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.

Supervenience and counterfactuals

On typical functionalist views of mind, what mental states a physical system has depends on counterfactual connections between physical properties in that system. But we can have two worlds that are exactly the same physically—have exactly the same tapestry of physical objects, properties and relations—but differ in what counterfactual connections hold between the physical properties. To see that, just imagine that one of the two worlds is purely physical, and in that world, w₁, striking a certain match causes a fire, and:

1. Were that match not struck, there would have been no fire.

But now imagine another world, w₂, which is physically exactly the same, but there is a nonphysical spirit who wants the fire to happen, and who will miraculously cause the fire if the match is not struck. But since the match is struck, the spirit does nothing. In w₂, the counterfactual (1) is false. (This is of course just a Frankfurt case.)

Thus physicalist theories where counterfactual connections are essential are incompatible with supervenience of the mental upon the physical.

I suppose one could insist that the supervenience base has to include counterfactual facts, and not just physical facts. But this is problematic. Even in purely physical worlds, counterfactual facts are not grounded in physical facts, but in physical facts combined with the absence of spirits, ghosts, etc. And in worlds that are only partly physical, counterfactual connections between physical facts may be grounded in the dispositions of non-physical entities.

Naturalists shouldn't be virtue ethicists

Virtue ethics is committed to this claim:

1. A choice of A is wrong if and only if a person who had the relevant virtues explanatorily prior to having chosen A and was in these circumstances would not have chosen A.

But (1) implies this generalization:

2. A person who has the relevant virtues explanatorily prior to a choice never chooses wrongly.

In my previous post I argued that Aristotelian Jews and Christians should deny (2), and hence (1).

Additionally, I think naturalists should deny (1). For we live in a fundamentally indeterministic world given quantum mechanics. If a virtuous person were placed in a position of choosing between aiding and insulting a stranger, there will always be a tiny probability of their choosing to insult the stranger. We shouldn’t say that they wouldn’t insult the stranger, only that they would be very unlikely to do so (this is inspired by Alan Hajek’s argument against counterfactuals).

And (2) itself is dubious, unless we have such a high standard of virtue that very few people have virtues. For in our messy chaotic world, very little is at 100%. Rare exceptions should be expected when human behavior is involved.

(Perhaps a dualist virtue ethicist who does not accept the Hebrew Scriptures could accept (1) and (2), holding that a virtuous soul makes the choices and is not subject to the indeterminacy of quantum mechanics and the chaos of the world.)

There is a natural way out of the above arguments, and that it so to change (1) to a probabilistic claim:

3. A choice of A is wrong if and only if a person who had the relevant virtues explanatorily prior to having chosen A and was in these circumstances would be very unlikely to have chosen A.

But (3) is false. Suppose that Alice is a virtuous person who has a choice to help exactly one of a million strangers. Whichever stranger she chooses to help, she does no wrong. But it is mathematically guaranteed that there is at least one stranger such that her chance of helping them is at most one in a million (for if p_n is her chance of helping stranger number n, then p₁ + ... + p_1000000 ≤ 1, since she cannot help more than one; given that 0 ≤ p_n for all n, it follows mathematically that for some n we have p_n ≤ 1/1000000). So her helping a particular such stranger is very unlikely to be chosen, but isn’t wrong.

Or for a less weighty case, suppose I say something perfectly morally innocent to start off a conversation. Yet it is very unlikely that a virtuous person would have said so. Why? Because there are so very many perfectly morally innocent ways to start off a conversation, it is very unlikely that they would have chosen the same one I did.

Christians and Jews should not be Aristotelian virtue ethicists

If virtue ethics is correct:

1. An choice is wrong if and only if a person with the relevant virtues and in these circumstances wouldn’t have made that choice. (Premise)

If Aristotelian virtue ethics is correct:

2. An adult lacking a virtue is defective. (Premise)

But:

3. Humans became defective because of the choice of Adam and Eve to eat the forbidden fruit. (Premise)

And it seems that:

4. Adam and Eve were adult humans when they chose to eat the forbidden fruit. (Premise)

Thus it seems:

