Definitions

In the previous post, I offered a criticism of defining logical consequence by means of proofs. A more precise way to put my criticism would be:

1. Logical consequence is equally well defined by (i) tree-proofs or by (ii) Fitch-proofs.

2. If (1), then logical consequence is either correctly defined by (i) and correctly defined by (ii) or it is not correctly defined by either.

3. If logical consequence is correctly defined by one of (i) and (ii), it is not correctly defined by the other.

4. Logical consequence is not both correctly defined by (i) and and correctly defined by (ii). (By 3)

5. Logical consequence is neither correctly defined by (i) nor by (ii). (By 1, 2, and 4)

When writing the post I had a disquiet about the argument, which I think amounts to a worry that there are parallel arguments that are bad. Consider the parallel argument against the standard definition of a bachelor:

6. A bachelor is equally well defined as (iii) an unmarried individual that is a man or as (iv) a man that is unmarried.

7. If (6), then a bachelor is either correctly defined by (iii) and correctly defined by (iv) or it is not correctly defined by either.

8. If logical consequence is correctly defined by one of (iii) and (iv), it is not correctly defined by the other.

9. A bachelor is not both correctly defined by (iii) and correctly defined by (iv). (By 9)

10. A bachelor is neither correctly defined by (iii) nor by (iv). (By 6, 7, and 10)

Whatever the problems of the standard definition of a bachelor (is a pope or a widower a bachelor?), this argument is not a problem. Premise (9) is false: there is no problem with saying that both (iii) and (iv) are good definitions, given that they are equivalent as definitions.

But now can’t the inferentialist say the same thing about premise (3) of my original argument?

No. Here’s why. That ψ has a tree-proof from ϕ is a different fact from the fact that ψ has a Fitch-proof from ϕ. It’s a different fact because it depends on the existence of a different entity—a tree-proof versus a Fitch-proof. We can put the point here in terms of grounding or truth-making: the grounds of one involve one entity and the grounds of the other involve a different entity. On the other hand, that Bob is an unmarried individual who is a bachelor and that Bob is a bachelor who is unmarried are the same fact, and have the same grounds: Bob’s being unmarried and Bob’s being a man.

Suppose one polytheist believes in two necessarily existing and essentially omniscient gods, A and B, and defines truth as what A believes, while her coreligionist defines truth as what B believes. The two thinkers genuinely disagree as to what truth is, since for the first thinker the grounds of a proposition’s being true are beliefs by A while for the second the grounds are beliefs by B. That necessarily each definition picks out the same truth facts does not save the definition. A good definition has to be hyperintensionally correct.

Logical consequence

There are two main accounts of ψ being a logical consequence of ϕ:

• Inferentialist: there is a proof from ϕ to ψ

• Model theoretic: every model of ϕ is a model of ψ.

Both suffer from a related problem.

On inferentialism, the problem is that there are many different concepts of proof all of which yield an equivalent relation of between ϕ and ψ. First, we have a distinction as to how the structure of a proof is indicated: is a tree, a sequence of statements set off by subproof indentation, or something else. Second, we have a distinction as to the choice of primitive rules. Do we, for instance, have only pure rules like disjunction-introduction or do we allow mixed rules like De Morgan? Do we allow conveniences like ternary conjunction-elimination, or idempotent? Which truth-functional symbols do we take as undefined primitives and which ones do we take as abbreviations for others (e.g., maybe we just have a Sheffer stroke)?

It is tempting to say that it doesn’t matter: any reasonable answers to these questions make exactly the same ψ be logical consequence of the same ϕ.

Yes, of course! But that’s the point. All of these proof systems have something in common which makes them ``reasonable’’; other proof systems, like ones including the rule of arbitrary statement introduction, are not reasonable. What makes them reasonable is that the proofs they yield capture logical consequence: they have a proof from ϕ to ψ precisely when ψ logically follows from ϕ. The concept of logical consequence is thus something that goes beyond them.

