Finetuning the multiverse

It has occurred to me that there is a kind of fine-tuning of multiverse theories.

If there is too large a variety of universes within a multiverse, we get skeptical problems. For instance, if all possible laws of nature are realized, then induction-friendly laws like “Gravity is always attractive” are either outnumbered by or at least do not outnumber nasty laws like “Gravity is attractive until the end of March 2026 and repulsive afterwards”. Or if there are too many kinds of material arrangements, we would expect scenarios with local order, like Boltzmann brains to outnumber or at least not be outnumbered by scenarios like brains arising by evolution.

On the other hand, if there are too few universes, then the laws by which universes are generated need to be themselves fine-tuned or else there won’t be life.

So, a multiverse theory needs to be tuned as to the variety of universes it supposes. I don’t have a a great argument that this tuning is highly improbable, but it might be. My intuition says, for what it’s worth, that the on naturalism we should either expect a single universe or a very wide and varied multiverse, and the latter is likely to engender skepticism.

A new Thomistic view of the Trinity

I think the historical Aquinas’ solution to how God is one in three persons is a relative identity view that posits two kinds of identity:

1. Individual identity: the lack of a real distinction, where real distinction might come from a difference of matter (as in material substances), a difference of form (as in angels) or an opposition of real relations (as in the Trinity).

2. Sameness of essence: having the numerically same substantial form.

While this is a relative identity account, it has an advantage over more standard relative identity accounts in that it nicely explains our intuitions about absolute identity. For on Aquinas’ solution, as far as we know, except where God is involved, the two identities are co-extensive. For Aquinas (like perhaps Aristotle) believes in individual forms of creatures, so all creatures that are distinct by individual identity have numerically distinct substantial forms, and conversely there are no mere creatures with two substantial forms. Thus when we are not dealing with God, we can afford to be ambiguous between the two kinds of identity, and hence talk as if there was absolute identity. (As far as I know, Thomas never makes this point.)

The point of this post, however, is to offer an alternative version of Aquinas’ two-identities account. The alternative is thoroughly Thomistic, but it’s not what Thomas has. Here it is. We have two kinds of identity. The first is individual identity, as in the historical Aquinas’ solution. The second is:

3. Sameness of esse: having the numerically same esse or act of being.

Just as in the historical Aquinas’ solution, the alternate account has the feature that apart from theological cases, the two kinds of identity are co-extensive. Distinct individual creatures have distinct acts of being, and no creature has two acts of being.

A disadvantage of this view is that “sameness of esse” doesn’t fit quite as well as “sameness of essence” with the Nicene Creed’s “homoousion”. But some things are worth noting. The previous English translation of the Nicene Creed in the Catholic Church actually had “sameness of being”. The word ousia is the abstract form of einai, to be. And in God essence and act of being are the same by divine simplicity. So I think the disadvantage can perhaps be overcome.

An advantage, however, is that sameness of esse seems even more identity-like and unifying than sameness of form. If x and y have the very same act of being, it is hard to deny that there is a real unity between x and y. Here is one way to see the point. On Platonist metaphysics, unlike on Thomastic metaphysics, you and I have sameness of form: we each have the numerically same Form of Humanity. Granted, Platonist metaphysics may be false. But the fact that you and I count as having sameness of form on Platonist metaphysics should make a little less happy about using that as an account of the unity in the Trinity. On the other hand, once we supplement Platonist metaphysics with an act of being, it is clear that you and I will get different acts of being, since each of us can exist without the other doing so.

Moreover, this account may help cast a light on the thorny question of why Aquinas at one point posits a human esse in the incarnate Christ distinct from the divine esse (though elsewhere he denies it). For suppose that there is only one esse in Christ and we accept the sameness-of-being account. Then we would be able say that the Jesus Christ as human is God, since the one and only esse in Jesus Christ, whether as human or as God, is the divine esse, and this we shouldn’t say (and indeed Aquinas denies it). Thus if we accept the sameness of esse account, we get an argument for the two esse view of the Incarnation. Of course, we can turn this around. We might say that the two esse view is false, and hence the sameness of esse account is false.

