Restricted composition and laws of nature

Ted Sider famously argues for the universality of composition on the grounds that:

1. If composition is not universal, then one can find a continuous series of cases from a case of no composition to a case of composition.

2. Given such a continuous series, there won’t be any abrupt cut-off in composition.

3. But composition is never vague, so there would have to be an abrupt cut-off.

Consider this argument that every velocity is an escape velocity:

4. If it’s not the case that every velocity is an escape velocity from a spherically symmetric body of some fixed size and mass, then one can find a continuous series of cases from a case of insufficiency to escape to a case of sufficiency to escape.

5. Given such a continuous series, there won’t be any abrupt cut-off in escape velocity.

6. But escape velocity is never vague, so there would have to be an abrupt cut-off.

It’s obvious that we should deny (5). There is an abrupt cut-off in escape velocity, and there is a precise formula for what it is: (2GM/r)^1/2 where G is the gravitational constant, M is the mass of the spherical body, and r is its radius. As the velocity of a projectile gets closer and closer to the (2GM/r)^1/2, the projectile goes further and further before turning back. When the velocity reaches (2GM/r)^1/2, the projectile goes out forever. There is no paradox here.

Why think that composition is different from escape velocity? Why not think that just as the laws of nature precisely specify when the projectile can escape gravity, they also precisely specify when a bunch of objects compose a whole?

My suspicion is that the reason for thinking the two are different is thinking that composition is something like a “logical” or maybe “metaphysical” matter, while escape is a “causal” matter. Now, universalists like David Lewis do tend to think that the whole is a free lunch, nothing but the “sum of the parts”, in which case it makes sense to think that composition is not something for the laws of nature to specify. But if we are not universalists, then it seems to me that it is very natural to think of composition in a causal way: when the xs are arranged a certain way, they cause the existence of a new entity y that stands in a composed-by relation to the xs, just as when a projectile has a certain velocity, that causes the projectile to escape to infinity.

Some may be bothered by the fact that laws of nature are often taken to be contingent, and so there would be a world with the same parts as ours but different wholes. That would bother one if one thinks that wholes are a free lunch. But if we take wholes seriously, it should no more bother us than a world where particles behave the same way up to time t₁, and then behave differently after t₁ because the laws are different.

David Lewis probably has a good reason to reject the above view, though. If the laws of composition are to match our intuitions about composition, they are likely to be extremely complex, and perhaps too complex to be part of the best system defining the laws on a Humean account of laws. But if we are not Humeans about laws, and think the simplicity of laws is merely an epistemic virtue, the explanatory power of laws of composition might make it reasonable to accept very complex such laws.

(I don’t endorse the view described above. I prefer, but still do not endorse, an Aristotelian alternative: when y is in a certain condition, it causes the existence of xs related to it in a composing way.)

More on full conditional probabilities and comparative probabilities

I claim that there is no general, straightforward and satisfactory way to define a total comparative probability with the standard axioms using full conditional probabilities. By a “straightforward” way, I mean something like:

1. A ≲ B iff P(A−B|AΔB) ≤ P(B−A|AΔB) (De Finetti)

or:

2. A ≲ B iff P(A|A∪B) ≤ P(B|A∪B) (Pruss).

The standard axioms of comparative probability are:

3. Transitivity, reflexivity and totality.

4. Non-negativity: ⌀ ≤ A for all A

5. Additivity: If A ∪ B is disjoint from C, then A ≲ B iff A ∪ C ≲ B ∪ C.

A “straightforward” definition is one where the right-hand-side is some expression involving conditional probabilities of events definable in a boolean way in terms of A and B.

To be “satisfactory”, I mean that it satisfies some plausible assumptions, and the one that I will specifically want is:

6. If P(A|C) < P(B|C) where A ∪ B ⊆ C, then A < B.

Definitions (1) and (2) are straightforward and satisfactory in the above-defined senses, but (1) does not satisfy transitivity while (2) does not satisfy the right-to-left direction of additivity.

