"On the same grounds"

Each of Alice and Seabiscuit is a human or a horse. But Alice is a human or a horse “on other grounds” than Seabiscuit is a human or a horse. In Alice’s case, it’s because she is a human and in Seabiscuit’s it’s because he’s a horse.

The concept of satisfying a predicate “on other grounds” is a difficult one to make precise, but I think it is potentially a useful one. For instance, one way to formulate a doctrine of analogical predication is to say that whenever the same positive predicate applies to God and a creature, the predicate applies on other grounds in the two cases.

The “on other/same grounds” operator can be used in two different ways. To see the difference, consider:

1. Alice is Alice or a human.

2. Bob is Alice or a human.

In one sense, these hold on the same grounds: (1) is grounded in Alice being human and (2) is grounded in Bob being human. In another sense, they hold on different grounds: for the grounds of (1) also include Alice’s being Alice while the grounds of (2) do not include Bob’s being Alice (or even Bob’s being Bob).

Stipulatively, I’ll go for the weaker sense of “on the same grounds” and the stronger sense of “on different grounds”: as long as there is at least one way of grounding “in the same way”, I will count two claims as grounded the same way. This lets me say that Christ knows that 2 + 2 = 4 on the same grounds as the Father does, namely by the divine nature, even though there is another way in which Christ knows it, which the Father does not share, namely by humanity.

Even with this clarification, it is still kind of difficult to come up with a precise account of “on other/same grounds”. For it’s not the case that the grounds are literally the same. We want to say that the claims that Bob is human and that Carl is human hold on the same grounds. But the grounding is literally different. The grounds of the former is Bob’s possession of a human nature while the grounds of the latter is Carl’s possession of a human nature. Moreover, if trope theory is correct, then the two human natures are numerically different. What we want to say is something like this: the grounds are qualitatively the same. But how exactly to account for the “qualitatively sameness” is something I don’t know.

There is a lot of room for interesting research here.

Grace and theories of time

1. All grace received is given through Christ’s work of salvation.
2. Christ’s work of salvation happened in the first centuries AD and BC.
3. One cannot give something through something that does not exist.
4. Abraham received grace prior to the first century BC.
5. So, Abraham’s grace was given through Christ’s work of salvation.
6. So, it was true to say that Christ’s work of salvation exists even when it was yet in the future.
7. So, presentism and growing block are false.

Junia/Junias and the base rate fallacy

I think it would be useful to apply more Bayesian analyses to textual scholarship.

In Romans 16:7, Junia or Junias is described as “famous among the apostles”. Without accent marks (which were not present in the original manuscript) it is not possible to tell purely textually if it’s Junia, a woman, or Junias, a man. Moreover, “among the apostles” can mean “as being an apostle” or “to the apostles”. There seems to be, however, some reason to think that the name Junia is more common than Junias in the early Christian population, and the reading of “among” as implying membership seems more natural, and so the text gets used as support for women’s ordination.

This post is an example of how one might go about analyzing this claim in a Bayesian way. However, since I am not a Biblical scholar, I will work with some made-up numbers. A scholarly contribution would need to replace these with numbers better based in data (and I invite any reader who knows more Biblical scholarship to write such a contribution). Nonetheless, this schematic analysis will suggest that even assuming that there really were female apostles, it is more likely than not that Junia/s is one.

Let’s grant that in the early Christian population, “Junia” outnumbers “Junias” by a factor of 9:1. Let’s also generously grant that the uses of “famous among” where the individual is implied to be a member of the group outnumber the uses where the individual is merely known to the group by a factor of 9:1. One might think that this yields a probably of 0.9 × 0.9 = 0.81 that the text affirms Junia/s to be an apostle.