Trinity, quaternity and naming God

One now-classic solution to some of the challenges of the doctrine of the Trinity is relative identity. On relative identity accounts of the Trinity, we do not heretically say, e.g., that the Father is absolutely identical with the Son, but instead we posit a same-essence (or same-God) relative identity relation, and say that the Father is same-essence as the Son. Now, most of the time proponents of relative identity views deny that there is any such thing as absolute identity at all. All identities are relative to a sortal. However, in their famous “Material Constitution and the Trinity”, Brower and Rea offer an account of the Trinity using relative identity within a system where there is such a thing as absolute identity, governed by the standard rules of first-order logic.

Any account of the Trinity that allows for absolute identity governed by classical logic faces an interesting problem. Suppose:

1. There is such a thing as absolute identity, denoted by “=” and governed by its standard first-order logic rules.

2. There is a name for God as such, “G”, and names for the Father, Son and Holy Spirit, respectively, “F”, “S” and “H”.

3. F ≠ S

4. S ≠ H

5. F ≠ H

6. For all x, if x = F or x = S or x = H while y = F or y = S or y = H, and if x = G, then y = G.

7. If x = x, y = y, z = z, w = w, x ≠ y, x ≠ z, y ≠ z, x ≠ w, y ≠ w and z ≠ w, then x, y, z and w are four, counting by absolute identity.

8. Then F, S, H and G are four, counting by absolute identity.

9. Heresy! In no sense is there a divine quaternity!

Premise (6) captures the idea that the persons of the Trinity have equality with respect to divinity, and hence if x and y are persons of the Trinity, then if either of x and y “is God” in some sense of “is” and “God”, then the other “is God” in the same sense.

Proof of 8 from 1–7:

1. If F = G or S = G or H = G, then F = G and S = G and H = G. By 6

2. If F = G and S = G, then F = S. By 1

3. So F ≠ G or S ≠ G. By 3 and ii

4. So F ≠ G and S ≠ G and H ≠ G. By i and iii

5. F = F, S = S, H = H. By 1

6. G = G. By 1

7. So, F, S, H and G are four, counting by absolute identity. By 3–5, iv–vi and 8

There is a way out of this, in van Inwagen’s work on the Trinity. Van Inwagen denies that there are names for the Father, Son, Spirit and God: there are just predicates, like “is divine” and “is a Father”.

This is overkill for getting out of the above. It is difficult for a Christian to completely deny the existence of divine names. After all, you become a Christian by having someone pour water over you while saying: “I baptize you in the name of the Father, and of the Son and of the Holy Spirit.”

I think the quaternity argument leaves Christians who want to uphold classical absolute identity with two options:

1. There is no name for God.

2. There are no names for the Father, Son or Holy Spirit.

Here, “name” means “proper name”, since that’s the kind of name that classical logic talks about.

Accepting both (a) and (b) is not, I think, an option. Of the two, I think (b) is the more problematic. Names are too central to human relationships with persons. Thus, I think, the Christian who wants to uphold classical absolute identity should accept (a). And there is some support in Christian tradition for denying that God has a name.

(This is not what Brower and Rea do. In footnote 7, they talk of “the name ‘God’”, albeit in the context of discussion of a heretical view, but as I read the footnote they have no reservation about “God” being a name.)

While (a) is the less problematic of the two options, I still think (a) is problematic. Grammatically, “God” and “the Trinity” and the tetragrammaton all function as proper names. While there is some precedent in the Christian tradition for denying that God has a name, that still seems a bit of a stretch.

I think that if one is going to adopt a relative-identity account of the Trinity, then one should either deny that there is such a thing as absolute identity or at least one should deny that it is governed by the rules of classical logic.

I tentatively propose the following modification to the classical theory of absolute identity as a way of getting out of the quaternity argument. Restrict the identity-introduction rule that says that for any name N we get to write down N = N to the case of what one might call hypostatic or individual names (or, perhaps, to non-essence names). Thus, we get to write “Father=Father” but not “God=God”. Now everything in the quaternity argument goes through, except we don’t get (vi) and and hence we can’t infer (vii) or (8).

A technical problem with brutal composition

Markosian once proposed brutal composition as an answer to the question of when a (proper) plurality of objects composes a whole: composition is a brute fact, where a brute fact is a fact F such that “it is not the case that F obtains in virtue of some other fact or facts.”

