. There do seem to be counterexamples to explanatory finitism. Here some I’m thinking about right now:

1. Why are there infinitely many primes? Because there are infinitely many primes bigger than two. Why are there infinitely many primes bigger than two? Because there are infinitely many primes bigger than three. And so on.

2. Alice has just gone to heaven. Why is she going to experience a day of bliss? Because she will have a day of bliss tomorrow. And also because she will have a day of bliss the day after tomorrow. And also because she will have a day of bliss the day after that. And so on.

3. Alice has just gone to heaven. Why is she going to experience bliss on every future day? Partly because she will have bliss tomorrow. And partly because she will have bliss the day after tomorrow. And so on.

Some Fitch-style arguments

Let strong physicalist supervenience (SPS) be the view that necessarily all truths supervene on first-order truth about the spatiotemporal arrangement of physical stuff.

Here is a plausible thesis:

1. Every first-order truth about the spatiotemporal arrangement of physical stuff is possibly known.

Add this premise:

2. If a conjunction is known, so are its conjuncts.

It follows from SPS, 1 and 2 that:

3. Every actual first-order truth about the spatiotemporal arrangement of physical stuff is actually known.

For suppose that p is any actual first-order truth about the spatiotemporal arrangement of physical stuff. Let q be the conjunction of all other actual first-order truths about the spatiotemporal arrangement of physical stuff. Let r be the conjunction of p and q. Then r is a first-order truth about the spatiotemporal arrangement of physical stuff. By 1, there is a possible world w where r is known. Since r contains all actual first-order truths about the spatiotemporal arrangement of physical stuff, w and the actual world have the same first-order truths about the spatiotemporal arrangement of physical stuff. By SPS, it follows that w and the actual world have the same truths. Thus, since it’s true at w that r is known, it’s actually true that r is known. And since r is actually known, so is p by 2. Hence, we have 3.

In exactly the same way, given:

4. Every first-order truth about the spatiotemporal arrangement of physical stuff is possibly explained

5. If a conjunction is explained, so are its conjuncts

and SPS, we get:

6. Every actual first-order truth about the spatiotemporal arrangement of physical stuff actually has an explanation.

But given physicalism, it is clear that 3 and 6 are false. Regarding 3, it is clear that if physicalism is true, nobody knowns how many hairs I have on my head. Regarding 6, given physicalism, there is no explanation of the sum total of physical reality. Thus:

7. Given either 1 and 2, or 4 and 5, at least one of physicalism and SPS is false.

It is tempting to turn this into an argument against physicalism by holding that:

8. If SPS is true, physicalism is true.

But actually I can imagine someone denying 8. Imagine a classical theist who thinks that everything other than God has to be physical. Such a classical theist will actually believe SPS. For on classical theism, facts about God supervene on facts about everything other than God, since classical theism includes divine simplicity according to which there are no contingent facts intrinsically about God, so any two worlds that differ in their facts about God will have to differ in something other than God. And according to our classical theist, everything other than God has to be physical. So any two worlds that differ must differ in facts about the arrangement of physical stuff, which is what SPS says. However, the classical theist is not a physicalist, because God is not physical on classical theism.

Still, I think an argument against the conjunction of physicalism and SPS is still interesting.

A test case for explanationism

Here is a test case for explanationist stories about initial priors, on which a more explanatory theory has a higher initial prior. Consider these hypotheses:

1. The universe is not created or governed by reason, has a finite lifetime, and has low initial entropy.

2. The universe is not created or governed by reason, has a finite lifetime, and has low final entropy.

Causation goes from past to future, so absent some kind of foresight involved in the initial conditions, initial conditions are more explanatory than final conditions. Now, presumably there is a nice one-to-one correspondence between worlds where (1) and (2) hold, so absent an explanationist bias in the priors, we shouldn’t have a preference between (1) and (2).

This is just a test case, but I have to confess I don’t have any direct intuition comparing the probabilities of (1) and (2). I like explanationism, so I have a theory-laden reason to assign a higher probability to (1) than to (2).

Optimalism and mediocritism

We can think of the optimalist theory of ultimate explanation as the claim:

1. Necessarily, that reality is for the best explains everything.

(I won’t worry in this post about two details. First, whether “reality” in (1) includes the principle of optimality itself—Rescher has suggested that it does, since it’s for the best that everything be for the best. Second, whether “reality” is all the detail of the world, or just the “core” of the world—the aspects not set by indeterministic causation.)