Here is a proof of my claim. If the definition is straightforward, then if A ≲ B, and A′ and B′ are events such that there is a boolean algebra isomorphism ψ from the algebra of events generated by A and B to the algebra of events generated by A′ and B′ such that ψ(A) = A′, ψ(B) = B′ and P(C|D) = P(ψ(C)|ψ(D)) for all C and D in the algebra generated by A and B, then A′ ≲ B′.

Now consider a full conditional probability P on the interval [0,1] such that P(A|[0,1]) is equal to the Lebesgue measure of A when A is an interval. Let A = (0,1/4) and suppose B is either (1/4,1/2) or (1/4, 1/2]. Then there is an isomorphism ψ from the algebra generated by A and B to the same algebra that swaps A and B around and preserves all conditional probabilities. For the algebra consists of the eight possible unions of sets taken from among A, B and [0,1] − (A∪B), and it is easy to define a natural map between these eight sets that swaps A and B, and this will preserve all conditional probabilities. It follows from my definition of straightforwardness that we have A ≲ B if and only if we have B ≲ B. Since the totality axiom for comparative probabilities implies that either A ≲ B or B ≲ A, so we must have both A ≲ B and B ≲ A. Thus A ∼ B. Since this is true for both choices of B, we have

7. (0,1/4) ∼ (1/4,1/2) ∼ (1/4, 1/2].

But now note that ⌀ < {1/2} by (3) (just let A = ⌀, B = {1/2} and C = {1/2}). The additivity axiom then implies that (1/4,1/2) < (1/4, 1/2], a contradiction.

I think that if we want to define a probability comparison in terms of conditional probabilities, what we need to do is to weaken the axioms of comparative probabilities. My current best suggestion is to replace Additivity with this pair of axioms:

8. One-Sided Additivity: If A ∪ B is disjoint from C and A ≲ B, then A ∪ C ≲ B ∪ C.

9. Weak Parthood Principle: If A and B are disjoint, then A < A ∪ B or B < A ∪ B.

Definition (2) satisfies the axioms of comparable probabilities with this replacement.

Here is something else going for this. In this paper, I studied the possibility of defining non-classical probabilities (full conditional, hyperreal or comparative) that are invariant under a group G of transformations. Theorem 1 in the paper characterizes when there are full conditional probabilities that are strongly invariant. Interesting, we can now extend Theorem 1 to include this additional clause:

6. There is a transitive, reflexive and total relation ≲ satisfying (4), (8) and (9) as well as the regularity assumption that ⌀ < A whenever A is non-empty and that is invariant under G in the sense that gA ∼ A whenever both A and gA are subsets of Ω.

To see this, note that if there is are strongly invariant full conditional probabilities, then (2) will define ≲ in a way that satisfies (vi). For the converse, suppose (vi) is true. We show that condition (ii) of the original theorem is true, namely that there is no nonempty paradoxical subset. For to obtain a contradiction suppose there is a non-empty paradoxical subset E. Then E can be written as the disjoint union of A₁, ..., A_n, and there are g₁, ..., g_n in G and 1 ≤ m < n such that g₁A₁, ..., g_mA_m and g_m + 1A_m + 1, ..., g_nA_n are each a partition of E.

A standard result for additive comparative probabilities in Krantz et al.’s measurement book is that if B₁, ..., B_n are disjoint, and C₁, ..., C_n are disjoint, with B_i ≲ C_i for all i, then B₁ ∪ ... ∪ B_n ≲ C₁ ∪ ... ∪ C_n. One can check that the proof only uses One-Sided Additivity, so it holds in our case. It follows from G-invariance that A₁ ∪ ... ∪ A_m ∼ E ∼ A_m + 1 ∪ ... ∪ A_n. Since E is the disjoint union of A₁ ∪ ... ∪ A_m with  A_m + 1 ∪ ... ∪ A_n, this violates the Weak Parthood Principle.

The epistemic force of beauty in laws of nature does not reduce to simplicity

Some people think that simplicity of laws of nature is a guide to truth, and some think beauty of laws of nature is. One might ask: Is the beauty of laws of nature a guide that goes beyond simplicity? Are there times when one could make epistemic decisions about the laws of nature on the basis of beauty where simplicity wouldn’t do the job?