Here is a nitpicky objection. Obviously:

1. The xs compose a whole if and only if there is a y such that the xs compose y.

Moreover, this doesn’t just happen to be true. For:

2. The fact that the xs compose a whole is the fact that there is a y such that the xs compose y.

And:

3. If it is a fact that there is a y such that the xs compose y, then there is an object y₀ such that the fact that there is a y such that the x compose y obtains in virtue of the fact that the xs compose y₀.

This follows from the Weak Existential Grounding principle:

4. If it is a fact that ∃yF(y), then there is at least one y₀ such that the fact that F(y₀) grounds the fact that ∃yF(y).

(Strong Existential Grounding says that the fact that ∃yF(y) is grounded in every instance. There are apparent counterexamples, but Weak Existential Grounding is hard to deny.) Assuming that “obtains in virtue of” is the same as “is grounded by”, we conclude from (2)–(4) that:

5. If the xs compose a whole, then there is a y₀ such that the fact that the xs compose a whole obtains in virtue of the fact that the xs compose y₀.

What can a brutalist say about this argument? I think one move would be to deny (2).

Instead, perhaps, we have a grounding relation between the facts that the xs compose a whole and that there is a y such that the xs compose y. If this grounding relation runs right-to-left, we have an immediate contradiction to the brutal composition thesis from (2): that the xs compose a whole obtains in virtue of there being a y such that the xs compose y.

So the grounding relation would have to run left-to-right. Thus we would have to have it that the xs composing a whole grounds there being a y such that the xs compose y. But likewise the particular fact that the xs compose y₀ grounds there being such a y. Now consider the two facts:

1. that the xs compose y₀

2. that the xs compose a whole.

If neither grounds the other, then the fact that there is a y such that the xs compose y is grounding-overdetermined by (i) and (ii). This is implausible. If (i) grounds (ii), we have contradicted brutal composition. That leaves the option that (ii) grounds (i). I am dubious. For it seems implausible to think that in general the fact that the xs compose a whole grounds that the xs compose y₀. For, plausibly, we can have cases where in one world the xs compose y₀ and in another world the xs compose y₁, where y₁ ≠ y₀. (Imagine that in one world a chair is made of the xs and in another a statue is.) But if the fact that p grounds the fact that q, then p entails q by Grounding Entailment.

The Sign of the Cross and the Trinity

A principle that both Catholic and Orthodox Christians appreciate is lex orandi, lex credendi: the law of prayer is the law of faith,

Now notice that we cross ourselves with the Trinity in a specific order: “In the name of the Father, and of the Son, and of the Holy Spirit.” This is probably the most common Christian prayer in the world. And it has an order: first, the Father is mentioned, then the Son, and then the Holy Spirit. The same order is found in the baptismal formula. While Eastern Christians tend to cross themselves right-to-left and Western Christians left-to-right, the order of the words is maintained (at least in the Churches where Trinitarian words are used: more details here).

It feels badly off to pray “In the name of the Father, and of the Holy Spirit, and of the Son.” Now, we might say that the feeling of its being “off” has a deflationary psychological explanation—we are just used to a specific order. Or, somewhat better, it is off insofar as it disconnects us from the practices of the community. But the universality of the order in the most common Christian prayer together with the theological methodology of lex orandi, lex credendi and the intuition that other orders aren’t as good should impel us to see theological significance in the order.

We might, of course, see a purely missional order here. First the Father is active in the world, then the Son, and then the Holy Spirit is sent. But I think we want more than just such a temporal ordering in our explanation. After all, while the Holy Spirit is especially sent in New Testament times, it seems right to talk of the Spirit’s activity in Old Testament times. And maybe even some of the explicit mentions of Ruakh Elohim, God’s Spirit, including at the beginning of Genesis, is already an indication of the economic involvement of the Spirit. It is at least better if we can ground what feels like an important feature of a central Christian prayer in something eternal.

One solution would be a subordinationist one: the Son is subordinate to the Father, and the Holy Spirit is subordinate to the Son. But that would be a heresy.

I propose that the filioque, whether formulated in terms of the Holy Spirit proceeding from the Father and the Son or from the Father through the Son, gives an elegant eternity-based account of what makes the order in the prayer appropriate. First, the Father, from whom the Son proceeds, and then the Holy Spirit who proceeds from the Father and/through the Son. The order is not that of subordination but of procession.