5. When Adam and Eve chose to eat the forbidden fruit, they were not lacking relevant virtues. (By 2–4)

6. Thus, persons (namely Adam and Eve!) with the relevant virtues and in their circumstances did choose to eat the forbidden fruit. (By 5)

7. Thus, their choice to eat the forbidden fruit wasn’t wrong. (1 and 6)

8. But their choice was wrong. (Premise)

9. Contradiction!

Here is one thing the classic virtue ethicist can question about this argument: the derivation of (5) depends on how we read premise (1). We could read (1) as:

10. A choice of A is wrong if and only if a person who had the relevant virtues explanatorily prior to having chosen A and was in these circumstances would not have chosen A

or as:

11. A choice of A is wrong if and only if a person who had the relevant virtues while having chosen A and was in these circumstances would not have chosen A.

If we opt for (10), the derivation of (5) works, and the argument stands. But if we opt for (11) then we can say that as soon as Adam and Eve chose to eat the fruit, they no longer counted as virtuous.

Could the virtue ethicist thus opt for (11) in place of (10)? I don’t think so. It seems central to virtue ethics that the right choices are ones that result from virtue. And that is what (10) captures. To a great extent (11) would trivialize virtue ethics, in that obviously in doing a bad thing one isn’t virtuous.

Subjective sameness of choice situations and Molinism

Suppose that Alice on a street corner sells Bob a “Rolex” for $15. Bob goes home and his wife Carla says: “You got scammed!” Bob takes the “Rolex” to a jeweller and finds that it is indeed a Rolex. He goes back to Carla and says: “No, I got a good deal!” But Carla says: “But if it was a fake, you would have bought it, too.”

Here is Carla’s reasoning behind her counterfactual. A typical fake would have looked the same as the real thing to Bob, and so the counterfactual situation where Bob is offered the fake would have been subjectively the same to Bob. And Carla subscribes to this principle:

1. If you were instead in a different situation that was subjectively identical to the one you were in, you would have chosen the same way.

I find (1) pretty plausible, and it seems to nicely come out as true on a Lewis-style account of counterfactuals in terms of similarity of worlds (with approximate match counting as similarity).

But I doubt a Molinist should accept (1). For the Molinist does not think that counterfactuals of free will are grounded in similarity of worlds (except of the trivial sort, where truth values of counterfactuals count as part of the similarity).

Plagiarism and causation

Suppose I write a paper and you write a paper of the same length. But then I plagiarize your paper using the following procedure. I look at the first character in our papers, and if it’s different, I erase (unless it’s a space) the character in my paper and write down the character you had in its place. And then I repeat for the second, third, and so on. I then submit the paper for publication.

It seems clear that I’ve plagiarized your paper in its entirety, even though some of the letters in my paper weren’t erased as by coincidence I originally had the same letter in the same place as you did—this will happen more often with more common letters like “e”.

But what if, by chance, your paper and my original paper were verbatim the same, and I never noticed this? Then the paper I submit for publication depends for all of its content counterfactually on the paper you wrote, but not a letter was changed from the paper that I wrote. If authorship is defined by causation, then the paper I am submitting is my own. If it’s defined by counterfactual dependence, it’s yours.

I don’t know which is the right answer.

A bad argument from hiddenness

Consider the following variant of the argument from hiddenness:

1. If God exists, no mature human is ignorant of God’s existence through no fault of their own.

2. Some mature humans are ignorant of God’s existence through no fault of their own.

3. So, God doesn’t exist.

It’s occurred to me that premises like (3) are either nonsense, or trivially false, or far beyond our capacity to know to be true.

For to evaluate whether some x is ignorant of God’s existence through no fault of their own requires asking something like this:

4. Would x still have been ignorant of God’s existence had x lived a morally perfect life?

But it does not seem likely that there is a sensible positive answer to (4). Here is a quick argument for this. Those who deny Molinism are going to say that either the proposition asked about in (4) has no truth value or that it is trivially false. And even some Molinists will say this about (4), because Molinists are committed to there being conditionals of free will only when the antecedent is maximally specified, while “x lived a morally perfect life” is too unspecified. The question is much like:

5. Had Napoleon been born in South America, would he still have been a great military leader?

There are many ways for Napoleon to have been in South America, and they are apt to result in different answers to the question about whether he was a great military leader.