I think so. Here is one case. Suppose we live in a Newtonian universe, and we are discovering fundamental forces. The first one has an inverse cube law. The second has an inverse cube law. These two laws account for most phenomena, but a few don’t fit. Scientists think there is a third fundamental force. For the third force law, we have two proposals that fit the data: an inverse square law and a slightly more complicated inverse cube law. It is, I think, quite reasonable to go for an inverse cube law by induction over the laws.

There is something indeed beautiful about the idea that the same power law applies to all the forces of nature. But if we just go with simplicity, we will go for an inverse square law. However, going for the inverse cube law seems clearly reasonable, and it is what beauty suggests—but not simplicity.

Here is another thought. Sometimes a fundamental law has some particularly lovely mathematical implications. For instance, a conservative force law is connected in a lovely way with a potential. But it need not be the case that a conservative force law is simpler than a non-conservative alternative. (It is true that a conservative force is the gradient of a potential. If the potential can be particularly simply expressed, this makes it easier to express the conservative force law. But we can have a case where the potential is harder to express than the force itself.)

The de Finetti definition of comparative probabilities in terms of conditional probabilities

Suppose we have a full conditional probability P(A∣B) defined for all pairs of events (stipulating that P(A∣⌀) = 1 if we wish). Two methods have been proposed for defining a probability comparison using conditional probabilities:

1. Pruss: A ⪅ B iff P(A∣A∪B) ≤ P(B∣A∪B).

2. De Finetti: A ⪅ B iff P(A−B∣AΔB) ≤ P(B−A∣AΔB), where AΔB = (A−B) ∪ (B−A) is the symmetric difference.

In a footnote in a paper, I wrote about the de Finetti ordering: “This ordering has the advantage that if A is a proper subset of B, then A < B, but it is somewhat harder to prove transitivity”.

Well, that was an understatement! It’s not just harder to prove transitivity: it’s impossible.

Define:

• Ω: all integers

• E₀: non-negative even integers

• E: positive even integers

• D: positive odd integers.

Let P be a full conditional probability such that:

3. P(E₀∣E₀∪D) = 1/2 = P(D∣E₀∪D)

4. P(D∣D∪E) = 1/2 = P(E|D∪E).

It is easy to see from (3) and (4) that because E₀ and D are disjoint, and so are D and E, then by the de Finetti definition (2) we have E₀ ⪅ D and D ⪅ E. (For disjoint A and B, the de Finetti definition of A ⪅ B is equivalent to the Pruss definition.) However, E₀ − E = {0}, E − E₀ = ⌀, and E₀ΔE = {0}, so P(E−E₀∣E₀ΔE) = 0 while P(E₀−E∣E₀ΔE) = 1, and thus we cannot have E₀ ⪅ E.

The only question is whether there actually is a full conditional probability satisfying (3) and (4). If there is, then the de Finetti comparison is not transitive in our case.

There is such a full conditional probability. Let Q_n(A) =  ∣ A ∩ [−n,n] ∣ /(2n+1), where $B$ is the cardinality of a set B. Then Q_n is a probability. Let Q be a limit of the Q_n along an ultrafilter. This is a finitely additive hyperreal probability which is non-zero for all non-empty sets. Define P(A∣B) as the standard part of Q(A∩B)/Q(B) for B non-empty. This is a full conditional probability. Moreover, P(A∣B) = lim Q_n(A∩B)/Q_n(B) whenever the latter limit is defined. That limit is defined in the cases of the events involved in (3) and (4), and it is easy to evaluate the limits and see that (3) and (4) are true.

I don’t know what I was thinking when I wrote that footnote. My guess is that I had in my mind a proof sketch that doesn’t work (I have some idea what that might have been).

Whew! I noticed this afternoon that Theorem 1 of this paper of mine was incompatible with the transitivity of the de Finetti comparison (2). This made me really worried that my Theorem 1 was false. But since the de Finetti comparison isn’t transitive, I can relax.