None of these are the definition of proof. This is just like the point we learn from Benacerraf that none of the set-theoretic “constructions of the natural numbers” like 3 = {0, 1, 2} or 3 = {{{0}}} gives the definition of the natural numbers. The set theoretic constructions give a model of the natural numbers, but our interest is in the structure they all have in common. Likewise with proof.

The problem becomes even worse if we take a nominalist approach to proof like Goodman and Quine do, where proofs are concrete inscriptions. For then what counts as a proof depends on our latitude with regard to the choice of font!

The model theoretic approach has a similar issue. A model, on the modern understanding, is a triple (M,R,I) where M is a set of objects, R is a set of relations and I is an interpretation. We immediately have the Benacerraf problem that there are many set-theoretic ways to define triples, relations and interpretations. And, besides that, why should sets be the only allowed models?

One alternative is to take logical consequence to be primitive.

Another is not to worry, but to take the important and fundamental relation to be metaphysical consequence, and be happy with logical consequence being relative to a particular logical system rather than something absolute. We can still insist that not everything goes for logical consequence: some logical systems are good and some are bad. The good ones are the ones with the property that if ψ follows from ϕ in the system, then it is metaphysically necessary that if ϕ then ψ.

A praise-blame asymmetry

There is a certain kind of symmetry between praise and blame. We praise someone who incurs a cost to themselves by going above and beyond obligation and thereby benefitting another. We blame someone who benefits themselves by failing to fulfill an obligation and thereby harming another.

But here is a fun asymmetry to note. We praise the benefactor in proportion to the cost to the benefactor. But we do not blame the malefactor in proportion to the benefit to the malefactor. On the contrary, when the benefit to the malefactor is really small, we think the malefactor is more to be blamed.

Realism about arithmetical truth

It seems very plausible that for any specific Turing machine M there is a fact of the matter about whether M would halt. We can just imagine running the experiment in an idealized world with an infinite future, and surely either it will halt or it won’t halt. No supertasks are needed.

This commits one to realism about Σ₁ arithmetical propositions: for every proposition expressible in the form ∃nϕ(n) where ϕ(n) has only bounded quantifiers, there is a fact of the matter whether the proposition is true. For there is a Turing machine that halts if and only if ∃nϕ(n).

But now consider a Π₂ proposition, one expressible in the form ∀m∃nϕ(m,n), where again ϕ(m,n) has only bounded quantifiers. For each fixed m, there is a Turing machine M_m whose halting is equivalent to ∃nϕ(m,n). Imagine now a scenario where on day n of an infinite future you build and start M_m. Then there surely will be a fact of the matter whether any of these Turing machines will halt, a fact equivlent to ∀m∃nϕ(m,n).

What about a Σ₃ proposition, one expressible in the form ∃r∀m∃nϕ(r,m,n)? Well, we could imagine for each fixed r running the above experiment starting on day r in the future to determine whether the Π₂ proposition ∀m∃nϕ(r,m,n) is true, and then there surely is a fact of the matter whether at least one of these experiments gives a positive answer.

And so on. Thus there is a fact of the matter whether any statement in the arithmetical hierarchy—and hence any statement in the language of arithmetic—is true or false.

This argument presupposes a realism about deterministic idealized machine counterfactuals: if I were to build such and such a sequence of deterministic idealized machines, they would behave in such and such a way.

The argument also presupposes that we have a concept of the finite and of countable infinity: it is essential that our Turing machines be run for a countable sequence of steps in the future and that the tape begin with a finite number of symbols on it. If we have causal finitism, we can get the concept of the finite out of the metaphysics of the world, and a discrete future-directed causal sequence of steps is guaranteed to be countable.