But that would be to commit the infamous base rate fallacy in statistical reasoning. We should think of a text that praises a Junia/s as “famous among the apostles” as like a positive medical test result for the hypothesis that the individual praised is a female apostle. The false positive rate on that test is about 0.19 given the above data. For to get a true positive, two things have to happen: we have to have Junia, probability 0.9, and we have to use “among” in the membership-implying sense, probability 0.9, with an overall probability of 0.81 assuming independence. So the false positive rate on the test is 1 − 0.81 = 0.19. In other words, of people who are not female apostles, 19 percent of them will score positive on tests like this.

But we have very good reason to think that even if there were any female apostles in the early church, they are quite rare. Our initial sample of apostles includes the 12 apostles chosen by Jesus, and then one more chosen to replace Judas, and none of these were women. Thus, we have reason to think that fewer than 1/13 of the apostles were women. So let’s assume that about 1/13 of the apostles were female. If there were any female apostles, they were unlikely to be much more common than that, since then that would probably have been more widely noted in the early Church.

Moreover, not everyone that Paul praises are apostles. “Apostle” is a very special position of authority for Paul, as is clear from the force of his emphases on his own status as one. Let’s say that apostles are the subjects of 1/3 of Pauline praises (this is something that it would be moderately easy to get a more precise number on).

Thus, the chance that a randomly chosen person that Paul praises is a female apostle even given the existence of female apostles is only about (1/13)×(1/3) or about three percent.

If we imagine Paul writing lots and lots of such praises, there will be a lot of Junia/s mentioned as “famous among the apostles”, some of whom will be male, some female, and some of whom will be apostles and some not.
All of these are the “positive test results”. Of these positive test results, the 97% percent of people praised by Paul who aren’t female apostles will contribute a proportion of 0.19 × 97%=18% of the positive test results. These will be false positives. The 3% people who are female apostles will contribute at most 3% of the positive test results. These will be true positives. In other words, among the positive test results, approximately the ratio 18:3 obtains between the false and true positives, or 6:1.

In other words, even assuming that some apostles are female, the probability that Junia/s is a female apostle is at most about 14%, once one takes into account the low base rate of women among apostles and apostles among those mentioned by Paul.

But the numbers above are made-up. Someone should re-do the analysis with real data. We need four data points:

• Relative prevalence of Junia vs. Junias in the early Christian population.

• Relative prevalence of the two senses of “famous among” in Greek texts of the period.

• Reasonable bounds on the prevalence of women among apostles.

• Prevalence of apostles among the subjects of Pauline praise.

And without such numbers and Bayesian analysis, I think scholarly discussion is apt to fall into the base rate fallacy.

Is eternalism compatible with the actualization of potentiality?

Every so often, someone claims to me that there is a difficulty in reconciling the Aristotelian idea of the actualization of potential with eternalism, the view that past, present and future are equally real. I am puzzled by this question, because I can’t see the difficulty. On the contrary, there is a tension between presentism, the view that only present things exist, and this Aristotelian thesis:

1. Some present events are the actualization of a no-longer present potentiality.

2. A non-existent thing is not actualized.

3. Therefore, some no-longer present potentialities exist.

4. Therefore, something that is no longer present exists.

5. Therefore, presentism is false.

One might say: Yes, the potentiality doesn’t exist, but it did exist, and it was actualized. But then:

6. Some present potentialities are actualized in not yet present events.

7. A non-existent thing does not actualize anything.

8. So, there exist some not yet present things.

9. So, presentism is false.

Of course, this is the old problem of transtemporal relations for presentism as applied to the actualization relation.

So, what about the question whether eternalists can have actualization of potentials? Here may be the problem. On eternalism plus Aristotelianism, it seems that the past unactualized potential exists even though it is now actualized. This seems to be a contradiction: how can an unactualized potential be actualized?

