The accuracy argument for probabilism

A standard scoring rule argument for probabilism—the doctrine that credence assignments should satisfy the axioms of probability—goes as follows. If s is a scoring rule on a finite probability space Ω, so that s(c)(ω) is the epistemic utility of credence assignment c at ω in Ω, and (a) s is strictly proper and (b) s is continuous, then for any credence c that does not satisfy the axioms of probability, there is a credence p that does satisfy them such that s(p)(ω) is better than s(c)(ω) for all ω. This means that it’s stupid to have a non-probabilistic credence c, since you could instead replace it with p, and do better, no matter what.

Here is a problem with the dialectics behind this argument. Let P be the set of all credence assignments that satisfy the axioms of probability. But suppose that I think that there is some nonempty set M of credence assignments that do not satisfy the axioms of probability but are rationally just as good as those in P. Then I will think there is some way of making decisions using credences in M, just as good as the way of making decisions using credences in P. The best candidate in the literature for this is to use a level set integral, which allows one to assign an expected value E_cU to any utility assignment U even if c is not a probability. Note that E_pU is the standard mathematical expectation with respect to p if p is a probability.

The argument for probabilism assumed two things about the scoring rule: strict propriety and continuity. Strict propriety is the claim that:

1. E_ps(p) > E_ps(c) whenever c is a credence other than p

for any probability p. In words, by the lights of a probability p, then we get the best expected epistemic utility if we make p be our credence.

Now, if I am not convinced by the argument that (1) should hold for any probability p and any credence c other than p, then I will be unmoved by the scoring rule argument for probabilism. So suppose that I am convinced. But recall that I think that credences in M are just as rationally good as the probabilities in P. Because of this, if I find (1) convincing for all probabilities p, I will also find it convincing for all credences p in M, where E_p is my preferred way of calculating expected utilities—say, a level set integral.

Thus, if I am convinced by the argument for strict propriety, I will just as much accept (1) for p in M as for p in P. But now we have:

Theorem 1. If E_p is strongly monotonic for all p ∈ P ∪ M and coincides with mathematical expectation for p ∈ P, and (1) holds for all p in P ∪ M, where M is non-empty, then s is not continuous on P.

(Strong monotonicity means that if U < V everywhere then E_pU < E_pV. The Theorem follows immediately from the Pettigrew-Nielsen-Pruss domination theorem.)

Suppose then that I am convinced that a scoring rule s should be continuous (either on P or on all of P ∪ M). Then the conclusion I am apt to draw is that there just is no scoring rule that satisfies all the desiderata I want: continuity as well as (1) holding for all p ∈ P ∪ M.

In other words, the only way the argument for probabilism will be convincing to me is if my reason to think (1) is true for all p in P is significantly stronger than my reason to think (1) is true for all p in M, and I have a sufficiently strong reason to think that there is a scoring rule that satisfies all the true rational desiderata on a scoring rule to conclude that (1) holding for all p in M is not among the true rational desiderata even though its holding for all p in P is.

And once I additionally learn about the difficulties in defining sensible scoring rules on infinite spaces, I will be less confident in thinking there is a scoring rule that satisfies all the true rational desiderata on a scoring rule.

Instrumental bads

Suppose that a process Q has a chance r of producing some non-instrumentally bad result B, and nothing else of relevance. That fact gives us reason not to actualize Q. But suppose Q is actualized. Is it bad?

Well, if it’s bad, it seems it’s only instrumentally bad. It is no worse to be killed by a well-aimed arrow than by a well-aimed bullet, even though in the case of the well-aimed arrow the process of a deadly projectile’s flight lasts longer. Yet if a process producing a bad result were non-instrumentally bad, it would be worse if it lasted longer.

So we now have four options:

1. Q is always instrumentally bad (whether or not B eventuates)

2. Q is never instrumentally bad

3. Q is instrumentally bad if and only if B eventuates

4. Q is instrumentally bad if and only if B does not eventuate.

Option (4) is crazy. Option (2) destroys the very idea of an instrumental bad. So that leaves options (1) and (3).

If we opt for option (1), then we can have a world that contains instrumental bads without any non-instrumental bads—just imagine that Q obtains, B does not eventuate, and nothing else that’s bad ever happens. This seems a little counterintuitive: instrumental bads are derivatively bad, but how can something be derivatively bad without anything that is non-derivatively bad?