But even if (4) has a truth value, perhaps because (4) is to be interpreted in some probabilistic way or because we have an expansive version of Molinism that makes (4) make sense, it is far beyond our epistemic powers to know the answer to (4) to be true. Here is why. Our lives are full of wrongdoing. Our lives would likely be unrecognizable had they been morally perfect. To ask what we would have thought and known in the counterfactual scenario where we live a morally perfect life is to ask about a scenario further from actuality than Napoleon’s being born in South America.

Now, that said, there are times when we can evaluate counterfactuals that involve a massive change to the antecedent on the basis of certain generalities. For instance, while we have no answer to (5), we do have a negative answer to:

6. Had Napoleon suffered a massive head injury rendering him incapable of interpersonal communication, would he still have been a great military leader?

Similarly, if an atheist had suffered a head injury removing the capacity for higher level thought, the shape of their life would have been very different, but at least we can say that they wouldn’t have been an atheist, because they wouldn’t have had the concepts necessary to form the belief that there is no God. So, indeed, sometimes counterfactuals that take us far afield can be evaluated sensibly on general grounds.

But I do not think we have good general grounds for a positive answer to (4), unless we have independent grounds to doubt the truth and rationality of theism:

7. There are no good grounds for reasonably believing in God, and a person who lives a morally innocent life won’t believe things groundlessly, so they won’t believe in God.

8. There are people who grow up in societies where there is no concept of God, and they would not be aware of God no matter what the shape of their lives would have been.

Obviously, (7) requires independent grounds to doubt the rationality of theism. And if God exists, then for all we know, he has a general practice of making those who are morally perfect be aware of him, so (8) is dubious if God exists.

Of course, an atheist might think (7) is true, but this is unlikely to be a helpful move in an argument against the existence of God. After all, similarly, a theist might think the following is true:

9. God will ensure that every morally perfect mature human is aware of him.

Indeed, a typical Christian thinks that there have only ever been one or two people—Jesus and maybe Mary—who have been morally perfect, and both candidates were aware of God.

Time and clocks

Einstein said that time is what clocks measure.

Consider an object x that travels over some path P in spacetime. How long did the travels of x take? Well, if in fact x had a clock traveling with it, we can say that the travels of x took the amount of time indicated on the clock.

But what if x had no clock with it? Surely, time still passed for x.

A natural answer:

• the travels of x took an amount of time t if and only if a clock would have measured t had it been co-traveling with x.

That can’t be quite right. After all, perhaps x would have traveled for a different amount of time if x had a clock with it. Imagine, for instance, that x went for a one-hour morning jog, but x forgot her clock. Having forgot her clock, she ended up jogging 64 minutes. But had she had a clock with her, she would have jogged exactly 60 minutes.

That seems, though, a really uncharitable interpretation of the counterfactual. Obviously, we need to fix the spacetime path P that x takes. Thus:

• the travels of x over path P took an amount of time t if and only if a clock would have measured t had it been co-traveling with x over the same path P.

But this is a very strange counterfactual if we think about it. Clocks have mass. Like any other massive object, they distort spacetime. The spacetime manifold would thus have been slightly different if x had a clock co-moving with it. In fact, it is quite unclear whether one can make any sense of “the same path P” in the counterfactual manifold.

We can try to control for the mass of the clock. Perhaps in the counterfactual scenario, we need to require that x lose some weight—that x plus the clock have the same mass in the counterfactual scenario as x alone had in the actual scenario. Or, more simply, perhaps we can drop x altogether from the counterfactual scenario, and suppose that P is being traveled by a clock of the same mass as x.

But we won’t be able to control for the mass of the clock if x is lighter than any clock could be. Perhaps no clock can be as light as a single electron, say.

I doubt one can fix these counterfactuals.

Perhaps, though, I was too quick to say that if x had no clock with it, time still passed for x. Ordinary material substances do have clocks in them. These clocks may not move perfectly uniformly, but they still provide a measure of length of time. Alice’s jog took 396,400 heartbeats. Bob’s education took up 3/4 of his childhood. Maybe the relevant clocks, then, are internal changes in substances. And where the substances lack such internal changes, time does not pass for them.