A first answer is that a potential is actualized at a time t provided that its actualization exists at t. Thus, the potential is unactualized at t₁ but actualized at a later time t₂, because its actualization exists at t₂ but not at t₁. But, the objector can continue, by eternalism at t₁ isn’t it the case that the actualization exists? Yes: but the eternalist distinguishes:

10. It is true at t₁ that B exists.

11. B exists at t₁.

Claim (11), for spatiotemporal objects, means something like this: the three-dimensional spacetime hypersurface corresponding to t = t₁ intersects B. Claim (10) means that B exists simpliciter, somewhere in spacetime (assuming it’s a spatiotemporal object). There is no contradiction in saying that the actualization doesn’t exist at t₁, even though it is true at t₁ that it exists simpliciter.

The second answer is that Aristotelianism does not need actualizations of unactualized potentials. Causation is the actualization of a potential. But Aristotle and Aquinas both believed in the possibility of simultaneous causation. In simultaneous causation, an event B is the actualization of a simultaneous potential A. At the time of the simultaneous causation, nobody, whether presentist or eternalist, can say that B is the actualization of unactualized potential, since then the potential would be actualized and unactualized at the same time. Thus, one can have causation, and actualization of potential, where the potential and the actualization are simultaneously real, and hence where the actualization is not of an unactualized potential. The eternalist could—but does not have to—say that transtemporal cases are like this, too: they are actualizations of a potential, but not of an unactualized potential.

Final and efficient causation

It is sometimes said that:

1. One can have p explain q and q explain p when the types of explanation are different.

I think (1) is mistaken, but in this post I want to focus not on arguing against (1), but simply on arguing against one particular and fairly common form of argument for (1):

2. In cases of Aristotelian final causation, it typically happens that y is a final cause of its own efficient cause.

3. If y is a final cause of x, then that y occurred finally explains that x occurred.

4. If x is an efficient cause of y, then that x occurred efficiently explains that y occurred.

5. So, it’s possible to have p explain q and q explain p when the types of explanation are final and efficient, respectively.

I want to argue that this argument fails (bracketing the interpretive question whether Aristotle or Aquinas accepts its premises).

First, explanation is factive: if p explains q, then both p and q are true. This is because explanations provide correct answers to why questions, and a false answer isn’t correct. But final explanations are not factive. I can offer an argument in order to convince you and yet fail to convince you. (Indeed, perhaps this post is an example.) Therefore, (3) is not always true. That doesn’t show that (3) is false in the case that the argument needs. But it is plausible that an action that fails for extrinsic reasons has exactly the same explanation as a successful action. The failed action cannot be explained by its achieving its goal, since it doesn’t achieve its goal. Therefore, the successful action cannot be explained in terms of its achieving its goal, either.

Second, efficient causation is a relation between tokens. If I turn on the lights in order to alert the burglars, then my token turning-on-the-lights is the efficient cause of the token alerting-the-burglars. But final causation is not a relation between tokens. For suppose that I fail to alert the burglars, say because the burglars are blindfolded (they were challenged to rob me blind, and parsed that phrase wrong) and don’t see the lights. Then there are infinitely many possible tokens of the alerting-the-burglars type any one of which would pretty much equally well serve my goals. For instance, I could alert the burglars at 10:44:22.001, at 10:44:22.002, etc. In the case of action failure, no one of these tokens can be distinguished as “the final cause”, the token I am aiming at. Indeed, if one particular possible token a₀ were the final cause, then if I happened to produce another token, say a₇, my action would have been a failure—which is absurd. Thus, either all the infinitely many possible tokens serve as the final causes of the action or none of them do. It seems wrong to say that there are infinitely many final causes of the action, so none of the tokens is.

Given that explanation of the failed action is the same as of the successful action, it follows that even in the successful case, none of the tokens provides the final cause.

Therefore, we should see final causation as a relation between a type, say alerting the burglars at some time or other near 10:44:22, and a token, say my particular turning on of the lights. But if so, then (2) is false, for it is false that in the successful case the same things are related by final and efficient causation: the final causation relates the outcome type with a productive token and efficient causation relates the productive token with the outcome token.

As I said, this doesn’t show that (1) is false, but it does show that efficient and final explanation do not provide a case of (1).