Degrees of gratitude

How grateful x should be to y for ϕing depends on:

1. The expected benefit to x

2. The actual benefit to x

3. The expected cost to y

4. The actual deontic status of y’ ϕing

5. The believed deontic status of y’s ϕing.

The greater the expected benefit, the greater the appropriate gratitude. Zeroing the expected benefit zeroes the appropriate gratitude: if someone completely accidentally benefited me, no gratitude is appropriate.

I think the actual benefit increases the expected gratitude, even when the expected benefit is fixed. If you try to do something nice for me, I owe you thanks, but I owe even more thanks when I am an actual beneficiary. However, zeroing the actual benefit does not zero the expected gratitude—I should still be grateful for your trying.

The more costly the gift to the giver, the more gratitude is appropriate. But zeroing the cost does not zero the expected gratitude: I owe God gratitude for creating me even though it took no effort. I think that in terms of costs, it is only the expected and not the actual cost that matters for determining the appropriate gratitude. If you bring flowers to your beloved and slip and fall on the way back from the florist and break your leg, it doesn’t seem to me that more gratitude is appropriate.

I think of deontic status here as on a scale that includes four ranges:

1. Wrong (negative)

2. Merely permissible (neither obligatory nor supererogatory) (zero)

3. Obligatory (positive)

4. Supererogatory (super positive)

In cases where both the actual and believed deontic status falls in category (i), no gratitude is appropriate. Gratitude is only appopriate for praiseworthy actions.

The cases of supererogation call for more gratitude than the cases of obligation, other things being equal. But nonetheless cases of obligatory benefiting also call for gratitude. While y might say “I just did my job”, that fact does not undercut the need for gratitude.

Cases where believed and actual deontic status come apart are complicated. Suppose that a do-not-resuscitate order is written in messy handwriting, and a doctor misreads it as a resuscitate order, and then engages in heroic effort to resuscitate, succeeds, and in fact benefits the patient. (Maybe the patient thought that they would not be benefited by resuscitation, but in fact they are.) I think gratitude is appropriate, even if the action was actually wrong.

There is presumably some very complicated function from factors (1)–(5) (and perhaps others) to the degree of appropriate gratitude.

I am really grateful to Juliana Kazemi for a conversation on relevant topics.

Against full panpsychism

I have access to two kinds of information about consciousness: I know the occasions on which I am conscious and the occasions on which I am not. Focusing on the second, we get this argument:

1. If panpsychism is true, everything is always conscious.

2. In dreamless sleep, I exist and am not conscious.

3. So, panpsychism is false.

One response is to retreat to a weaker panpsychism on which everything is either conscious or has a conscious part. On the weaker panpsychism, one can say that in dreamless sleep, I have some conscious parts, say particles in my big toe.

But suppose we want to stick to full panpsychism that holds that everything is always conscious. This leaves two options.

First, one could deny that we exist in dreamless sleep. But if we don’t exist in dreamless sleep, then it is not possible to murder someone in dreamless sleep, and yet it obviously is.

Second, one could hold that we are conscious in dreamless sleep but the consciousness is not recorded to memory. This seems a dubious skeptical hypothesis. But let’s think about it a bit more. Presumably, the same applies under general anaesthesia. Now, while I’m far from expert on this, it seems plausible that the brain functioning under general anaesthesia is a proper subset of my present brain functioning. This makes it plausible that my experiences under general anaesthesia are a proper subset of my present wakeful experiences. But none of my present wakeful experiences—high level cognition, sensory experience, etc.—are a plausible candidate for an experience that I might have under general anaesthesia.

Being known

The obvious analysis of “p is known” is:

1. There is someone who knows p.

But this obvious analysis doesn’t seem correct, or at least there is an interesting use of “is known” that doesn’t fit (1). Imagine a mathematics paper that says: “The necessary and sufficient conditions for q are known (Smith, 1967).” But what if the conditions are long and complicated, so that no one can keep them all in mind? What if no one who read Smith’s 1967 paper remembers all the conditions? Then no one knows the conditions, even though it is still true that the conditions “are known”.