Given that only truths explain, (1) entails:

2. Necessarily, reality is for the best.

Notice that one could accept (2) without accepting (1). One might, for instance, be a Leibnizian and think that there is a two-fold structure to ultimate explanation: first, God’s existence is explained by the ontological argument and, second, God creates the best contingent reality. On this account everything is for the best, but that everything is for the best is not the ultimate explanation, because it does not explain why God exists. Or one might think that reality is necessary and brute, and it brutely has to be like it is. And as a very suprising but non-explanatory matter of fact the way it is is in fact optimal.

I am emphasizing this, because I want to problematize (1). Grant (2). Why should we think that the fact that everything is for the best in fact explains everything?

Suppose that modal fatalism is true, and that it so happens that reality is exactly mid-way between the worst and the best possibility, and is in the only option mid-way between the worst and best. (I assume one can talk of options for reality even given modal fatalism. Otherwise, optimalism falls apart. The “options for reality” may be something like narrowly logically possible worlds.) Then:

3. Necessarily, reality is exactly middling.

Now suppose a “mediocritist” said: “And that reality is necessarily exactly middling explains why reality is what it is.” But why would we buy that? Or suppose that reality is necessarily the only one that is exactly 56.4% of the way up between the worst and the best (where worst would count as 0% of the way up and best as 100%)? Surely we wouldn’t conclude that its being exactly at 56.4% explains why it is the way it is. But if not, then why should its being at 50% explain it, as on mediocritism, or its being at 100% explain it, as on optimalism?

I think what the optimalist ought to say at this point is that analogously to non-Humean pushy laws of nature, there are non-Humean pushy laws of metaphysics. One of these laws is that everything is for the best. It is the pushiness of this metaphysical law that explains reality. But there is something rather odd about pushy laws prior to all beings—they seem really problematically ungrounded.

Explanatory principlism

There are three views about ultimate explanations of reality:

1. Nihilism: There is no ultimate explanation of reality.

2. Onticism: The ultimate explanation of reality involves one or more beings.

3. Principlism: The ultimate explanation of reality involves principles rather than beings.

Nihilism is the standard view among atheists. Onticism is the standard view among theists. The main examples of principlism are axiarchism and optimalism, on which reality is explained by its having the kind of value it does. But other combinations are possible: Rescher was an principlist theist, since he thought that God’s existence could be explained by the fact that it’s for the best that God exists.

I want to say a little about what I think is wrong about principlism. Start with the observation that if truthmaker maximalism is true, principlism cannot get off the ground, because whatever principle helps explain the world is made true by the existence of some being, and if an explanatory proposition is made true by a being, that being is certainly “involved” in the explanation.

Truthmaker maximalism is false. However, I think that a neighboring grounding view is true:

4. Being grounds truth (BGT): All truths are grounded in a combination of what there isn’t, what there is and how what is is.

Given BGT, we have very good reason to reject principlism. For presumably all the principles involved in the explanation of reality are true. If any of them are grounded in what there is or how what is is, then we have onticism rather than principlism: onticism is compatible with principles being involved in explanation, as long as beings are also involved. Thus the only way to defend principlism is to say that what gives the ultimate explanation are principles made true solely by the nonexistence of certain entities. And it is very implausible that the nonexistence of certain entities is the ultimate explanation of our reality, rich in being as it is.

The one version of principlism I can think of that I can reconcile with BGT would be an explanation of our reality in terms of the non-existence of beings that would prevent this reality. (This is kind of like the idea in the Scotus argument that nothing can prevent the existence of God, so God exists.) But this is dubious. Mere lack of preventers of x is not enough to explain the existence of x. One would need some kind of basic principle of plenitude on which everything not prevented must exist.

An argument against the Thomistic primary/secondary causation account of strong providence

Most people agree that one cannot have circularity in the order of explanation when one keeps the type of explanation fixed. Some like me think one cannot have circularity in the order of explanation at all. I argued for this thesis in my previous post today. Now I want to draw an interesting application.

On one influential (and I think exegetically correct, pace Eleonore Stump) reading of Aquinas, God decides what our free choices will be. Our free choices cannot be determined by created causes, but they are determined by God. This is because God’s causation is primary causation which is of a different sort from the secondary causation which is creaturely causation. God can primarily cause you to freely secondarily cause something, and this is how providence and free will interact. Often the analogy between an author and a character is given: the author decides what the character will freely do and this does not infringe on the character’s freedom.