Counterfactuals and the randomness objection to libertarianism

The randomness objection to libertarian free will says that undetermined choices could only be random rather than reason-governed.

I want to consider a bit of a reply to this. Suppose that you are choosing between A and B. You have a reason R for A and a reason S for B, and you freely end up choosing A. I think the following will be true, and I think the libertarian can say that they are true as well:

1. If the reason R for A were stronger, you'd still have chosen A.
2. If the reason S for B were weaker, you'd still have chosen A.
3. If the reason R for A were noticeably weaker, you might not have chosen A.
4. If the reason S for B were noticeably stronger, you might not have chosen B.

If this is right, then there is a real counterfactual dependence of your action on the reasons. The dependence isn't as precise as the compatibilist's dependence. The compatibilist's story may allow for precise values of strengths of reasons that produce counterfactuals like (3) and (4) with quantitative antecedents and "would" rather than "might". Still, I don't think anything so precise is needed for reasons-governance of our actions.

So, I think that if I am right that the libertarian can reasonably affirm (1)-(4), then the randomness objection fails. Of the four, I don't think there is any difficulty with (3) and (4): even if there were pure randomness, we would expect (3) and (4) to be true. So the question is: Can the libertarian affirm (1) and (2)? And I think (1) and (2) are in the same boat, so really the question is whether the libertarian can affirm (1)?

And I say: Why not? At the same time, I know that when I've talked with fellow libertarians about this, they've been pretty sceptical about counterfactuals like (1). Their main reason for scepticism was van Inwagen's re-run argument: In indeterministic free choice situations, if you repeated the same circumstances, you'd get different results. And you'd expect to get different results in repeat runs even if you somewhat raised the strength of the reasons for A.

I agree with the re-run intuition here, but I don't see it as incompatible with (1). The re-run intuition is about what we would get at a later time, albeit in a situation that is qualitatively the same. But (1) is about what would have happened at the time you made the original choice, albeit in a situation that was tweaked to favor A more.

Procreation and love

Start with this consequence of the essentiality of origins:

1. The identity of an individual depends on the exact time of her coming-into-existence.

(It's very plausible that it depends on coarse-grained time of coming-into-existence: I couldn't have come into existence a hundred years earlier. But there is also a well-known argument from that for the claim that it depends on the exact time.) Now, it seems that no action people can take prior to the origination of an individual determines the exact time of her coming-into-existence. There isn't any true counterfactual of the form of the form: "If we were to do A, then an individual would come into existence exactly at t." In fact, for any action we can take, quantum indeterminism will ensure a cluster of infinitely (if time is continuous) many or at least very many close-together possible resulting conception times. Moreover, each exact time will have a very tiny, perhaps infinitesimal or zero, probability of being the actual time of origination. Hence it seems:

2. There is no true probabilistic counterfactual of the form: "If we were to do A, then an individual would be not unlikely to come into existence exactly at t."

(Molinists will, of course, disagree. But Molinism is false, I take it.) Thus:

3. People cannot procreate out of love for the particular individual who would likely come into existence from the procreation.

For there is no particular individual such that she would likely come into existence from the procreation.

Now, notice that God in creating an individual--either ex nihilo or in cooperating with the joining of an egg and sperm--is under no such constraints. Thus it is possible for God to act in such a way that were he to act so, an individual would come into existence at a precise time. And the same is true for other features of the origination of the individual besides the time of its occurrence. So God could create out of love for the particular individual who would come into existence from the creation.

This fact suggests a weakness in premise (2) of my above argument. It could be that God has already decided what individual would result if a couple chose to procreate, and has resolved in particular that sperm and egg would meet at a particular time, as long as the couple chooses the procreate. God, thus, is resolved to control all the quantum phenomena to ensure a particular circumstance of origination. Thus it's up to the couple whether someone comes into existence, but there is a particular someone who would exist if the couple procreated. It would not be surprising if this were to happen in special cases, and it would be possible for it to happen in all cases. I don't know if it does, though.

The upshot of the above argument, thus, is that unless special theological assumptions hold, it is not possible for a couple to decide whether to procreate out of love for the particular child that would result.