Thus, (1) is not necessary for a proposition to be known. Nor is this a rare case. I expect that more than half of the mathematics articles from half a century ago contain some theorem or at least lemma that is known but which no one knows any more.

I suspect that (1) is not sufficient either. Suppose Alice is dying of thirst on a desert island. Someone, namely Alice, knows that she is dying of thirst, but it doesn’t seem right to say that it is known that she is dying of thirst.

So if it is neither necessary nor sufficient for p to be known that someone knows p, what does it mean to say that p is known? Roughly, I think, it has something to do with accessibility. Very roughly:

2. Somebody has known p, and the knowledge is accessible to anyone who has appropriate skill and time.

It’s really hard to specify the appropriateness condition, however.

Does all this matter?

I suspect so. There is a value to something being known. When we talk of scientists advancing “human knowledge”, it is something like this “being known” that we are talking about.

Imagine that a scientist discovers p. She presents p at a conference where 20 experts learn p from her. Then she publishes it in a journal when 100 more people learn it. Then a Youtuber picks it up and now a million people know it.

If we understand the value of knowledge as something like the sum of epistemic utilities across humankind, then the successive increments in value go like this: first, we have a move from zero to some positive value V when the scientist discovers p. Then at the conference, the value jumps from V to 21V. Then after publication it goes from 21V to 121V. Then given Youtube, it goes from 121V to 100121V. The jump at initial discovery is by far the smallest, and the biggest leap is when the discovery is publicized. This strikes me as wrong. The big leap in value is when p becomes known, which either happens when the scientist discovers it or when it is presented at the conference. The rest is valuable, but not so big in terms of the value of “human knowledge”.

Epistemically paternalistic lies

Suppose Alice and Bob are students and co-religionists. Alice is struggling with a subject and asks Bob to pray that she might do fine on the exam. She gets 91%. Alice also knows that Bob’s credence in their religion is a bit lower than her own. When Bob asks her how she did, she lies that she got 94%, in order to boost Bob’s credence in their religion a bit more.

Whether a religion is correct is very epistemically important to Bob. But whether Alice got 91% or 94% is not at all epistemically important to Bob except as evidence for whether the religion is correct. The case can be so set up that by Alice’s lights—remember, she is more confident that the religion is correct than Bob is—Bob can be expected to be better off epistemically for boosting his credence in the religion. Moreover, we can suppose that there is no plausible way for Bob to find out that Alice lied. Thus, this is an epistemically paternalistic lie expected to make Bob be better off epistemically.

And this lie is clearly morally wrong. Thus, our communicative behavior is not merely governed by maximization of epistemic utility.

More on averaging to combine epistemic utilities

Suppose that the right way to combine epistemic utilities across people is averaging: the overall epistemic utility of the human race is the average of the individual epistemic utilities. Suppose, further, that each individual epistemic utility is strictly proper, and you’re a “humanitarian” agent who wants to optimize overall epistemic utility.

Suppose you’re now thinking about two hypotheses about how many people exist: the two possible numbers are m and n, which are not equal. All things considered, you have credence 0 < p₀ < 1 in the hypothesis H_m that there are m people and 1 − p₀ in the hypothesis H_n that there are n people. You now want to optimize overall epistemic utility. On an averaging view, if H_m is true, if your credence is p₁, your contribution to overall epistemic utility will be:

• (1/m)T(p₁)

and if H_m is false, your contribution will be:

• (1/n)F(p₁),

where your strictly proper scoring rule is given by T, P. Since your credence is p₁, by your lights the expected value after your changing your credence to p₀ will be:

• p₀(1/m)T(p₁) + (1−p₀)(1/n)F(p₁) + Q(p₀)

where Q(p₀) is the contribution of other people’s credences, which I assume you do not affect with your choice of p₁. If m ≠ n and T, F is strictly proper, the expected value will be maximized at

• p₁ = (p₀/m)/(p₀/m+(1−p₀)/n) = np₀/(np₀+m(1−p₀)).