But now observe this (which was brought home to me by a paper of one of our grad students). On this picture, God will presumably sometimes providentially make earlier actions happen because of later ones. Thus, God may want you to perform some heroic self-sacrifice in ten years. So, right now God prepares you for this by having you freely engage in small self-sacrifices now. In the “because” corresponding to the explanatory order of providence and primary causation, we thus have:

1. You engage in small self-sacrifices because you will engage in a great self-sacrifice.

However, divine primary causation does not undercut secondary causation, and we have the standard Aristotelian story of habituation at the level of secondary causation in light of which we have:

2. You will engage in a great self-sacrifice because you are engaging in small self-sacrifices.

These explanations form a heterotypic explanatory loop (i.e., we have explanations of two different sorts in opposite directions). But if I am right that no explanatory loop is possible, the above story is not possible. However, there is nothing to rule out the above story if the above Thomistic account of primary and secondary causation’s role in providence is correct. Hence, I think we should reject that account.

If no homotypic circles of explanation, no heterotypic ones either

Most people agree that one cannot have circularity in the order of explanation when one keeps the type of explanation fixed, i.e., there are no homotypic circles of explanation. Some like me think one cannot have circularity in the order of explanation at all. Why? One intuition might be that explanations of all types are still explanations, and so the circularity is still an explanatory circularity. :-) (Yes, that begs the question.) More seriously, heterotypic explanations (namely, explanations of different types) can be combined, sometimes chainwise (A explains B and B explains C, and thereby A explains C) and sometimes in parallel (A explains B and C explains D so A-and-C explains B-and-D). This means that the types of explanation are not quite as separate as they might seem.

Here is an argument building on the second intuition. We need two concepts. First, we can talk of two sets of explanatory relations as independent, namely without any interaction between the explanatory relations in the two sets. Second, given two type of explanation₁ and explanation₂, I will say that explanation_1|2 is a type of explanation where explanation₁ and explanation₂ are combined in parallel.

1. If circularity in explanation is possible, it is possible to have a two-item heterotypic explanatory loop.

2. If it is possible to have a two-item heterotypic explanatory loop, it is possible to have two independent two-item heterotypic explanatory loops where each loop involves the same pair of explanation types as the other loop.

3. Necessarily, if A explains₁ B and C explains₂ D, and the two explanatory relations here are independent, then A-and-C explains_1|2 B-and-D.

4. Necessarily, the relations explains_1|2 and explains_2|1 are the same.

5. It is not possible to have a circle of explanations of the same type.

6. Suppose circularity in explanation is possible. (Assume for reductio)

7. There is a possible world w, such that at w: there are A, B, C and D such that A explains₁ B, B explains₂ A, D explains₁ C and C explains₂ D, and the above explanatory relations between A and B are independent of the above explanatory relations between C and D. (6,1,2)

8. At w: A-and-C explains_1|2 B-and-D. (3,7)

9. At w: B-and-D explains_2|1 A-and-C. (3,7)

10. At w: B-and-D explains_1|2 A-and-C. (4,9)

11. At w: there is a circle of explanations of type 1|2. (8,10)

12. Contradiction! (5,11)

13. So, circularity in explanation is impossible.

I think the most problematic premise in this argument is (4). However, if (4) is not true, we have a vast multiplication in types of explanation.

A violation of the Principle of Sufficient Reason?

I have a number of times over my career claimed that in the ordinary course of life, we don’t take seriously the hypothesis that something we can’t find an explanation for has no explanation.

Well, I now had an opportunity for observing what happens psychologically to me when I can’t find an explanation.

A couple of days ago, my wife found a significant pool of water in the morning on the top surface of our clothes dryer. When I looked at it, it was like 250ml or more. If it were on the floor or on the washer, I would expect it was from a washer-related leak. If our clothes dryer had a water connection for steaming clothes, a leak would make sense (ChatGPT 3.5 suggested this hypothesis). If the quantity were lower, it could easily be from wet clothes put carelessly on top of the dryer or condensation. If there was wetness in the cabinets above the dryer, it would likely be a leak in one of the many containers of cleaning, photo-developing and other chemicals stored there. If the ceiling showed a discoloration above the dryer, it would be a leak from upstairs. If the liquid smelled, it might be urine from the cat sneaking in.

But none of these apply, to the point where my best four explanations are all hard to believe:

1. a family member sleepwalking with a glass of water, wandering into the laundry room, spilling the water, and walking away,

2. God doing a miracle just to impress on me that there are more things in heaven and earth than are dreamed of in my natural philosophy,

3. a very precisely aimed horizontal leak from one of the faucets in the room, none of which are above the dryer (the next morning, there were slight leaks in two faucets in the room, but the leaks were a non-directional wetness rather than a jet aimed at a precise target).