That suggests we should go for option (3): a process that has a chance of leading to a non-instrumental bad is bad only when the non-instrumental bad eventuates.

But now imagine Molinism is true. Suppose that God knows that Q, if actualized, would not lead to B, even though it has a non-zero chance r of doing so. In that case, the fact that Q has a chance r > 0 of leading to B is no reason for God not to actualize Q. But that something is bad is always a reason not to actualize it. If instrumental bads are an exception for this, then instrumental bads aren’t bads.

Now, I think Molinism is false. But whether (3) is true should not, it seems, depend on whether Molinism is true. So if (3) is false on Molinism, it is simply false.

So we seem to be stuck!

Maybe the right move is this. Fake money isn’t money and merely instrumental bads aren’t bad. This allows us an escape from the Molinism argument. For if merely instrumental bads aren’t bad, there is no problem about the fact that the Molinist God has no reason not to produce them.

Another move might be to say that (3) is true, but disproves Molinism. This doesn’t strike me as right, but maybe it’s defensible.

Until this is resolved, one really shouldn’t be running any arguments that depend on instrumental bads being actually bad.

The intrinsic badness of certain future tensed facts on presentism

It is bad that tomorrow someone will be in intense pain. On eternalism, we can easily explain this: tomorrow’s pain is just as real as today’s. But on presentism and growing block, future pains don’t exist.

Presumably, the presentist and growing blocker will say that the tensed fact of there being an intense pain tomorrow is bad, and this bad tensed fact presently exists.

Is this badness of the future tensed fact about the pain an instrumental or non-instrumental badness? If it’s instrumental, it is not clear what it could be instrumental to. The main candidate (apart from special cases where there is an obvious candidate, such as when the pain leads to despair) is that the fact that there will be a pain tomorrow is instrumental to tomorrow’s pain. But the fact that tomorrow there will be pain won’t cause that pain—otherwise, it would be trivial that every future event has a cause.

So the present badness of there being a pain tomorrow would be non-instrumental. But now imagine two scenarios with finite time lines.

• Scenario A: There is a mindless universe with a day of random particle movement, followed by the formation of a brain which has intense pain for a minute, followed by the end of time.

• Scenario B: There is a mindless universe with a century of random particle movement, followed by the formation of a brain which has intense pain for a minute, followed by the end of time.

Let’s suppose we find ourselves at the last moment of time in one scenario or the other. Then in Scenario A, there was a day of the obtaining of a “future pain fact”, and in Scenario B, there was a century of the obtaining of a “future pain fact”. If a future pain fact is a non-instrumentally bad thing, then there was non-instrumentally bad stuff in Scenario B for a much longer period of time than in Scenario A, and so Scenario B is much worse than Scenario A with respect to future pain. But that seems mistaken: the greater length of time during which there is a future pain fact does not seem any reason to prefer one scenario over another.

Should the A-theorist talk of tensed worlds?

For this post, suppose that an A-theory of time is true, so there is an absolute present. If we think of possible worlds as fully encoding how things can be so that:

1. A proposition p is possible if and only if p holds at some world,

then we live in different possible worlds at different times. For today a Friday is absolutely present and tomorrow a Saturday is absolutely present, and so how things are is different between today and tomorrow (or, in terms of propositions, that it’s Saturday is false but possible, so there must be a world where it’s true). In other words, given (1), the A-theorist is forced to think of worlds as tensed, as centered on a time.

But there is something a little counterintuitive about us living in different worlds at different times.

However, the A-theorist can avoid the counterintuitive conclusion by limiting truth at worlds to propositions that cannot change their truth value. The most straightforward way of doing that is to say:

2. Only propositions whose truth value cannot change hold at worlds

and restrict (1) to such propositions.

This, however, requires the rejection of the following plausible claim:

3. If (p or q) is true at a world w then p is true at w or q is true at w.

For the disjunction that it’s Friday or it’s not Friday is true at some world, since it’s a proposition that can’t change truth value, but neither disjunct can be true at a world by (2).

Alternately, we might limit the propositions true at a world to those expressible in B-language. But if our A-theorist is a presentist, then this still leads to a rejection of (3). For on presentism, the fundamental quantifiers quantify over present things, and the quantifiers of B-language are defined in terms of them. In particular, the B-language statement “There exist (tenselessly) dinosaurs” is to be understood as the disjunction “There existed, exist or will exist dinosaurs.” But if we have (3), then worlds will have to be tensed, because different disjuncts of “There existed, exist or will exist dinosaurs” will hold at different times. A similar issue comes up for growing block.