Thomson's lamp and two counterfactauls

Thomson's lamp toggles each time you press the button and nothing else affects its state. The lamp is on at noon, and then a supertask consisting of infinitely many button presses that completes by 1 pm, and the question is whether the light is on or off at 1 pm. There is no contradiction yet. But now add these two claims:

1. The state of the lamp at 1 pm would not be affected by shifting the times at which the button presses happen, if (a) all the button presses happen between noon and 1 pm, and (b) we ensure that no two button presses happen simultaneously.
2. If we removed one button press from the sequence of button presses between noon and 1 pm, the state of the lamp at 1 pm would not change.

Given this intuition, we do have a problem. Suppose that our sequence of supertask button presses occurs at 12:30, 12:45, 12:52.5, and so on. Then shift this sequence of button presses forward in time, so that now the sequence is at 12:45, 12:52.5, 12:45.25,and so on. By (1) this wouldn't affect the outcome, but by (2) it would as we will have gotten rid of the first button press. That's a contradiction.

So if we think Thomson's lamp is possible--which I do not--we need to deny at least one of the two counterfactuals. I think the best move would be simply to deny both (1) and (2), on the grounds that the connection between the state of the lamp at 1 pm and the button presses must be indeterministic.

Is divine hiddenness an evil?

There is a discussion in the literature on whether the argument from divine hiddenness is a special case of the argument from evil. Here is an interesting difference. While many evils, like suffering and death, are evils no matter whether there is a God, divine hiddenness—the state of affairs of some epistemically virtuous agents not believing in God—can only be an evil if God exists. Indeed, if there is no God, then non-belief in God is intrinsically good.

This has an interesting implication. Say that an evil is "unjustified" provided that God does not (or would not?) have good moral reason to allow it. Then divine hiddenness is not an unjustified evil. For:

1. Either God exists or God does not exist.
2. If God exists, there are no unjustified evils.
3. If God does not exist, divine hiddenness is not an evil.
4. So, either way, divine hiddenness is not an unjustified evil.

This means that the hiddenness argument cannot be a special case of the argument from apparently unjustified evils. It's tempting to run the argument in counterfactual mode and make it a close parallel to the argument from apparently unjustified evil by contending:

5. If there were a God, then hiddenness would be an unjustified evil.

But (5) is implausible in light of the fact that

6. If there were a God, there would be no unjustified evils.

In fact, the following argument is valid and only the first premise is controversial:

7. Divine hiddenness is an evil.
8. If God doesn't exist, divine hiddenness is not an evil.
9. So, God exists.

Hiddenness is not the only case where this logical issue comes up. It comes up in the case of any state of affairs that has the property that necessarily it is an evil only if God exists. In all such cases, the state of affairs simply cannot be an unjustified evil, and one has an argument from the state of affairs being an evil to God existing. Blasphemy against God is another such state of affairs.

(And if it could be shown that, necessarily, there is only evil if God exists, then this would mean that all arguments from evil are in this boat.)

None of this means that there is no argument from hiddenness for atheism that's worth discussing. It just means that one shouldn't see it as just a special case of the problem of evil. A much more cogent formulation of the hiddenness argument is this simple one (inspired by Trent Dougherty):

10. P(this pattern of hiddenness | God) << P(this pattern of hiddenness | no God).
11. So, probably, no God.

Here, "this pattern of hiddenness" includes all the relevant features of this world that might be taken to justify hiddenness if God existed.

That said, one needs to be cautious about (10). For a part of this pattern of hiddenness is that some people have thought long and hard about whether God exists. And it's not clear that they would have been likely to think long and hard about whether God exists if there were no God.

Kant and Lewis on our freedom

Kant (on one reading) holds that the initial conditions of the universe and the laws of nature depend on us (noumenally speaking). This reconciles determinism with freedom: sure, our actions are determined by the laws and initial conditions, but the laws and initial conditions are up to us. Kant also thinks that a further merit of this view is that one can blame people whose misdeeds come from a bad upbringing, because noumenally speaking they were responsible for their own upbringing.