4. a family member spilling water (from what?) on the dryer and forgetting all about it.

(A plumber called in for the faucet leaks could think of no explanation, except to note that there are many plumbing problems given our current Texas freeze.)

What is my psychology about this? I can’t get myself to believe any of (a)–(d), or even their disjunction. I find myself strongly pulled to just forget the event, to pretend to myself that the event was but a dream, and it now seems to me that that is one way in which we cope with unexplained events. But of course my wife remembers the event, and I can’t get myself to take seriously the idea that we both had the same dream (plus there was no waking up after it—after we cleaned up the spill, I launched into other activities rather than finding myself back in bed).

What about this option?

5. The event has an explanation: it violates the Principle of Sufficient Reason.

I also can’t take (e) seriously. But do I take (e) less seriously than the options in (a)–(d)? Speaking of subjective feelings, I don’t think I feel much more incredulous about (e) than about (a)–(d).

So what do I really think? I guess:

6. There is a mundane explanation and I am not smart enough to think of it.

A dilemma for best-systems accounts of laws

Here is a dilemma for best-systems accounts of laws.

Either:

1. law-based scientific explanations invoke the lawlike generalization itself as part of the explanation, or

2. they invoke the further fact that this generalization is a law.

Thus, if it is a law that all electrons are charged, and Bob is an electron, on (1) we explain Bob’s charge as follows:

3. All electrons are charged.

4. Bob is an electron.

5. So and that’s why Bob is charged.

But on (2), we replace (3) with:

6. It is a law that all electrons are charged.

Both options provide the Humean with problems.

If it is just the lawlike generalization that explains, then the explanation is fishy. The explanation of why Bob is charged in terms of all electrons being charged seems too close to explaining a proposition by a conjunction that includes it:

7. Bob is charged because Bob is charged and Alice is charged.

Indeed both (3)–(5) and (7) are objectionably cases of explaining the mysterious by the more mysterious: the conjunction is more mysterious than its conjunct and the universal generalization is more mysterious than its instances.

On the other hand, suppose that our explanation of why Bob is charged is that it’s a law that all electrons are charged. This sounds correct in general, but is not appealing on a best-systems view. For on a best-systems view, what the claim that it’s a law that all electrons are charged adds to the claim that all electrons are charged is that the generalization that all electrons are charged is sufficiently informative and brief to make it into the best system. But the fact that it is thus informative and brief does not help it explain anything.

Moreover, if the problem with (3)–(5) was that universal generalizations are too much like conjunctions, the problem will not be relieved by adding more conjuncts to the explanation, namely that the generalization is sufficiently informative and brief.

On two problems for non-Humean accounts of laws

There are three main views of laws:

• Humeanism: Laws are a summing up of the most important patterns in the arrangement of things in spacetime.

• Nomism: Laws are necessary relations between universals.

• Powerism: Laws are grounded in the essential powers of things.

The deficiencies of Humeanism are well known. There are also deficiencies in nomism and powerism, and I want to focus on two.

The first is that they counterintuitively imply that laws are metaphysically necessary. This is well-known.

The second is perhaps less well-known. Nomism and powerism work great for fundamental laws, and for those non-fundamental laws that are logical deductions from the fundamental laws. But there is a category of non-fundamental laws, which I will call impure laws, which are not derivable solely from the fundamental laws, but from the fundamental laws conjoined with certain facts about the arrangement of things in spacetime.

The most notorious of the impure laws is the second law of thermodynamics, that entropy tends to increase. To derive this from the fundamental laws, we need to add some fact about the initial conditions, such as that they have a low entropy. The nomic relations between universals and the essential powers of things do not yield the second law of thermodynamics unless they are combined with facts about which universals are instantiated or which things with which essential powers exist.

A less obvious example of an impure law seems to be conservation of energy. The necessary relations between universals will tell us that in interactions between things with precisely such-and-such universals energy is conserved. And it might well be that the physical things in our world only have these kinds of energy-conserving universals. But things whose universals don’t conserve energy are surely metaphysically possible, and the fact that such things don’t exist is a contingent fact, not grounded in the necessary relations between universals. Similarly, substances with causal powers that do not conserve energy are metaphysically possible, and the non-existence of such things is at best a contingent fact. Thus, to derive the law of conservation of energy, we need not only the fundamental laws grounded in relations between universals or essential powers, but we also need the contingent fact that conservation-violators don’t exist.