The filioque

Suppose we model the Trinity as three vertices with arrows between them representing relations of procession, such that:

1. No two vertices have two arrows between them and arrows only go between distinct vertices.

2. The arrows define the vertices in the sense that there is no way to relabel the vertices while preserving which vertex has an arrow to which vertext (i.e., there is no non-trivial directed graph automorphism).

There are seven different structures possible subject to (a).

Condition (b) is a way of capturing part of the traditional Western idea that the Trinity is relational, so that the distinctions between the persons arise from the processional structure. Four of the graphs violate this condition: in (1) we can rearrange the labels in any way without changing the relational structure; in (4) and (5) we can swap A and C; in (7), we can rotate the whole graph by 120 degrees.

Aquinas captures the relationality idea by the principle that person x is distinguished from person y if and only if there is a (real) processional relation between x and y. I think this principle is too strong in one way and too weak in another way. It is too strong, because (2) would allow the persons to be distinguished, even though there is no processional relation between A and C (or B and C): A would be the unique person from whom someone proceeds; B would be the unique person that proceeds; C would be the unique person unrelated by procession. It is too weak, because in (7) we have a processional relation between each person, but it fails to distinguish the three persons. Of course, (2) and (7) have other flaws.

The following options satisfy condition (b): (2), (3) and (6).

Graph (2) is theologically unacceptable, as it has a person not related to any other. And so while the relationality in (2) would be sufficient to define the persons, because C would be unrelated to A and B, the graph does not represent a fully relational Trinity. Besides which, (2) has two persons, A and C, who do not proceed from another, and the East and West agree that only the Father does not proceed.

Graph (3) is interesting. If we adopted this model of the Trinity, we would have to take A to be the Father, since everyone agrees that the Father does not proceed for anyone. Moreover, since everyone agrees that the Son proceeds from the Father, we would hae to

In graph (3), we would have to take A to be the Father, since the Father does not proceed from any other person, as the East and West agree. Likewise, the East and West agree that the Son comes from the Father. So B would have to be the Son. That would make the Holy Spirit be C, and proceed only from the Son. Neither the East nor the West will like this. And it would violate the principle that the Son eternally has nothing beyond what the Father has, excepting what follows from the Son’s being generated by the Father, since now the Son has the Spirit proceeding from him, but the Father does not. I suppose one might try to reconcile (3) with the Western view by saying that the Spirit proceeds mediately from the Father and immediately from the Son. I don’t like this.

That leaves graph (6), the Western view of the Trinity, with A being the Father, C being the Son and B being the Holy Spirit.

This isn’t really a great argument against the Eastern view of the Trinity, because the East will deny (b), on the grounds that one can distinguish the persons not merely by the directions of the procession, but by the types of processions, with generation and spiration being different types. One can think of this Eastern move as allowing the arrows to have different colors, and then instead of (b) having the condition that there is no way to relabel the vertices while preserving which vertex has an arrow of what color to what vertex.

The point here is rather to allow us to state with some precision what principles force the western view. Namely, we have:

Theorem: Suppose:

1. There are exactly three persons in the Trinity

2. The (binary) procession relation is asymmetric: if y proceeds from x, x does not proceed from y

3. At most most one person does not proceed from another

4. There is one person from whom both of the other persons proceed

5. There is no way to relabel the persons while preserving the procession relations.

Then there is a way of respectively naming the persons “Father”, “Son” and “Holy Spirit” such that it is correct to say: “The Father does not proceed, the Son proceeds from the Father, and the Holy Spirit proceeds from the Father and the Son.”

Paley's watch and a caveman

I was teaching Paley, for the nth time, and it hit me to wonder how I would react to finding the watch if I was a caveman. There is one thing I would be sure of: the watch is not the work of a human—the difference between the watch and the most complex human technology of the time would be too vast (think of a bow and arrow—impressive as that is). Would I think it the work of a non-human designer? I am not sure. Supposing that I already thought that nature was the product of a non-human designer, it might be reasonable to think that the same designer produced the watch. The intricacy of the watch’s insides might have some resemblance to the intricacy of the internal parts of animals. But if I did not already think nature around me to be the work of design, I doubt the watch would move me. Whatever stories, if any, I told myself about nature, I would try to extend to the watch. Maybe it sprouted from the ground?