If m > n, then p₁ < p₀ and if m < n, then p₁ > p₀. In other words, as long as n ≠ m, if you’re an epistemic humanitarian aiming to improve overall epistemic utility, any credence strictly between 0 and 1 will be unstable: you will need to change it. And indeed your credence will converge to 0 if m > n and to 1 if m < n. This is absurd.

I conclude that we shouldn’t combine epistemic utilities across people by averaging the utilities.

Idea: What about combining them by computing the epistemic utilities of the average credences, and then applying a strictly proper scoring rule, in effect imagining that humanity is one big committee and that a committee’s credence is the average of the individual credences?

This is even worse, because it leads to problems even without considering hypotheses on which the number of people varies. Suppose that you’ve just counted some large number nobody cares about, such as the number of cars crossing some intersection in New York City during a specific day. The number you got is even, but because the number is big, you might well have made a mistake, and so your credence that the number is even is still fairly low, say 0.7. The billions of other people on earth all have credence 0.5, and because nobody cares about your count, you won’t be able to inform them of your “study”, and their credences won’t change.

If combined epistemic utility is given by applying a proper scoring rule to the average credence, then by your lights the expected value of the combined epistemic utility will increase the bigger you can budge the average credence, as long as you don’t get it above your credence. Since you can really only affect your own credence, as an epistemic humanitarian your best bet is to set your credence to 1, thereby increasing overall human credence from 0.5 to around 0.5000000001, and making a tiny improvement in the expected value of the combined epistemic utility of humankind. In doing so, you sacrifice your own epistemic good for the epistemic good of the whole. This is absurd!

I think the idea of averaging to produce overall epistemic utilities is just wrong.

Adding or averaging epistemic utilities?

Suppose for simplicity that everyone is a good Bayesian and has the same priors for a hypothesis H, and also the same epistemic interests with respect to H. I now observe some evidence E relevant to H. My credence now diverges from everyone else’s, because I have new evidence. Suppose I could share this evidence with everyone. It seems obvious that if epistemic considerations are the only ones, I should share the evidence. (If the priors are not equal, then considerations in my previous post might lead me to withhold information, if I am willing to embrace epistemic paternalism.)

Besides the obvious value of revealing the truth, here are two ways to reason for this highly intuitive conclusion.

First, good Bayesians will always expect to benefit from more evidence. If my place and that of some other agent, say Alice, were switched, I’d want the information regarding E to be released. So by the Golden Rule, I should release the information.

Second, good Bayesians’ epistemic utilities are measured by a strictly proper scoring rule. But if Alice’s epistemic utilities for H are measured by a strictly proper (accuracy) scoring rule s that assigns an epistemic utility s(p,t) to a credence p when the actual truth value of H is t, which can be zero or one. By definition of strict propriety, the expectation by my lights of what Alice’s epistemic utility for a given credence should be is strictly maximized when that credence equals my credence. Since Alice shares the priors I had before I observed E, if I can make E evident to her, her new posteriors will match my current ones, and so revealing E to her will maximize my expectation of her epistemic utility.

So far so good. But now suppose that the hypothesis H = H_N is that there exist N people other than me, and my priors assign probability 1/2 to there being N and 1/2 to its being n, where N is much larger than n. Suppose further that my evidence E ends up significantly supporting hypothesis H_n, so that my posterior p in H_N is smaller than 1/2.

Now, my expectation of the total epistemic utility of other people if I reveal E is:

• U_R = pNs(p,1) + (1−p)ns(p,0).

And if I conceal E, my expectation is:

• U_C = pNs(1/2,1) + (1−p)ns(1/2,0).

If we had N = n, then it would be guaranteed by strict propriety that U_R > U_C, and so I should reveal. But we have N > n. Moreover, s(1/2,1) > s(p,1): if some hypothesis is true, a strictly proper accuracy scoring rule increases strictly monotonically with the credence. If N/n is sufficiently large, the first terms of U_R and U_C will dominate, and hence we will have U_C > U_R, and thus I should conceal.