This raises an interesting potential research problem. The Pruss definition of ⪅ does not satisfy the additivity axiom for comparative probabilities, namely that if C is disjoint from A ∪ B, then A ≲ B if and only if A ∪ C ≲ B ∪ C (it only preserves the left-to-right implication). The de Finetti definition does satisfy the additivity axiom, which is what I liked about it.

I suspect there is no good definition of comparative probabilities in terms of full conditional probabilities that satisfies the additivity axiom. (One reason for this intuition has to do with the fact that in Figure 1 here there are entries with a Yes in column 3 and a No in column 5.)

So, I now wonder: Is there some good combination of a definition of comparative probabilities in terms of full conditional probabilities with some weakened version of the additivity axiom?

A really bad moral dilemma

Here would be a really bad kind of moral dilemma:

• It is certain that unless you murder one innocent person now, you will freely become a mass murderer, but if you do murder that innocent person, you will freely repent of it later and live an exemplary life.

If compatibilism is true, such dilemmas are possible—the world could be so set up that these unfortunate free choices are inevitable. If compatibilism is false, such dilemmas are impossible, absent Molinism.

We might have a strong intuition that such dilemmas are impossible. If so, maybe that gives us another reason to reject compatibilism and Molinism.

Proportionality and deterrence

There are many contexts where a necessary condition of the permissibility of a course of action is a kind of proportionality between the goods and bads resulting from the course of action. (If utilitarianism is true, then given a utilitarian understanding of the proportionality, it’s not only necessary but sufficient for permissibility.) Two examples:

• The Principle of Double Effect says it is permissible to do things that are foreseen to have a basic evil as an effect, if that evil is not intended, and if proportionality between the evil effect and the good effects holds.

• The conditions for entry into a just war typically include both a justice condition and a proportionality condition (sometimes split into two conditions, one about likely consequences of the war and the other about the probability of victory).

But here is an interesting and difficult kind of scenario. Before giving a general formulation, consider the example that made me think about this. Country A has a bellicose neighbor B. However, B’s regime while bellicose is not sufficiently evil that on a straightforward reading of proportionality it would be worthwhile for A to fight back if invaded. Sure, one would lose sovereignty by not fighting back, but B’s track record suggests that the individual citizens of A would maintain the freedoms that matter most (maybe this is what it would be like to be taken over by Alexander the Great or Napoleon—I don’t know enough of history to know), while a war would obviously be very bloody. However, suppose that a policy of not fighting back would likely result in an instant invasion, while a policy of fighting back would have a high probability of resulting in peace for the foreseeable future. We can then imagine that the benefits of likely avoiding even a non-violent takeover by B outweigh the small risk that despite A’s having a policy of armed resistance B would still invade.

The general case is this: We have a policy that is likely to prevent an unhappy situation, but following through on the policy violates a straightforward reading of proportionality if the unhappy situation eventuates.

One solution is to take into account the value of follwing through on the policy with respect to one’s credibility in the future. But in some cases this will be a doubtful justification. Consider a policy of fighting back against an invader—at least initially—even if there is no chance of victory. There are surely many cases of bellicose countries that could successfully take over a neighbor, but judge that the costs of doing so are too high given the expected resistance. But if the neighbor has such a policy, then in case the invasion nonetheless eventuates, whatever is done, sovereignty will be lost, and the policy will be irrelevant in the future. (One might have some speculation about the benefits for other countries of following through on the policy, but that’s very speculative.)

One line of thought on these kinds of cases is that we need to forego such policies, despite their benefits. One can’t permissibly act on them, so one can’t have them, and that’s that. This is unsatisfying, but I think there is a serious chance that this is right.

One might think that the best of both worlds is to make it seem like one has the policy, but not in fact have it. A problem with this is that it might involve lying, and I think lying is wrong. But even aside from that, in some cases this may not be practicable. Imagine training an army to defend one’s country, and then having a secret plan, known only to a very small number of top commanders, that one will surrender at the first moment of an invasion. Can one really count on that surrender? The deterrent policy is more effective the fiercer and more patriotic the army, but those factors are precisely likely to make them fight despite the surrender at the top.