This is making me think that the objection—dismissed by Paley—that the reason we think the watch to be the product of design is that we’ve seen similar things made has more force than he grants it. It is pretty clear to me that the fact that we’ve seen things of “this sort” (understood broadly enough to included non-watches but not so broadly as to include the works of our cave-dwelling ancestors) made is an important ingredient of our inference that the watch was made.

However, I also don’t think this undercuts Paley’s argument. For it may be that seeing really complex things made opens our eyes to the possibility of an explanation that would not occur to a caveman. Indeed, here is a second thought experiment. Suppose we find something with the mechanical and functional complexity of a watch, but (a) we have never seen an artifact with this kind of function, and (b) we can determine conclusively that the item was not made by a human. How would we get (b)? Maybe the item came from space, buried in the center of a meteorite. I think I would be pretty completely confident that the item had a designer (presumably an alien). And I would surely be right.

So I think that seeing really complex things made is important to inferring design—but nonetheless the inference may be a sound one. When we find a watch on a heath, we are not just inferring that a human designed the watch, and hence the watch was designed. For even if we knew no human designed the watch, we would be confident the watch was designed. It’s just that our “explanatory imagination” has an advantage over the caveman’s.

But I am no caveman. I could be very wrong when trying to put myself in the sandals of one.

A variant of the preface paradox

Imagine being given a list generated as follows, without being told how it was generated:

1. A randomly chosen set of 2000 mutually consistent propositions that you justifiedly believe, subject to the constraint that they are consistent.

Given your fallibility, it’s highly probable that the list contains multiple falsehoods. As you look over the list, you say to yourself about each one: “Yup, that sounds right.” However, given what you know about your fallibility, you should also say: “I think some of these are false, though.” You will thus incline to disbelieve the conjunction of these propositions.

But suppose instead you were handed a list generated as follows, again without being told how it was generated:

2. A randomly chosen set of 2000 propositions that you know.

This list would look to you just like the first one. Thus, you would say about it exactly what you would have said about the first one. To each one, you would say: “Yup, that sounds right.” But then on reflection you would say: “I think some of these are false, though.” You will thus incline to disbelieve the conjunction of these propositions, just as in the first case. However, while in the first case you are correct in disbelieving, in the second you are mistaken in disbelieving.

But in any case, if you incline to disbelieve something, then you don’t know it. Hence, you do not know the conjunction of the 2000 randomly chosen propositions that you know, and so knowledge is not closed under conjunction.

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).

How has Aquinas not proved the Trinity?

St Thomas holds that we cannot know by natural reason that God is a Trinity. However, he also endorses a version of St Augustine’s account of the Trinity, on which God has a mental Word or concept of himself that he generates mentally, and then there is a Love that is responsive to that Word. Why is this not a proof by natural reason?

On this point, Thomas is sadly brief in the Summa Theologiae: “Again, the similarity with our intellect does not sufficiently prove anything about God, since the intellect is not found univocally in God and in us.”

But Aquinas thinks that we can literally talk about God, even if we do so by means of analogy. He has argued, by reason, that God exists and has all perfections. One of these perfections is knowledge of everything, and thus of himself. Does this not commit Aquinas to being able to say on the basis of natural reason that God generates his self-knowledge, a self-knowledge that, by divine simplicity, must then be God himself rather than a mere creature? And does not God love himself under the description that this knowledge provides?

I think Aquinas would (with perhaps minor cavils) endorse all the claims in my previous paragraph as knowable by natural reason. How is this not a proof of the Trinity? I think the answer is this. Focus on the generation of the Son—the case of the Holy Spirit will presumably be similar. We do not know by reason whether the relation of generating self-knowledge is a real relation in God. If it is a real relation it will individuate the knower from the knowledge, thereby ensuring that there are at least two persons in God. But it needs to be a real relation in God, rather than a merely conceptual one. And here Thomas’s brief remark about univocity seems apposite. Aquinas thinks that what makes our language about God analogical rather than univocal is that the grounding of claims about God is radically different from the grounding of similar claims about creatures. Thus, when we say Socrates is wise, this is grounded by the inherence in Socrates of an accident of wisdom, while when we say God is wise, this is grounded by the identity between God and his wisdom. Because of this, even if it turns out that in us the generation of self-knowledge is a real relation (but not one between persons, since we are not simple, and hence our self-knowledge does not need to be of one essence with us), since the grounding structure of claims about God is radically different, as far as unaided human reason goes, it may not be grounded in a real relation in God—though it might be. We need God’s revelation to know if it is a real relation or not. And indeed God has revealed that it is.