Finally, the special sciences (geology, biology, etc.) are surely full of impure laws. Some of them perhaps even merely local ones.

One might bite the bullet and say that the impure laws are not laws at all. But that makes the nomist and powerist accounts inadequate to how “law” gets used in science.

The Humean stands in a different position. If they can account for fundamental laws, impure laws are easy, since the additional grounding is precisely a function of patterns of arrangement. The Humean’s difficulty is with the fundamental laws.

There is a solution, and this is for the nomist and powerist to say that “law of nature” is spoken in many ways, analogically. The primary sense is the fundamental laws that the theories nicely account for. But there are also non-fundamental laws. The pure ones are logical consequences of the fundamental laws, and the impure ones are particularly important consequences of the fundamental laws conjoined with important patterns of things in nature. In other words, impure laws are to be accounted for by a hybrid of the non-Humean theory and the Humean theory.

Now let’s come back to the other difficulty: the necessity worry. I submit that our intuitions about the contingency of laws of nature are much stronger in the case of impure laws than fundamental laws or pure non-fundamental laws. It is not much of a bullet to bite to say that matching charges metaphysically cannot attract—it is quite plausible that this is explained by thevery nature of charge. It is the impure laws where contingency is most obvious: it is metaphysically possible for entropy to decrease (funnily enough, many Humeans deny this, because they define the direction of time in terms of the increase of entropy), and it is metaphysically possible for energy conservation to be violated. But on our hybrid account, the contingency of impure laws is accounted for by the Humean element in them.

Of course, we have to check whether the objections to Humeanism apply to the hybrid theory. Perhaps the most powerful objection to a Humean account of laws is that it only sums up and does not explain. But the hybrid theory can explain, because it doesn’t just sum up—it also cites some fundamental laws. Moreover, it may be the case that the patterns that need to be added to get the impure laws could be initial conditions, such as that the initial entropy is law or that no conservation-violators come into existence. But fundamental law plus initial conditions is a perfectly respectable form of explanation.

Laws of nature are hyperintensional

Are the laws of nature hyperintensional? I.e., if p and q are logically equivalent, could it be that one of them is a law of nature and the other is not?

I am inclined to think so.

Argument 1: The laws of nature in our world do not make reference to particular substances. But if p is a law of nature, then let q be the proposition that p is true and either Biden is president or Biden is not presiden. Then p and q are logically equivalent, but q is not a law as it makes reference to a particular substance.

Argument 2: The laws of nature in our world are first-order. But any first-order proposition p is logically equivalent to the second-order proposition that p is true.

Argument 3: Plausibly, the values of fundamental constants like the fine-structure constant α are a part of the laws of nature. But now imagine that it turns out that the infinitely many significant digits of α express the infinite list of all arithmetical propositions and their truth values in some specific simple encoding scheme. There are two possibilities. Supposing that it is a law of nature that the digits of α have this curious property, then after verifying this property for a sufficiently large number of digits, we could know which of the remaining arithmetical propositions are true simply by measuring α to a high degree of precision. But if the law of nature is simply the brute fact that the digits are 0.007297352569…, and it just happens that these digits encode arithmetical truths in that encoding scheme, then we wouldn’t know truths by just measuring α. (Compare: Imagine a machine where you input an arithmetical proposition, and the machine flips a coin to yield an output of “True” and “False”. Even if we are so lucky that the machine always gives the right answer, that answer wouldn’t be knowledge. It would be just luck.) This means that there is a difference between having a law that says that the digits of α are determined by the arithmetical truths according to that encoding scheme and having an infinite law that simply states the digits, even though the two laws are logically equivalent (assuming the truths of arithmetic are logically necessary; if not, replace the truths of arithmetic by any sequence of hard to know logically necessary truths).

Argument 4: Laws of nature figure in explanations, but explanation is hyperintensional. The correct explanation of why the apple fell down is not that F = Gm₁m₂/r² and either Biden is president or Biden is not president, but simply that F = Gm₁m₂/r².

Argument 5: One of our best accounts of laws of nature is the Lewis-Ramsey best-systems model. But on that model it is very natural to identify the laws of nature with the axioms of the best system, and not just with propositions equivalent to the axioms of the best system.

Final note: I wonder, though, whether there is a unique proposition that expresses any given law of nature. Is there really a fact of the matter whether the law is F = Gm₁m₂/r² or F = m₁m₂(G/r²)?