Another move is this. Perhaps proportionality itself takes into account not just the straightforward computation of costs and benefits, but also the value of remaining steadfast in reasonably adopted policies. I find this somewhat attractive, but this approach has to have limits, and I don’t know where to draw them. Suppose one has invented a weapon which will kill every human being in enemy territory. Use of this weapon, with a Double Effect style intention of killing only the enemy soldiers, is clearly unjustified no matter what policies one might have, but a policy to use this weapon might be a nearly perfect protection against invasion. (Obviously this connects with the question of nuclear deterrence.) I suppose what one needs to say is that the importance of steadfastness in policies affects how proportionality evaluation go, but should not be decisive.

I find myself pulled to the strict view that policies we should not have policies acting on which would violate a straightforward reading of proportionality, and the view that we should abandon the straightforward reading of proportionality and take into account—to a degree that is difficult to weigh—the value of following policies.

Another argument on the interpretation of Matthew 5:32 and 19:9

Mark (10:11-12) and Luke (16:18) have rather simple and straightforward statements on divorce and remarriage: if you divorce and remarry, you’re in adultery. A standard interpretation is the Strict View:

• (SV) Divorce does not actually remove the marriage, and so if you remarry, you’re still married to the previous party, and hence are committing adultery.

It’s usual in the Christian tradition to restrict this to consummated Christian marriage, and I will take that for granted.

However, Matthew has a more complex set of prohibitions:

• Matthew 5:32: Anyone who divorces his wife, except on account of porneia, makes her commit adultery, and anyone who marries a divorced woman commits adultery.

• Matthew 19:9: Anyone who divorces his wife, except due to porneia, and marries another commits adultery.

There are several puzzles here. First, unlike in Mark and Luke, we have exceptions for porneia, a generic term for sexual immorality. There are two main interpretations of this exception:

1. Except when the wife has committed sexual immorality (most commonly, adultery).

2. Except when the “marriage” constitutes sexual immorality.

Reading (1) supports the Less Strict View:

• (LSV) Except when your spouse has committed adultery, divorce does not actually remove the marriage, and so if you remarry, you’re still married to the previous party, and hence are committing adultery.

Reading (2) is based on the observation that not every legal marriage is genuinely a marriage: the Romans, for instance, might have allowed a couple to marry despite their being too closely related from the Christian point of view. In such a case, their “marriage” is not a real marriage but incest, a form of sexual immorality, and divorce is not only permissible, but a very good idea. Note that on reading (2), we can but need not suppose that Jesus verbally included the exception—the inspired author might have added it for clarification because the issue came up for converts, much as we put things in square brackets within a quote to clarify the author’s meaning (there were no brackets in Greek, of course).

Reading (2) has the advantage that it explains how all three Gospels can be inspired, even though Mark and Luke have unqualified statements of SV, since on reading (2) it is true that divorcing one’s wife and remarrying is never permitted, but it is permissible, of course, to divorce one’s partner in an immoral sexual relationship that non-Christian society may call “marriage”. Note that the Greek for “his wife” can literally just mean “his woman”, which makes the disambiguation especially appropriate.

But I want to turn towards a different and more complex argument for SV. Notice that in Matthew 5:32, instead of us being told that the man who divorces his wife (or woman) commits adultery, we are oddly told that he makes her commit adultery. But being a betrayed spouse does not constitute adultery! What’s going on? Well, the good interpretations that I’ve seen note that the social context is a society where it is very difficult to be a woman without a husband. There will thus be significant social pressure to marry or become a concubine, either of which would constitute adultery against the first husband. The realities of the day were such that very likely she would succumb to the pressure, and the first husband would have caused her to commit adultery, and thereby he would have earned himself something worse than a millstone about the neck (Matthew 18:6). This reading also nicely explains why Matthew 5:32, unlike the three other texts, does not mention the man marrying another. For the woman is going to be exposed to the social pressure to join herself to another man whether or not her (first) husband marries another.