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.

More on strong open-mindedness

For the last couple of days I have been exploring what I like to call strongly open-minded accuracy scoring rules. It’s well known that every proper scoring rule is open-minded in the sense that it never requires you to reject free information: the expected epistemic utility of updating on the free information is always at least as good as your current expected epistemic utility. It’s strictly open-minded provided that in non-trivial cases (i.e., when the information has a non-zero probability of having statistical relevance to the credences you are scoring) you are required to accept the free information.

Now there are two reasons why one might accept free information about some proposition q. First, you might be wrong about q: your credence may be high while q is false or your credence might be low while q is true. Second, even if you are right about q, the free information may boost your credence in the right direction. I say that a scoring rule is strongly open-minded provided that it licenses you to accept and update on the free information even if you disregard the first consideration. We can then tack on “strictly” if it requires you to do so in non-trivial cases. In the case of a strongly open-minded scoring rule, your acceptance of free information is not a sign of doubt in your propositions—it is not a way of hedging your bets—and thus is arguably compatible with faith in the propositions being evaluated.

A strongly open-minded scoring rule can also be characterized in the following way. There is a more ordinary kind of epistemic paternalism where I might have reason to block another from receiving free information on the grounds that this information could mislead due to the fact that the other has different likelihoods from the ones I think are right. For instance, if too many people have an unjustified mistrust of Dr. Smith such that they are likely to believe the opposite of what Dr. Smith’s experiments reveal, there is reason to give a grant to someone else, because Dr. Smith’s experiments are likely to lead people away from the truth, for no fault of Dr. Smith’s. Call this likelihood-based paternalism. But there is another kind of motivation of the refusal of free information for another, which we might call pure-risk-based paternalism. Even if someone else has the same likelihoods as you do—trusts Dr. Smith just as you do—perhaps the risk that Dr. Smith’s experiments will, by pure chance, provide evidence away from the truth is enough to justify not funding these experiments.

I’ve been collecting results about these issues. Here’s what I seem t have so far, though I have to emphasize that sometimes the proofs are just in my head and I might be wrong. I will specialize on scoring rules for a single proposition, given as a pair of functions T and F, where T(x) is the value of having credence x when the proposition is true and F(x) is the value of having credence x when the proposition is false.

1. A scoring rule sometimes calls for pure-risk-based paternalism if and only if it is not strongly open-minded.

2. A scoring rule that’s strongly open-minded is open-minded.

3. A scoring rule (T,F) is (strictly) strongly open-minded if and only if xT(x) and (1−x)F(1−x) are both (strictly) convex.

4. The logarithmic scoring rule is strictly strongly open-minded. The Brier and spherical rules are not strongly open-minded.

5. If a proper scoring rule is generated by the Schervisch-style integral representation T(x) = T(1/2) + ∫_x^1/2(1−t)b(t)dt and F(x) = F(1/2) + ∫_1/2^xtb(t)dt and b is sufficiently differentiable, then the scoring rule is strongly open-minded if and only if the derivative of log b(x) lies between (3x−2)/[x(1−x)] and (3x−1)/[x(1−x)].

6. A strongly open-minded scoring rule whose logarithm is sufficiently differentiable is unbounded.

7. [Item deleted as I discovered it to be false.]

8. For any credences p and r such that 1/2 < r and p < r, there is a strictly proper scoring rule and a situation where the scoring rule calls for the individual with credence r to have purely-risk-based epistemic paternalism for that hypothesis.

Algorithmic priors and human nature

One promising way to define priors is with algorithmic probability, such as Solomonoff priors. The idea is that we have a language L (say, one based on Turing machines), and we imagine generating random descriptions in L in a canonical way. E.g., add an end-of-string symbol to L and randomly and independently generate symbols until you hit the end-of-string symbol, and then conditionalize on the string uniquely describing a situation, and take the probability of a specific situation s to be the probability of that a random description so generated describes s.