Bidirectionality in means and ends

I never seem to tire of this action-theoretic case. You need to send a nerve signal to your arm muscles because there is a machine that detects these signals and dispenses food, and you’re hungry. So you raise your arm. What is your end? Food. What is your means to the food? Sending a nerve signal. But what is the means to the nerve signal?

The following seems correct to say: You raised your arm in order that a nerve signal go to your arm. What has puzzled me greatly about this case in the past is this. The nerve signal is a cause of the arm’s rising, and the effect can’t be the means to the cause. But I now think I was confused. For while the nerve signal is a cause of the arm’s rising, the nerve signal is not a cause of your raising your arm. For your raising your arm is a complex event C that includes an act of will W, a nerve signal S, and the rising of the arm R. The nerve signal S is a part, but not a cause, of the raising C, though it is a cause of the rising R.

So it seems that the right way to analyze the case is this. You make the complex event C happen in order that its middle part S should happen. Thus we can say that you make C happen in order that its part S should happen in order that you should get food. Then C is a means to S, and S is a means to food, but while S is a causal means to food, C is a non-causal means to S. But it’s not a particularly mysterious non-causal means. It sometimes happens that to get an item X you buy an item Y that includes X as a part (for instance, you might buy an old camera for the sake of the lens). There is nothing mysterious about this. Your obtaining Y is a means to your obtaining X, but there is no causation between the obtaining of Y and the obtaining of X.

Interestingly, sometimes a part serves as a means to a whole, but sometimes a whole serves as a means to the part. And this can be true of the very same whole and the very same part in different circumstances. Suppose that as a prop for a film, I need a white chess queen. I buy a whole set of pieces to get the white queen, and then throw out the remaining pieces in the newly purchased set to avoid clutter. Years later, an archaeologist digs up the 31 pieces I threw out, and buys my white queen from a collector to complete the set. Thus, I acquired the complete set to have the white queen, while the archaeologist acquired the white queen to have the complete set. This is no more mysterious than the fact that sometimes one starts a fire to get heat and sometimes one produces heat to light a fire.

Just as in one circumstances an event of type A can cause an event of type B and in other circumstances the causation can go the other way, so too sometimes an event of type A may partly constitute an event of type B, and sometimes the constitution can go the other way. Thus, my legal title to the white queen is constituted by my legal title to the set, but the archaeologist’s legal title to the set is partly constituted by legal title to the white queen.

There still seems to be an oddity. In the original arm case, you intend your arm’s rise not in order that your arm might rise—that you don’t care about—but in order that you might send a nerve signal. Thus, you intend something that you don’t care about. This seems different from buying the chess set for the sake of the queen. For there you do care about your title to the whole set, since it constitutes your title to the queen. But I think the oddity can probably be resolved. For you only intend your arm’s rising by intending the whole complex event C of your raising your arm. Intending something you don’t care about as part of intending a whole you do care about is not that unusual.

Partial and complete explanations

1. Any explanation for an event E that does not go all the way back to something self-explanatory is merely partial.

2. A partial explanation is one that is a part of a complete explanation.

3. So, if any event E has an explanation, it has an explanation going all the way back to something self-explanatory. (1,2)

4. Some event has an explanation.

5. An explanation going back to something self-explanatory involves the activity of a necessary being.

6. So, there is an active necessary being. (4,5)

I am not sure I buy (1). But it sounds kind of right to me now. Additionally, (3) kind of sounds correct on its own. If A causes B and B causes C but there is no explanation of A, then it seems that B and C are really unexplained. Aristotle notes that there was a presocratic philosopher who explained why the earth doesn’t fall down by saying that it floats on water, and he notes that the philosopher failed to ask the same question about the water. I think one lesson of Aristotle’s critique is that if it is unexplained why the water doesn’t fall down it is unexplained why the earth falls down.

Choices on a spectrum

My usual story about how to reconcile libertarianism with the Principle of Sufficient Reason is that when we choose, we choose on the basis of incommensurable reasons, some of which favor the choice we made and others favor other choices. Moreover, this is a kind of constrastive explanation.

This story, though it has some difficulties, is designed for choices between options that promote significantly different goods—say, whether to read a book or go for a walk or write a paper.

But a different kind of situation comes up for choices of a point on a spectrum. For instance, suppose I am deciding how much homework to assign, how hard a question to ask on an exam, or how long a walk to go for. What is going on there?