Acknowledgments: I am grateful to Tim Pawl for discussion of these questions.

Truly going beyond the binary in marriage?

There is an interesting sense in which standard polygamy (i.e., polygyny or polyandry) presupposes the binarity of marriage. In standard polygamy, there is one individual, A, who stands in a marriage relationship to each of a plurality of other individuals, the Bs. But the marriage relationships themselves are binary: A is married to each of the Bs, and the Bs are not married to each other (they have a different kind of relationship).

The same would be true with more complex graph theoretic structures than the simple star-shaped structure of polygyny and polyandry (with A at the center and the Bs at the periphery). If Alice is married to Bob and Carl, and Bob and Carl are each married to Davita, the quadrilateral graph-theoretic structure of this relationship is still constituted by a four binary marriage relationships.

Thus, in these kinds of cases, what we would have are not a plural marriage, but a plurality of binary marriages with overlap. This, I think, makes for more precise terminology. The moral and political questions normally considered under the head of “plural marriages” are about the possibility or morality of overlap between binary marriages.

To truly go beyond the binary would require a relationship that irreducibly contains more than two people, a relationship not constituted by pairwise relationships. I think a pretty good case can be made that even if one accepts overlapping binary marriages, as in standard polygamy, as genuine marriages (I am not sure one should), irreducibly non-binary relationships would still not be marriages (just as unary relationships wouldn't be). The structure of the relationship is just radically different.

God and analogy

According to Aquinas, whenever we correctly say something non-negative of God, we speak analogically.

It is correct to say that Socrates is wise and God is wise. But being humanly wise and divinely wise are different—the most fundamental difference being that, by divine simplicity, God doesn’t have his wisdom, but is his wisdom. But this leads to:

1. The predicate “is humanly wise or is divinely wise” applies literally to both Socrates and God.

And yet this disjunctive predicate is not negative, so (1) seems to provide a counterexample to Aquinas’ theory.

But this is too fast. Claim (1) only provides a counterexample to Aquinas’ theory if:

2. Applying analogically and applying literally are incompatible.

But I think Aquinas can, and should, say that (2) is false. If he does that, then he can affirm both (1) and:

3. The predicate “is humanly wise or is divinely wise” applies analogically to both Socrates and God.

In fact, I think Aquinas can say that the relevant kind of analogical application of predicates is a special case of literal predication.

I think that Aquinas is not really making a claim about literal and non-literal use of words when he is talking of analogical predication. Instead, I think he is making a claim about grounding, somewhat like:

4. The predicate “is F” is used analogically between entities x and y just in case the propositions that x is F and that y is F have a relevantly different grounding structure.

On this account, disjunctive predicates like “is a human or a dog” are used analogically: for the grounding structure of the proposition that Alice is a human or a dog is that it’s grounded in Alice being a human, while the grounding structure of the proposition that Fido is a human or a dog is that it’s grounded in Fido being a dog. And similarly, “is humanly wise or is divinely wise” is used analogically, since in the case of Socrates the grounds of applicability are Socrates having wisdom and in the case of God the grounds are God being (his) wisdom.

Notice that on this story, Aquinas’ claim about analogical predication is not so much a linguistic claim as a metaphysical claim about the truth grounds.

The story makes clear why negative predicates are not used analogically: for the grounding structure of the truth that God is not a bicycle and the truth that Alice is not a bicycle is relevantly the same—both are grounded in not being arranged bicycle-wise.

So far, our reconstruction of Aquinas’ theory of predication is:

5. A predicate that applies to God is negative or is used analogically.

But that’s not quite right. Here is one counterexample: “is not a bicycle or is both a bicycle and a non-bicycle.” This predicate is not negative but disjunctive. But it applies to God and to Socrates in the same way—by both not being bicycles.

I think the issue here is this. Just as analogical predication is a metaphysical and not linguistic notion, so negative predication is a metaphysical and not linguistic notion. We might say something like this:

6. The predicate “is F” is used negatively of entity x just in case what grounds x being F is the non-obtaining of some state of affairs.