The intuition behind this technical argument is this. If I reveal the evidence, I decrease people’s credence in H_N. If it turns out that the number of people other than me actually is N, I have done a lot of harm, because I have decreased the credence of a very large number N of people. Since N is much larger than n, this consideration trumps considerations of what happens if the number of people is n.

I take it that this is the wrong conclusion. On epistemic grounds, if everyone’s priors are equal, we should release evidence. (See my previous post for what happens if priors are not equal.)

So what should we do? Well, one option is to opt for averaging rather than summing of epistemic utilities. But the problem reappears. For suppose that I can only communicate with members of my own local community, and we as a community have equal credence 1/2 for the hypothesis H_n that our local community of n people contains all agents, and credence 1/2 for the hypothesis H_n + N that there is also a number N of agents outside our community much greater than n. Suppose, further, that my priors are such that I am certain that all the agents outside our community know the truth about these hypotheses. I receive a piece of evidence E disfavoring H_n and leading to credence p < 1/2. Since my revelation of E only affects the members of my own commmunity, depending on which hypothesis is true, if p is my credence after updating on E, the relevant part of the expected contribution to the utility of revealing E with regard to hypothesis H_n is:

• U_R = p((n−1)/n)s(p,1) + (1−p)((n−1)/(n+N))s(p,0).

And if I conceal E, my expectation contribution is:

• U_C = p((n−1)/n)s(1/2,1) + (1−p)((n−1)/(n+N))s(p,0).

If N is sufficiently large, again U_C will beat U_R.

I take it that there is something wrong with epistemic utilitarianism.

Bayesianism and epistemic paternalism

Suppose that your priors for some hypothesis H are 3/4 while my priors for it are 1/2. I now find some piece of evidence E for H which raises my credence in H to 3/4 and would raise yours above 3/4. If my concern is for your epistemic good, should I reveal this evidence E?

Here is an interesting reason for a negative answer. For any strictly proper (accuracy) scoring rule, my expected value for the score of a credence is uniquely maximized when the credence is 3/4. I assume your epistemic utility is governed by a strictly proper scoring rule. So the expected epistemic utility, by my lights, of your credence is maximized when your credence is 3/4. But if I reveal E to you, your credence will go above 3/4. So I shouldn’t reveal it.

This is epistemic paternalism. So, it seems, expected epistemic utility maximization (which I take it has to employ a strictly proper scoring rule) forces one to adopt epistemic paternalism. This is not a happy conclusion for expected epistemic utility maximization.

An example of a value-driven epistemological approach to metaphysics

1. Everything that exists is intrinsically valuable.

2. Shadows and holes are not intrinsically values.

3. So, neither shadows nor holes exist.

Incompleteness

For years in my logic classes I’ve been giving a rough but fairly accessible sketch of the fact that there are unprovable arithmetical truths (a special case of Tarski’s indefinability of truth), using an explicit Goedel sentence using concatenation of strings of symbols rather than Goedel encoding and the diagonal lemma.

I’ve finally revised the sketch to give the full First Incompleteness theorem, using Rosser’s trick. Here is a draft.

What numbers could be

Benacerraf famously argued that no set theoretic reduction can capture the natural numbers. While one might conclude from this that the natural numbers are some kind of sui generis entities, Benacerraf instead opts for a structuralist view on which different things can play the role of different numbers.

The argument that no set theoretic reduction captures the negative numbers is based on thinking about two common reductions. On both, 0 is the empty set ⌀. But then the two accounts differ in how the successor sn of a number n is formed:

1. sn = n ∪ {n}

2. sn = {n}.