So on the most popular A-theories (presentism and growing block), we have to either allow that we inhabit different worlds at different times or deny (3). I think the better move is to allow that we inhabit different worlds at different times.

Mill on injustice

Mill thinks that:

1. An action is unjust if society has a utility-based reason to punish that actions of that type.

2. An action is wrong if there is utility-based reason not to peform that action.

Mill writes as if the unjust were a subset of the wrong. But it need not be. Suppose that powerful aliens have a weird religious view on which dyeing one’s hair green ought to be punished with a week in jail, and they announce that any country that refuses to enforce such a punishment as part of the criminal code will be completely annihilated. In that case, according to (1), dyeing one’s hair green is unjust. But it is not guaranteed to be wrong according to (2). The pleasure of having green hair could be greater than the unpleasantness of a week in jail, depending on details about the prison system and one’s aesthetic preferences.

The problem with (1), I think, is that utility-based reasons to punish actions of some type need have little to do with moral reasons, utilitarian or not, against actions of that type.

The three big mysteries of the concrete world

There are three big mysterious aspects of the concrete world around us:

• the causal

• the mental

• the normative.

The three mysteries are interwoven. Teleology is the domain of the interplay of the causal and the normative. And the mental always comes along with the normative, and often with the causal.

There is no hope of reducing the normative or the mental to the causal. Some have tried to reduce the normative to the mental, either via relativism (reducing to the finite mental) or Plantingan proper functionalism (reducing to the divine mental), neither of which appears particularly appealing in the end. I’ve toyed with reducing the mental to the normative, but while there is some hope of making progress on intentionality in this way, I doubt that there is a solution to the problem of consciousness in this direction.

Theism provides an elegant non-reductive story on which the three mysterious aspects of concrete reality are all found interwoven in one perfect being, and indeed follow from the perfection of that perfect being.

I wonder, too, if there is some way of seeing the three mysteries as reflective of the persons of the Trinity. Maybe the Father, the ultimate source of the other persons, is reflected in causality. The Son, the Logos, in the mental. And the Spirit, the loving concord of the Father and the Son may be reflected in the normative. But such analogies can be drawn in many ways, and I wouldn’t be very confident of them.

Necessity and the open future

Suppose the future is open. Then it is not true that tomorrow Jones will freely mow the lawn. Moreover, it is necessarily not true that Jones will freely mow the lawn, since on open future views it is impossible for an open claim about future free actions to be true. But what is necessarily not true is impossible. Hence it is impossible that Jones will freely mow the lawn. But that seems precisely the kind of thing the open futurist wishes to avoid saying.

Two difficulties for wavefunction realism

According to wavefunction realism, we should think of the wavefunction of the universe—considered as a square-integrable function on R³ⁿ where n is the number of particles—as a kind of fundamental physical field.

Here are two interesting consequences of wavefunction realism. First, it seems like it should be logically possible for the fundamental physical field to take any logically coherent combination of values on R³ⁿ. But now imagine that the initial conditions of the wavefunction “field” are have it take a combination of values that is not a square-integrable function, either because it is nonmeasurable or because it is measurable but non-square-integrable. Then the Schroedinger equation “wouldn’t know” what to do with the wavefunction. In other words, for quantum physics to work, given wavefunction realism, we need a very special initial combination of values of the “wavefunction field”. This is not a knockdown argument, but it does suggest an underexplored need for fine-tuning of initial conditions.

Second, the solutions to the Schroedinger equation, understood distributionally, are only defined up to sets of measure zero. In other words, even though the Schroedinger equation is generally considered to be deterministic (any indeterminism in quantum mechanics comes in elsewhere, say in collapse), nonetheless the solutions to the equation are underdetermined when they are considered as square-integrable fields on R³ⁿ—if ψ(⋅,t) is a solution for a given set of initial conditions, so is any function that differs from ψ(⋅,t) only on a set of measure zero. Granted, any two candidates for the wavefunction that differ only on a set of measure zero provide the exact same empirical predictions. However, it is still troubling to think that so much of physical reality would be ungoverned by the laws. (There might be a solution using the lifting theorem mentioned in footnote 6 here, though.)