Thus, “is not a bicycle or is both a bicycle and non-bicycle” is used negatively of both God and Socrates, because what grounds its application in both cases are respectively the non-obtaining of the states of affairs of God being arranged bicycle-wise and of Socrates beng arranged bicycle-wise. On the other hand, the disjunctive predicate “is Athenian or not Greek” is used negatively of God and non-negatively of Socrates. Interestingly, this case shows that the disjunction in (5) is not exclusive. For “is Athenian or not Greek” is used both negatively of God and is used analogically between Socrates and God, since the structure of the grounds of application is relevantly different.

The problems haven’t all gone away. A necessary condition for “is F” to be used analogically of God and a creature is that “is F” applies to God and a creature, and hence a predicate that applies only to God cannot be used analogically. But suppose that in fact no one other than God knows whether the Continuum Hypothesis is true. Then the predicate “knows whether the Continuum Hypothesis is true” is not used analogically, since it only applies to God. But then we have a counterexample to (5).

We could try to modalize (4): a predicate is used analogically provided that it could have one ground as applied to God and another as applied to something other than God. But, again, it’s not hard to come up with a counterexample: “knows that 2 + 2 and is not a creature.” For that predicate can only apply to God.

We could also weaken (5) to merely apply to those predicates that apply (or could apply) to both God and a creature. This may seem to be an undue weakening: now one can escape from Aquinas’ doctrine of analogical predication simply by saying things that only apply (or could only apply) to God. But perhaps one can supplement the weakened (5) with:

7. Any predicate that applies to God is built out of predicates that apply both to God and to a creature.

I am not too happy about this.

Gunk, etc.

If we think parts are explanatorily prior to wholes, then gunky objects—objects which have parts but no smallest parts—involve a vicious explanatory regress. But if one takes the Aristotelian view that wholes are prior to parts, then the regress involved in gunky objects doesn’t look vicious at all: the whole is prior to some parts, these parts are prior to others, and so on ad infinitum. It’s just like a forward causal regress: today’s state causes tomorrow, tomorrow’s causes the next day’s, and so on ad infinitum.

On the other hand, on the view that parts are explanatorily prior to wholes, upward compositional regresses are unproblematic: the head is a part of the cow, the cow is a part of the earth, the earth is a part of the solar system, the solar system is a part of the Orion arm, the Orion arm is a part of the Milky Way, the Milky Way is a part of the Local Group, and this could go on forever. The Aristotelian, on the other hand, has to halt upward regresses at substances, say, cows.

This suggests that nobody should accept an ontologically serious version of the Leibniz story on which composition goes infinitely far both downward and upward, and that it is fortunate that Leibniz doesn’t accept an ontologically serious version of that story, because only the monads and their inner states are to be taken ontologically seriously. But that's not quite right. For there is a third view, namely that parthood does not involve either direction of dependence: neither do parts depend on wholes nor do wholes depend on parts. I haven't met this view in practice, though.

Leibniz on infinite downward complexity

Leibniz famously thinks that ordinary material objects like trees and cats have parts, and these parts have parts, and so on ad infinitum. But he also thinks this is all made up of monads. Here is a tempting mental picture to have of this:

• Monads, …, submicroscopic parts, microscopic parts, macroscopic parts, ordinary objects.

with the “…” indicating infinitely many steps.

This is not Leibniz’s picture. The quickest way to see that it’s not is that organic objects at each level immediately have primary governing monads. There isn’t an infinite sequence of steps between the cat and the cat’s primary monad. The cat’s primary monad is just that, the cat’s primary monad. The cat is made up of, say, cells. Each cell has a primary monad. Again, there isn’t an infinite sequence of steps between the cat and the primary monads of the cells: there might turn out to be just two steps.

In fact, although I haven’t come across texts of Leibniz that speak to this question, I suspect that the best way to take his view is to say that for each monad and each object partly constituted by that monad, the “compositional distance” between the monad and the object is finite. And there is a good mathematical reason for this: There are no infinite chains with two ends.