Lewis holds that freedom is compatible with determinism, and in a deterministic world had one acted otherwise, the laws would have been different.

Everybody agrees that the view I ascribe to Kant is crazy (though not everybody agrees that the ascription is correct). But Lewis's view is supposed to be much saner than Kant's.

How? The obvious suggestion is that Lewis only makes the laws depend counterfactually on our actions (assuming determinism) while Kant makes the laws depend explanatorily on our actions. But that suggestion doesn't work, since Lewis's best-systems account of laws makes the laws depend on the law-governed events, and so it makes the laws depend not just counterfactually on our actions but also explanatorily: the laws' being as they are is grounded in part in our actions. So both accounts make the laws explanatorily depend on us.

Admittedly, Kant also makes the past, not just the laws, depend on our actions. But that's also true for Lewis, albeit to a smaller degree, because of his doctrine of small miracles...

Causal probability and counterfactuals

The causal probability of an event B on an event A is cP_A(B)=∑_KP(K)P(B|AK), where the Ks are a partition based on the relevant dependency hypotheses compatible with A. (Compare to P(B|A)=∑_KP(K|A)P(B|AK).) A standard proposal in the literature is that

1. the degree of the assertibility of an indicative "If A, then B" is equal to the conditional probability P(B|A) of B on A.

Consider the parallel thesis that

2. the degree of assertibility of a subjunctive conditional or counterfactual A→B is equal to the causal probability cP_A(B) of B on A.

This thesis would unify the Stalnaker and Lewis (and Skyrms) approaches to causal decision theory as closely as possible. For according to the Stalnaker version, the causal expected value of an option A is:

3. EV(A) = ∑_BU(BA)P(A→B),

where the sum is over a partition based on outcomes. On the Lewis/Skyrms approach, it will be:

4. EV(A) = ∑_BU(BA)cP_A(B).

Now, if A→B has truth value, then the degree of asssertibility of A→B is equal to P(A→B), and hence by (2) we have P(A→B)=cP_A(B). And so the two formulae are equivalent. If, on the other hand, A→B has no truth value, then P(A→B) in (3) makes no sense. But we can replace it with Assertibility(A→B), which is basically the most natural replacement for P(A→B) when A→B has no truth value, and the revised (3) will come to the same thing as (4). So that's nice.

Notice, however, that this approach may not be compatibility with Molinism. For according to Molinism, God knows some conditionals of free will A→B, where B is a free action and A is a maximally specific set of antecedents, for sure. If P is God's probabilities, then in such cases:

5. 1=cP_A(B)=∑_KP(K)P(B|AK).

But because A is maximally specific, it will be compatible with only one relevant dependency hypothesis, say K₀, describing how B depends on A. So 1=P(K₀)P(B|AK₀). It follows that P(B|AK₀)=1 and P(K₀)=1. But now we see that there is a dependency hypothesis K₀ such that, together with A, it probabilistically necessitates B. But that can't be acceptable to a libertarian.

A problem for Lewis's account of counterfactuals

Let's say I have a bag of fifty ordinary chrome steel balls. They are of the same size up to a tolerance of 0.01 mm, but they do exhibit minor size variations below that tolerance. So one of these steel balls is smallest. On Lewis-type accounts of counterfactuals, we have to say that:

1. If one of the balls were made of brass, it would be the smallest of them.

For worlds where the smallest of the balls is made of brass are more like our world than worlds where another of the balls is made of brass (keeping everything else equal) since the area of spatiotemporal mismatch is smaller when it is the smaller ball that is made of brass.

But while we intuitively think that (1) might turn out to be true, it shouldn't turn out to be true simply because of such size considerations.

This is a non-temporal version of the coat thief problem (see, e.g., p. 42 here).

From ease to counterfactuals?

Consider the concept of how easy it is for a proposition to be made true, given how things are. It is by far easiest for propositions that are already true: nothing more needs to happen. It is hardest for self-contradictory propositions, like that Socrates is not Socrates: there is no way at all for it to happen. Contingently false propositions that require changes that go far back in time are going to be harder to be made true than ones that don't. And we can talk of the ease of p being made false as just the ease of not-p being made true. So, we can offer this account of counterfactuals:

• p→q holds if and only if it is easier for p to be made true than for the material conditional p⊃q to be made false.