On the first account, the number 5 is equal to the set {0, 1, 2, 3, 4}. On the second account, the number 5 is equal to the singleton {{{{{⌀}}}}}. Benacerraf thinks that we couldn’t imagine a good argument for preferring one account over another, and hence (I don’t know how this is supposed to follow) there can’t be a fact of the matter about why one account—or any other set-theoretic reductive account—is correct.

But I think there is a way to adjudicate different set-theoretic reductions of numbers. Plausibly, there is reference magnetism to simpler referrents of our terminology. Consider an as consisting of a set of natural numbers, a relation <, and two operations + and ⋅, satisfying some axioms. We might then say that our ordinary language arithmetic is attracted to the abstract entities that are most simply defined in terms of the fundamental relations. If the only relevant fundamental relation is set membership ∈, then we can ask which of the two accounts (a) and (b) more simply defines <, + and ⋅.

If simplicity is brevity of expression in first order logic, then this can be made a well-defined mathematical question. For instance, on (a), we can define a < b as a ∈ b. One provably cannot get briefer than that. (Any definition of a < b will need to contain a, b and ∈.) On the other hand, on (b), there is no way to define a < b as simply. Now it could turn out that + or ⋅ can be defined more simply on (b), in a way that offsets (a)’s victory with <, but it seems unlikely to me. So I conjecture that on the above account, (a) beats (b), and so there is a way to decide between the two reductions of numbers—(b) is the wrong one, while (a) at least has a chance of being right, unless there is a third that gives a simpler reduction.

In any case, on this picture, there is a way forward in the debate, which undercuts Benacerraf’s claim that there is no way forward.

I am not endorsing this. I worry about the multiplicity of first-order languages (e.g., infix-notation FOL vs. Polish-notation FOL).

Theistic Humeanism?

Here’s an option that is underexplored: theistic Humeanism. There are two paths to it.

The path from orthodoxy: Start with a standard theistic concurrentism: whenever we have a creaturely cause C with effect E, E only eventuates because God concurs, i.e., God cooperates with the creaturely causal relation. Now add to this a story about what creaturely causation is. This will be a Humean story—the best I know is the David Lewis one that reduces causation to laws and laws to arrangements of stuff. Keep all the deep theistic metaphysics of divine causation.

The path from heterodoxy: Start with the metaphysics of occasionalism. Don’t change any of the metaphysics. But now add a Humean analysis of creaturely causation in terms of regularities. Since the metaphysics of occasionalism affirms regularities in the world, we haven’t changed the metaphysics of occasionalism, but have redescribed it as actually involving creaturely causation.

The two paths meet in a single view, a theistic Humeanism with the metaphysics of occasionalism and the language of concurrentism, and with creaturely causation described in a Humean way.

This theistic Humeanism is more complex than standard non-theistic Humeanism, but overcomes the central problem with non-theistic Humeanism: the difficulty of finding explanation in nature. If the fact that heat causes boiling is just a statement of regularity, it does not seem that heat explains boiling. But on theistic Humeanism, we have a genuine explanatory link: God makes the water boil because God is aware of the heat.

There is one special objection to theistic Humeanism. It has two causal relations, a divine one and a creaturely one. But the two are very different—they don’t both seem to be kinds of causation. However, on some orthodox concurrentisms, such as Aquinas’s, there isn’t a single kind of thing that divine and creaturely causation are species of. Instead, the two stand in an analogical relationship. Couldn’t the theistic Humean say the same thing? Maybe, though one might also object that Humean creaturely causation is too different from divine causation for the two to count as analogous.

I suppose the main objection to theistic Humeanism is that it feels like a cheat. The creaturely causation seems fake. The metaphysics is that of occasionalism, and there is no creaturely causation there. But if theistic Humeanism is a cheat, then standard non-theistic Humeanism is as well, since they share the same metaphysics of creaturely causation. If non-theistic Humeanism really does have causation, then our theistic Humeanism really does have creaturely causation. If one has fake causation, so does the other. I think both have fake causation. :-)