If this is right, then the right way to express Leibniz’s infinite depth of complexity idea is not that there is infinite compositional distance between an ordinary object and its monads, but rather than there is no upper bound on the compositional distance between an ordinary object and its monads. For each ordinary object o and each natural number N, there is a monad m which is more than N compositional steps away from o.

Fundamental mereology

It is plausible that genuine relations have to bottom out in fundamental relations. E.g., being a blood relative bottoms out in immediate blood relations, which are parenthood and childhood. It would be very odd indeed to say that a is b’s relative because a is c’s relative and c is b’s relative, and then a is c’s relative because a is d’s relative and d is c’s relative, and so on ad infinitum. Similarly, as I argued in my infinity book, following Rob Koons, causation has to bottom out in immediate causation.

If this is right, then proper parthood has to bottom out in what one might call immediate parthood. And this leads to an interesting question that has, to my knowledge, not been explored much: What is the immediate parthood structure of objects?

For instance, plausibly, the big toe is a part of the body because the big toe is a part of the foot which, in turn, is a part of the body. And the foot is a part of the body because the foot is a part of the leg which, in turn, is a part of the body. But where does it stop? What are the immediate parts of the body? The head, torso and the four limbs? Or perhaps the immediate parts are the skeletal system, the muscular system, the nervous system, the lymphatic system, and so on. If we take the body as a complex whole ontologically seriously, and we think that proper parthood bottoms out in immediate parthood, then there have to be answers to such questions. And similarly, there will then be the question of what the immediate parts of the head or the nervous system are.

There is another, more reductionistic, way of thinking about parthood. The above came from the thought that parthood is generated transitively out of immediate parthood. But maybe there is a more complex grounding structure. Maybe particles are immediately parts of the body and immediately parts of the big toe. And then, say, a big toe is a part of the body not because it is a part of a bigger whole which is more immediately a part of the body, but rather a big toe is a part of the body because its immediate parts are all particles that are immediately parts of the body.

Prescinding from the view that relations need to bottom out somewhere, we should distinguish between fundamental parts and fundamental instances of parthood. One might have one without the other. Thus, one could have a story on which we are composed of immediate parts, which are composed of immediate parts, and so on ad infinitum. Then there would be fundamental instances of the parthoood relation—they obtain between a thing and its immediate parts—but no fundamental parts. Or one could have a view with fundamental parts while denying that there are any fundamental instances of parthood.

In any case, there is clearly a lot of room for research in fundamental mereology here.

Taste and cross-cultural encounters

After visiting the British Museum yesterday, I find it rather hard to take seriously the argument for the relativity of beauty from the diversity of taste. It seems clear that just as C. S. Lewis has argued for a moral core cutting across cultures, one can argue that there is an aesthetic core across cultures.

There is, however, an interesting apparent difference between the diversity of taste and the diversity of morals. I think a cross-cultural encounter involving a difference of taste regarding the best cultural artifacts—by each culture’s own standards—should typically lead to a broadening of taste. But a cross-cultural moral encounter should not typically lead to a broadening of morals. Very often, it should lead to a narrowing of morals: for instance, one culture learning from the other that slavery sex is wrong.

Why this difference? I think it may come from a difference in quantifiers.

As Aquinas already noted (in a somewhat different way), to be morally good, an action has to be good or neutral with respect to every relevant dimension of moral evaluation. If it is good with respect to courage and kindness and generosity, but it is bad with respect to justice (Robin Hood?), then the action is plain wrong. Thus as new dimensions of moral evaluation are discovered, as can happen in cross-cultural encounter, we get a narrowing of the actions that we classify as morally good.

On the other hand, for an item to be beautiful, it only needs to be beautiful with respect to some relevant dimensions of beauty. A musical performance is still beautiful on the whole even if the orchestra is dressed in dirty rags, and a painting can be beautiful even if it reeks of oil. Thus as we discover new dimensions of beauty, we get a broadening of the pieces that we classify as beautiful.