This yields the Lewis-Stalnaker account of counterfactuals provided that we stipulate that a is easier to be made true than b if and only if there is a world where a holds which is closer than every world where b holds.

But we need not make this stipulation. We might instead take the easier to be made true relation as more fundamental. (And while we might define a closeness relation in terms of it—say, by saying that w₁ is closer than w₂ iff <w₁ is actual> is easier to be made true than <w₂ is actual>—depending on which axioms easier to be made true satisfies, that might not yield an account equivalent to the Lewis-Stalnaker one.)

On some assumptions, this is a variant of the central idea in yesterday's post.

A sufficient condition for a subjunctive conditional

Start with the idea of grades of necessity. At the bottom, say[note 1], lie ordinary empirical claims like that I am typing now, which have no necessity. Higher up lie basic structural claims about the world, such as that, say, there are four dimensions and that there is matter. Perhaps higher, or at the same level, there are nomic claims, like that opposite charges attract. Higher than that lie metaphysical necessities, like that nothing is its own cause or that water is partly composed of hydrogen atoms. Perhaps even higher than that lie definitional necessities, and higher than that the theorems of first order logic. This gives us a relation: p<q if and only if p is less necessary than q.

Let → indicate subjunctive conditionals. Thus "p→q" says that were it that p, it would be that q. Let ⊃ be the material conditional. Thus "p⊃q basically says that p is false or q is true or both. Then, the following seems plausible:

1. If ~p<(p⊃q), then p→q.

I.e., if the material conditional has more necessity than the denial of its antecedent, the corresponding subjunctive conditional holds.

Suppose it's a law of nature that dropped objects fall. Then the material conditional that if this glass is dropped, then it falls is nomic and hence more necessary than the claim that this glass is not dropped, and the subjunctive holds: were the glass dropped, it would fall.

Moreover, the subjunctives that (1) can yield hold non-trivially, if there are grades of necessity beyond metaphysical necessity (on my view, those are somewhat gerrymandered necessities), and this yields non-trivial per impossibile conditionals. Let p be the proposition that water is H₃O, and let q be the proposition that a water molecule has four atoms. Then ~p<(p⊃q), because p⊃q is a definitional truth while ~p is a merely metaphysical necessity. Hence were p to hold, q would hold: were water to be H₃O, a water molecule would have four atoms.

I wonder if the left-hand-side of (1) is necessary for the non-trivial holding of its right-hand-side.

Accidental generalizations that accidentally support counterfactuals

One of the long-standing problems in philosophy of science is how to distinguish accidental generalizations, such as that all the coins in Sam's pocket are nickels when this is a mere coincidence, from non-accidental generalizations, such as that all electrons are charged.

A standard observation is that non-accidental generalizations support counterfactuals. If there were another electron, it too would be charged. But it is not true that if there were another coin in the pocket, it too would be a nickel. Or so the story goes.

But in fact one can make accidental generalizations accidentally support counterfactuals as well. Sam and Jenny are on a lifeboat. Sam has five nickels in his pocket and Jenny has two nickels and three dimes, all due to chance. And that's all the money around for miles. Jenny accidentally drops the three dimes in the sea. Hours pass. At this point the following counterfactual seems true:

• If another coin came to be present in Sam's pocket, it would be one of Jenny's nickels.

This is intuitively correct, and obviously right on Lewisian semantics. After all, worlds where a quarter comes into existence ex nihilo in Sam's pocket, or one of Jenny's dimes flies out of the ocean, or even ones where several hours back Jenny did not drop the dime in the ocean and now gives it to Sam, are all more distant from our world than a world where Jenny just hands Sam another nickel and he puts it in his pocket. And so, plausibly:

• If another coin came to be present in Sam's pocket, it would be a nickel.

So, the accidental generalization that Sam has five nickels has started to support counterfactuals. But it's still accidental, and its support of counterfactuals is but an accident.

Non-triviality of conditionals

Here's a rough start of a theory of non-triviality of conditionals.