These kinds of priors are rather appealing for science, since they appear to be induction-friendly, as they assign high probabilities to compressible—more briefly expressible—situations. Thus, if our situations are distributions of color among ravens, monochromatic distributions get much higher probability as they can be much more briefly described, like ∀x(B(x)) or ∀x(W(x)).

Philosophically, I think the big problem is with the choice of the language. It would be nice if we could let L be a language that cuts nature exactly at the joints. But we don’t know that language. And absent that language, we need something arbitrary.

Here is a particular version of the problem. Take Kuhn’s division of science into ordinary and revolutionary science. One aspect of this division is that in ordinary science, we have a scientific language, and are discovering things within it. In that case, it is reasonable to take L to be that language. However, when we are doing revolutionary science and creating new paradigms, we cannot do that. The new paradigms either cannot be described in the old language or their description is unwieldy in a way that does not do justice to the plausibility of the new paradigm. Indeed, much of the point of revolutionary science is to create a language within which the description of the world is simpler, and then argue that this language is therefore more likely to cut nature at the joints.

Another version of this problem is what language L we choose when we are generating fundamental priors. Practically speaking, we cannot use a scientific language that cuts nature at the joints, because we have not yet discovered it. If this was merely a practical concern, we could try to say that this doesn’t matter: the fundamental priors are ones that we ought to have rather than any that we actually have or could have—perhaps ought does not imply can. But the concern is not merely practical. For one of the main points of our inductive reasoning is to discover what concepts cut nature at the joints, and this is largely an empirical enterprise. If the right fundamental priors were to reflect the joints in nature, then the enterprise wouldn’t make much sense, as we would be obligated to have already completed much of the enterprise before we started it.

So, I think, we have to say L does not always cut nature at the joints, and yet this generating appropriate priors for us. But we still need a constraint on L. After all, we could imagine a language that thwarts our empirical enterprise, such as one where only fairies can be described briefly and anything else requires very long descriptions, so we have very high priors for fairies and very low priors for everything else. What will be the constraint? Practically, we pretty much have to start with some ordinary human language. I think our ideal should not be far from what is practical. Thus, I propose, if we are going to go with algorithmic priors, we should choose L to be a language that fits well with our human nature as communicators. This is an anthropocentric choice, and I think human epistemology is rightly anthropocentric.

But why think that the anthropocentric choice is apt to lead to truth? There are two stories to be told here. First, it may be that human nature requires a measure of trust in human nature. Second, that trust is vindicated if we are created by a good God who loves the truth.

Anselm and Brouwer

I was reading Anselm’s replies to Gaunilo, and was struck by this:

Furthermore: if it can be conceived at all, it must exist. For no one who denies or doubts the existence of a being than which a greater is inconceivable, denies or doubts that if it did exist, its non-existence, either in reality or in the understanding, would be impossible. For otherwise it would not be a being than which a greater cannot be conceived. But as to whatever can be conceived, but does not exist – if there were such a being, its non-existence, either in reality or in the understanding, would be possible. Therefore if a being than which a greater is inconceivable can be even conceived, it cannot be nonexistent.

Let → indicate subjunctive conditionals. Let E!(x) say that x exists.

1. E!(God) → □E!(God).

2. ∀x[if Conceivable(x) and  ∼ E!(x), then: E!(x) → ⋄ ∼ E!(x)].

3. So, not: (Conceivable(God) and  ∼ E!(God)).

4. So, if Conceivable(God), then E!(God).

The ∀x quantifier in (2) is problematic, since it ranges over beings that don’t exist. Perhaps we can read it substitutionally. Let’s suppose we can finesse this issue.

What interests me in (2) is that big conditional in it is most plausibly as seen as a special case of:

5. If C(p) and q, then p → ⋄q,

where C(p) says “conceivably p”, which may or may not be the same as “possibly p”.

We can prove (5) from the Brouwer Axiom

6. If q, then □⋄q,

where L is necessity, and the following principle about subjunctive conditionals:

7. If C(p) and Lr, then p → r,

namely that necessities would still hold no matter what conceivable things happened (to get (5) from (6) and (7), let r be ⋄q). Principle (7) is very plausible if conceivability is possibility: if a possible thing happened, anything necessary would still be true. It’s less plausible if conceivability is not possibility.