Well, here is a model that applies to a number of cases. There are two incommensurable goods one better served as one goes in one direction in the spectrum and the other better served as one goes in the other direction in the spectrum. Let’s say that we can quantify the spectrum as one from less to more with respect to some quantity Q (amount of homework, difficulty of a question or length of a walk), and good A is promoted by less of Q and incommensurable good B is promoted by more of Q. For instance, with homework, A is the student’s having time for other classes and for non-academic pursuits and B is the student’s learning more about the subject at hand. With exam difficulty, A may be avoiding frustration and B is giving a worthy challenge. With a walk, A is reducing fatigue and B is increasing health benefits. (Note that the claim that A is promoted by less Q and B is promoted by more Q may only be correct within a certain range of Q. A walk that is too long leads to injury rather than health.)

So, now, suppose we choose Q = Q₁. Why did one choose that? It is odd to say that one chose Q on account of reasons A and B that are opposed to each other—that sounds inconsistent.

Here is one suggestion. Take the choice to make Q equal to Q₁ to be the conjunction of two (implicit?) choices:

1. Make Q at most Q₁

2. Make Q at least Q₁.

Now, we can explain choice (a) in terms of (a) serving good A better than the alternative, which would be to make Q be bigger than Q₁. And we can explain (b) in terms of (b) serving good B better than the alternative of making Q be smaller.

Here is a variant suggestion. Partition the set of options into two ranges R₁, consisting of options where Q < Q₁ and R₂, where Q > Q₁. Why did I choose Q = Q₁? Well, I chose Q over all the choices in R₁ because Q better promotes B than anything in R₁, and I chose Q over all the choices in R₂ because Q better promotes A than anything in R₁.

On both approaches, the apparent inconsistency of citing opposed goods disappears because they are cited to explain different contrasts.

Note that nothing in the above explanatory stories requires any commitment to there being some sort of third good, a good of balance or compromise between A and B. There is no commitment to Q₁ being the best way to position Q.

The structure of morality

In physics, we hope for the following unification: there is a small set of simple laws, and all the rest of physics derives logically from these laws and the contingencies of the arrangement of stuff.

In ethics, a similar ideal has often manifested itself. While I have a hope for the ideal being realized in physics, I have come to be more pessimistic about the ideal in ethics. Instead, I think we can have a looser unificatory structure. We can have a multilevel hierarchy of more general laws, and then more specific laws that specify or implement the more general laws.

I suspect the looser structure is what we have in Aquinas’s Natural Law. At the highest level we have the general law that the good is to be pursued and the bad to be avoided. This is then specified into three laws about promoting the goods of existence, species-specific life and reason. These three laws, I think, are then further specified.

There is thus a structure to the moral law, but it is not a deductive structure. The higher level laws make the lower level laws fitting, but do not necessitate them.

A cosmological argument from the Hume-Edwards Principle

The Hume-Edwards Principle (HEP) says:

1. If you’ve explained every item in a collection, you’ve explained the whole collection of items.

This sounds very plausible, but powerful counterexamples have been given. For instance, suppose that exactly at noon, cannonball is shot out of a cannon. The collection C of cannonball states after noon has the property that each state in C is explained by an earlier state in C (e.g., a state at 12:01:00 is explained by a state at 12:00:30). By the Hume-Edwards Principle, this would imply that C is self-explanatory. But it plainly is not: it requires the cannon being fired at noon to be explained.

But I just realized something. All of the effective counterexamples to the Hume-Edwards Principle involve either circular causation or infinite causal regresses. We can now argue:

2. HEP is necessarily true.

3. If circular causation is possible, counterexamples to HEP are possible.

4. If infinite causal regresses are possible, counterexamples to HEP are possible.

5. So, neither circular causation nor infinite causal regresses are possible.

6. If there is no first cause, there is a causal circle or an infinite causal regress.

7. So, there is a first cause.

Similarly, it is very plausible that if infinite causal regresses are impossible, then causal finitism, the thesis that nothing can have an infinite causal history, is true. So, we get an argument from HEP to causal finitism.

Dialectically, the above is very odd indeed. HEP was used by Hume and Edwards to oppose cosmological arguments. But the above turns the tables on Hume and Edwards!

Objection: Not every instance of causal regress yields a counterexample to HEP. So it could be that HEP is true, but some causal regresses are still possible.

Response: It’s hard to see how there is sufficient structural difference between the cannonball story and other regresses to allow one to deny the cannonball story, and its relatives, while allowing the kind of regresses that are involved in Hume’s response to cosmological arguments.