A material conditional "if p then q" is trivially true provided that (a) the only reason that it is true is that p is false or (b) the only reason that it is true is that q is true or (c) the only reasons that it is true are that p and q are true.

A subjunctive conditional "p □→ q" is trivially true provided that (a) the only reason that it is true is that p and q are both true or (b) the only reason that it is true is that p is impossible or (c) the only reason that it is true is that q is necessary or (d) the only reasons that it is true are that p is impossible and q is necessary.

For instance, "If it is now snowing in Anchorage, then it is now snowing in the Sahara" understood as a material conditional is trivially true, because the falsity of the antecedent (I just checked!) is the only reason for the conditional to be true. The contrapositive "If it not now snowing in the Sahara, then it is not now snowing in Anchorage" is trivially true, since it is true only because of the truth of the consequent. On the other hand, "If I am going to meet the Queen for dinner tonight, I will wear a suit" is non-trivially true. It is true not just because its antecedent is false--there is another explanation.

Likewise, "Were horses reptiles, then Fermat's Last Theorem would be false" and "Were Fermat's Last Theorem false, horses would be mammals" are "Were I writing this, it would not be snowing in Anchorage" are trivially true, in virtue of impossibility of antecedent, necessity of consequent and truth of antecedent and consequent, respectively. But "Were horses reptiles, either donkeys would be reptiles or there would no mules" is non-trivally true--there is another explanation of its truth besides the impossibility of antecedent, namely that reptiles can't breed with mammals and mules are the offspring of horses and donkeys.

Divine omnirationality, reward and punishment

Omnirationality is the divine attribute in virtue of which when God does A, he does it for all the non-preempted reasons that in fact favor his doing A. (Here is an example of a reason preempted by a higher order reason: God promises me that as a punishment, he won't hear my prayers for the next hour; then that I ask God for something creates a preempted reason for him.) He does not choose only some of the relevant reasons and act on those, in the way a human being might.

One consequence of omnirationality is that when I pray for an event F, and F is good and in fact takes place, then I can safely conclude that F took place in part as a result of prayer. For a request is always a good reason to do something good, and while in principle the reason could be preempted, in fact it seems very unlikely that there was a preempting reason in this case. At this same time, in this case we cannot say that the good took place entirely as a result of prayer, because the very fact that it was a good was also, presumably, a non-preempted reason for God to bring it about.

Here is another example. Suppose Job leads a virtuous life in such a way that there is good reason for Job to have good things bestowed on him as a reward for the virtuous life. And suppose that, in fact, good things befall Job. Then we can confidently say that they befell Job in part in order to reward Job. For by hypothesis, God has a reason (not a conclusive one, as we learn from the Book of Job!) to bless Job, and the reason seems unlikely to be preempted, so when he blesses Job, he does so in part because it rewards Job.

The flip side of this is that, by omnirationality, if a sinner who has not been forgiven for a sin has a bad thing happen to her whose magnitude is not disproportionate to the sin, that bad thing happens to her at least in part as a divine punishment, unless some sort of preemption applies, since God has a reason to punish.

Forgiveness, of course, would preempt. But I assumed here the sin was unforgiven. Maybe one could claim that the redemptive events of the New Testament changed everything, preempting all of God's reasons to punish, but that does not seem to be the message of the New Testament. It really does seem that God's reasons to punish unforgiven sin are not preempted even in New Testament times. This does not, of course, mean that all evils that happen to people are best seen as divine punishments. First of all, forgiveness of a sin preempts, and probably annuls, the reasons of justice. Second, even when the justice of the matter is a non-preempted reason for God to allow the evil to befall, it need not be the most important one. God's desire to use the evil to reform the sinner or to glorify himself in a deeper way, may be a more important reason, sometimes to the point where it would be misleading, and maybe even false, to say that the evil befell because the person sinned—we could only say that the evil befell in small part because the person sinned.

Finally, as Jesus himself warns, that an evil befalls A and does not befall B does not imply that A was more worthy of the evil than B. For God may have had many additional reasons for allowing the evil to befall A and keeping it from B besides the merits of the wo.

We can try to probe more deeply by asking counterfactual questions: Would God still have had the evil befall A had A not sinned? But I think such counterfactual questions tend not to have answers.