And, of course, the Brouwer Axiom is controversial, albeit not quite as much as S5. I initially hoped that the use of the subjunctive conditional in (2) allowed Anselm to get by with something weaker than Brouwer. But not so if the route to (2) goes through (5) and possibility implies conceivability (PIC). For we get Brouwer from (5), PIC and the very plausible principle:

8. If p is possible, then we do not have p → r and p →  ∼ r.

For suppose that contrary to Brouwer we are at a world where q is true, but ⋄q is false at some accessible world. By PIC,  ∼ ⋄q is conceivable. Let p be  ∼ ⋄q. Then C(p) and  ∼ p. But clearly  ∼ ⋄q →  ∼ ⋄q. If we had (5), we would have  ∼ ⋄q → ⋄q, and contradict (8).

So, if Anselm’s argument for (2) goes through (5), we don’t have an improvement over Brouwer. But we can still get (2), given some very plausible assumptions, and the following special case of (5):

9. If C(∼q) and q, then  ∼ q → ⋄q.

And I feel that (9) has some plausibility above (6), at least if conceivability is the same as possibility. For suppose q is true and  ∼ q is possible. Then it seems somewhat plausible that q is possible in the  ∼ q-worlds that are closest to the actual world. Maybe. But maybe there is some way to derive Brouwer from (9) and additional plausible premises.

Does everything in time change?

Over the last two days, I’ve been thinking critically about Aquinas’ First Way. Central to my thinking, and especially yesterday’s post, was the idea that you could have an unmoved mover who is in time but isn’t pure act and who isn’t God. Such an unmoved mover constantly and unchangingly exercises—perhaps mentally—one and the same causal power to make something else move.

But I now wonder if this is possible. Suppose a demiurge that exists in time and has the power to make Bob rotate, and constantly exercises this power. Could this demiurge be unchanging? After all, at noon the demiurge is actively rotating Bob at noon, and at 1 pm the demiurge is actively rotating Bob at 1 pm. We can easily and coherently suppose that the demiurge engages in qualitatively the same activity at 1 pm as at noon. That was the intuition that was driving my thinking about this. But can we coherently add that it is the numerically same activity? For if it’s not numerically the same activity at 1 pm as at noon, then the demiurge has undergone a change, from engaging in activity a₁₂ to engaging in activity a₁₃, even if the two activities are exactly alike.

I am not sure, but I feel a pull to thinking that rotating Bob at 1 pm is a different thing from rotating Bob at noon, assuming that the agent is in time. I don’t just mean that it has different effects—which it does, since spinning-at-noon is a different effect from spinning-at-one—but that the activity of causing rotation is itself different. Maybe the pull comes from this thought. Perdurantists think that substances exist at different times by having different temporal parts at them. Perdurantism is likely false for substances. But whether or not it is true for substanes, it seems very plausible for events and activities. What made World War II exist on each day between September 1, 1939 and September 2, 1945 is that there were hostilities on each day, hostilities that are a part of World War II. Even if on two successive days the hostilities happened to be exactly alike, they would have been numerically different hostilities. If this is right in general, then the activity of rotating Bob at 1 pm is numerically different from that of rotating Bob at noon.

Furthermore, I think existence is a kind of activity. This is most obvious in the case of living things, given the Aristotelian idea that life is the existence of the living and life is an activity, but I think is true in general. Thus a thing that exists in time over a lifetime engages in a sequence of numerically different activities—existing at t₁, existing at t₂, and so on. And hence it changes. And intrinsically so. If so, then everything that exists in time must always change.

If the suggestion that there are no unchanging activities that last over time, then we can escape my worry yesterday that perhaps the sequence of moved movers in the First Way leads to a mover that is unchanging with respect to the activity of moving the next mover in the sequence but is still changing in some other coincidental respect. For the activity of moving the next mover in the sequence would have to change over time, and so the mover would be changing in respect of of its moving the next item in the sequence.

But perhaps not. For we might admit that in all the cases we are familiar with, activity only perdures over time, and there is always something numerically different happening at different times, but say that we could still imagine a being where the numerically same activity is temporally multilocated. And such a being could everlastingly rotate Bob with the activity of spinning Bob being genuinely unchanging.

I don’t know.