Final remark: What led me to the above line of thought was reflecting on scenarios like the following. Imagine a lamp with a terrible user interface: you need to press the button infinitely many times to turn the lamp on, and once you do, it stays on despite further presses. Suppose now that in an infinite past, Alice was pressing the button once a day. Then the lamp was always on. Now I find myself with two intuitions. On the one hand, it seems to me that there is no explanation in the story as to why the lamp was always on: “It’s always been like that” just isn’t an explanation. On the other hand, we have a perfectly good explanation why the lamp was on n days ago: because it was on n + 1 days ago, and another button press doesn’t turn it off. And I found the second intuition pushing back against the first one, because if every day’s light-on state has an explanation, then there should be an explanation of why the lamp was always on. And then I realized this intuition was based on somehow finding HEP plausible—despite having argued against HEP over much of my philosophical career. And then I realized that one could reconcile HEP with these arguments by embracing causal finitism.

Samuel Clarke on our ignorance of the essence of God

At times I am made uncomfortable by this objection to arguments for the existence of God: there feels like there is something fishy about inferring the existence of a being about which we know so very little. It may be that theism is the only reasonable explanation of the universe’s existence, but if we know so very little about that explanation, can the inference to the truth of that explanation be a genuine version of inference to best explanation?

Newton's disciple Samuel Clarke has a nice answer to this objection:

There is not so mean and contemptible a plant or animal, that does not confound the most enlarged understanding upon earth; nay, even the simplest and plainest of all inanimate beings have their essence or substance hidden from us in the deepest and most impenetrable obscurity.

In other words, all our ordinary day-to-day inferences are to things whose essence is hidden.

It may be thought that now that we know about DNA, we do know the essences of plants and animals. But even if that is true, which I am sceptical of, it doesn’t matter: for belief in plants and animals was quite reasonable even before our superior science. And even this day, our knowledge of the essences of the fundamental entities of physics (e.g., particles, fields, wavefunctions) is basically nil. All we know is some facts about the effects of these entities.

Are free actions a counterexample to the PSR?

I’ve argued somewhat as follows in the past:

1. Necessarily, no one is responsible for a brute fact—an unexplained contingent fact.

2. Necessarily, someone is responsible for every free decision or free action.

3. So, it is impossible for a free decision or free action to be a brute fact.

But then:

4. Necessarily, a counterexample to the Principle of Sufficient Reason (PSR) is a brute fact.

5. So, no free decision or free action can be a counterexample to the PSR.

One may imagine someone, however, arguing that although a free decision or a free action cannot be a counterexample to the Principle of Sufficient Reason, a contrastive report, such as that x freely chose to do A rather than B, could be a counterexample to the Principle of Sufficient Reason. But notice that if x freely chose to do A rather than B, then x is responsible for choosing to do A rather than B. Similarly if x freely chose to do A for reason R rather than B for S, then x is responsible for doing so. Freedom implies responsibility. But no one is responsible for a brute fact, so such contrastive reports cannot be reports of a brute fact.

Objection 1: Incompatibilism is true, and on incompatibilism it is obvious that no possible explanation can be given for why x freely chose to do A for R rather than B for S. Hence the Principle of Sufficient Reason is false.

Response: Given that no one is responsible for what has no explanation, if the “no possible explanation” claim is correct, then free will is impossible. Thus, rather than showing that the PSR is false, the argument would show that if incompatibilism is true, free will is impossible. As a libertarian, I think free will is possible (and actual). But it is important to keep clear on what it is that is really endangered by the argument: it is free will and not the PSR.

Objection 2: Freedom is a necessary but not sufficient condition for responsibility.

Response: I am not sure about this. When I think about what other conditions we need to add to freedom to yield responsibility, the only one I can think of is something like knowledge of what is at stake. But it is arguable that without knowledge of what is at stake, a choice is not free. Moreover, even if one does not know what is at stake with A and B beyond what is contained in the respective reasons R and S, one will still be responsible for choosing A for R rather than B for S if one chooses freely for these reasons. One just won’t be responsible for the further aspects, beyond those captured by R and S, that one does not know.

But let’s grant for the sake of argument that other conditions need to be added to freedom to yield responsibility. If so, then the claim has to be that free but non-responsible decisions or actions or contrastive reports thereof are a counterexample to the PSR although free and responsible ones are not. In other words, one has to hold that the alleged additional conditions that need to be added to freedom to yield responsibility are what secures explicability. But given that the most plausible candidate for the other conditions is knowledge of what is at stake, this is implausible. For a free action based on mistaken or limited knowledge is no less explicable than an action based on full knowledge, once one takes into account the agent’s epistemic deficiency.