A way forward on the normalizability problem for the Fine-Tuning Argument

The Fine-Tuning Argument claims that the life-permitting ranges of various parameters are so narrow that, absent theism, we should be surprised that the parameters fall into those ranges.

The normalizability objection is that if a parameter ξ can take any real value, then any finite life-permitting range of values of ξ counts as a “narrow range”, since every finite range is an infinitesimal portion of the full range from −∞ to ∞. Another way to put the problem is that there is no uniform probability distribution on the set of real numbers.

There is, however, a natural probability distribution on the set of real numbers that makes sense as a prior probability distribution. It is related to the Solomonoff priors, but rather different.

Start with a language L with a finite symbol set usable for describing mathematical objects. Proceed as follows. Randomly generate finite strings of symbols in L (say, by picking independently and uniformly randomly from the set of symbols in L plus an “end of string” symbol until you generate an end of string symbol). Conditionalize on the string constituting a unique description of a probability measure on the Lebesgue measurable subsets of the real numbers. If you do get a unique description of a probability measure, then choose a real number according to this distribution.

The result is a very natural probability measure P_L (a countable weighted sum of probability measures on the same σ-algebra with weights adding to unity is a probability measure) on the Lebesgue measurable subsets of the real numbers.

We can now in principle evaluate the fine-tuning argument using this measure.

The problem is that this measure is hard to work with.

Note that using this measure, it is false that all narrow ranges have very small probability. For instance, consider the intuitively extremely narrow range from 10¹⁰⁰⁰ to 10¹⁰⁰⁰. Supposing that the language is a fairly standard mathematical language for describing probability distributions, we can specify a uniform distribution on the 0-length interval from 10¹⁰⁰⁰ to 10¹⁰⁰⁰ as U[10¹⁰⁰⁰, 10¹⁰⁰⁰], which is 23 characters of LaTeX, plus an end of string. Using 95 ASCII characters, plus the end of string character, P_L of this interval will be at least 96⁻²⁴ or something like 10⁻⁴⁸. Yet the size of the range is zero. In other words, intuitively narrow ranges around easily describable numbers, like 10¹⁰⁰⁰, get disproportionately high probability.

But that is how it should be, as we learn from the fact that the exponent 2 in Newton’s law of gravitation had better have a non-zero prior, even though the interval from 2 to 2 has zero length.

Whether the Fine-Tuning Argument works with P_L for a reasonable choice of L and for a particular life-permitting range of ξ is thus a hard question. But in any case, for a fixed language L where we can define a map between strings and distributions, we can now make perfectly rigorous sense of the probability of a particular range of possibilities for ξ. We have replaced a conceptual difficulty with a mathematical one. That’s progress.

Further, now that we see that there can be a reasonable fairly canonical probability on infinite sets, the intuitive answer to the normalizability problem—namely, “this range seems really narrow”—could constitute a reasonable judgment as to what answer would be returned by one’s own reasonable priors, even if these are not the same as the probabilities given above.

Oh, and this probability measure solves the tweaked problem of regularity, because it assigns non-zero probability to every describable event. I think this is even better than my modified Solomonoff distribution.

Improving on Solomonoff priors

Let’s say that we want prior probabilities for data that can be encoded as a countably infinite binary sequence. Generalized Solomonoff priors work as follows: We have a language L (in the original setting, it’ll be based on Turing machines) and we generate 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 an infinite binary sequence). Typically the set of possible descriptions in L is countable and we get a nice well-defined probability measure on the space of all countably infinite binary sequences, which favors those sequences that are simpler in the sense of being capable of a simpler encoding.

Here is a serious problem with this method. Let N be the set of all binary sequences that cannot be uniquely described in L. Then the method assigns prior probability zero to N, even though most sequences are in N. In particular, this means that if we get an L-indescribable sequence—and most sequences generated by independent coin tosses will be like that—then no matter how much of it we observe, we will be almost sure of the false claim that the sequence is L-describable.

Here, I think, is a better solution. Use a language L that can give descriptions of subsets of the space Ω of countably infinite binary sequences. Now our (finitely additive) priors will be generated as follows. Choose a random string of symbols in L and conditionalize on the string giving a unique description of a subset. If the subset S happens to be measurable with respect to the standard (essentially Lebesgue) measure on infinite binary sequences (i.e., the coin toss measure), then randomly choose a point in S using a finitely additive extension of the standard measure to all subsets of S. If the subset S is not measurable, then randomly choose a point in S using any finitely additive measure that assigns probability zero to all singletons.

For a reasonable language L, the resulting measure gives a significant probability to an unknown binary sequence being indescribable. For Ω itself will typically be easily described, and so there will be a significant probability p that our random description of a subset will in fact describe all of Ω, and the probability that we have an indescribable sequence will be at least p.

It wouldn’t surprise me if this is in the literature.

On a twist on too-many-thinkers arguments

One of the ways to clinch a too-many-thinkers argument (say, Merricks’ argument against perdurantism, or Olson’s argument for animalism) is to say that the view results in an odd sceptical worry: one doesn’t know which of the many thinkers one is. For instance, if both the animal and the person think, how can you know that you are the animal and not the person: it seems you should have credence 1/2 in each.

I like too-many-thinkers arguments. But I’ve been worried about this response to the sceptical clinching: When the animal and the person think words like “I am a person”, the word “I” refers to the person, even when used by the animal, and hence both think the truth. In other words, “I” means something like: the person colocated with the the thinker/speaker.

But I think I have a good response to this response. It would be a weird limitation on our language if it did not allow speaker or thinker self-reference. Even if in fact “I” means the person colocated with the the thinker/speaker, we should be able to stipulate another pronoun, “I*”, one that refers just to the thinker/speaker. And it would be absurd to think that one not be able to justifiably assert “I* am a person.”

Functionalism and maximalism

It is widely held that consciousness is a maximal property—a property F such that, “roughly, … large parts of an F are not themselves F.” Naturalists have used maximality, for instance, to respond to Merricks’ worry that on naturalism, if Alice is conscious, so is Alice minus a finger, as they both have a brain sufficient for consciousness (see previous link). There are also the sceptical consequences, noted by Merricks, arising from thinking our temporal parts to be consciousness.

But functionalists cannot hold to maximalism. For imagine a variant on the Chinese room experiment where the bored clerk processes Chinese characters with the essential help of exactly one stylus and one wax tablet. The functionalist is committed to the clerk plus the stylus and tablet—call that clerk-plus—being conscious, as long as the stylus and tablet are essential to the functioning of the system. But if the clerk-plus is conscious, the clerk is not by maximalism. For consciousness is a maximal property, and the clerk is a large part of the clerk-plus. But it is absurd to think that the clerk turns into a zombie as soon as he starts to process Chinese characters.

Perhaps, though, instead of consciousness being maximal, the functionalist maximalist can say that maximally specific phenomenal types of consciousness—say, feeling such and such a sort of boredom B—are maximal. The clerk feels B, but clerk-plus is, say, riveted by reading the Romance of the Three Kingdoms. There is no violation of maximality with respect to the clerk’s feeling bored, because clerk-plus isn’t bored.

That could be the case. But it could also so happen that at some moment clerk-plus feels B as well. After all, the same feeling of boredom can be induced by different things. The Romance has slow bits. It could happen that clerk-plus is stuck in a slow bit, and for a moment clerk and clerk-plus lose sight of the details and are aware of nothing but their boredom—the qualitatively same boredom. And that violates maximality for specific types of consciousness.

If maximalism is needed for a naturalist theory of mind and if functionalism is our best naturalist theory of mind, then the best naturalist theory fails.