Note that this reading of “makes her commit adultery” prima facie works on both readings of the porneia exception. On the reading where the porneia is the wife’s adultery against her husband, obviously if she is already committing adultery, by divorcing her he isn’t making her commit adultery. On the reading where the porneia is constituted by the immorality of the first “marriage”, because the woman wasn’t really married to the man, if she goes and marries another, she isn’t committing adultery.

Nonetheless, there is a serious problem for this reading of “makes her commit adultery” on the Less Strict View and reading (1). While Matthew 5:32 does not talk of the man marrying another, often the man will marry another. So now imagine this story. There is a valid marriage between Alice and Bob with no adultery, but Bob divorces Alice, and marries Charlene. At this point, Bob is committing adultery against Alice on both SV and LSV. Thus, if LSV is correct, then Alice is entitled to divorce Bob and marry another, say Dave. But if she avails herself of this, she isn’t committing adultery. In other words, if LSV is correct, in many cases the first wife will be able to avoid committing adultery without going against social pressures: she need only wait for her first husband to marry, and then the “except on account of porneia” clause on interpretation (1) frees her (and since he’s already legally divorced her, she doesn’t need to do any legal paperwork). (Of course, there will still be less common cases where she is stuck, namely when the man fails to remarry. But such a case wouldn’t be the rule, and Matthew 5:32 implies that leading the woman to adultery is the rule rather than an exception.)

On SV, the problem for the reading of “makes her commit adultery” entirely disappears. Whether or not the man remarries, there is social pressure for the divorced wife to marry, and in doing so, she would be committing adultery against the man.

Interestingly, there is a historically represented view that avoids the Strict View, allows our interpretation of “makes her commit adultery” and avoids the above interpretative problem, namely the quite awful Asymmetric View:

• (AV) A woman is not permitted to remarry after a divorce, whether or not the first husband committed adultery against her, but a man is permitted to remarry after a divorce if, and only if, the first wife committed adultery against him.

Additionally, AV also explains why neither of the texts in Matthew has an exception for porneia in the “anyone who marries a divorced woman” clause, a minor weak point for LSV. (On SV and reading (2) of porneia, we just note that one need not repeat a parenthetical clarification every time.)

In fact, while there was controversy in the early centuries of Christianity over remarriage and divorce following adultery, I understand that it was mainly a controversy between advocates of SV and AV, not between advocates of SV and LSV. However, AV was rightly lambasted by St. Jerome for being sexist, and I assume almost nobody wants to defend it now.

Thus to sum up my argument for SV:

1. One of SV, AV and LSV is true, as they are the historically plausible Christian views on marriage.

2. The right interpretation of “makes her commit adultery” is the social pressure interpretation.

3. This interpretation is incompatible with LSV.

4. AV is false.

5. Therefore, SV is true.

Events and the unreality of time

When I think about McTaggart’s famous argument against the A-theory of time—the theory that it is an objective fact about the universe what time it is—I sometimes feel like it’s just a confusion but sometimes I feel like I am on the very edge of getting it, and that there is something to the argument. When I try to capture the latter feeling in an argument that actually has a chance of being sound, I find it slipping away from me.

So for the nth time in my life, let me try again to make something of McTaggart style arguments. Last night I gave a talk at University of North Texas. When I gave the talk, it was present, and afterwards it became past, and every second that talk is receding another second into the past, becoming more and more past, “older and older” we might say. There is something odd about this, however, since the talk doesn’t exist now. Something that no longer exists can’t change anymore. So how can the talk recede into the further past, how can it become older and older?

Well, we do have a tool for making sense of this. Things that no longer exist can’t really change, but they can have Cambridge change, change relative to something else. Suppose a racehorse is eventually forgotten after its death. The horse isn’t, of course, really changing, but there is real change elsewhere.

More generally, we learn from McTaggart that events can’t really change, but can only change relative to real change in something other than events. The reasoning above shows that events can’t really change in their A-determinations. And they can’t change in their intrinsic non-temporal features, as McTaggart rightly insists: it is eternally true that my talk was about God and mathematics; all the flaws in the talk eternally obtain; etc. So if events can’t really change, but only relatively to real change elsewhere, and yet all of reality is just events, then there is no change.

But reality isn’t just events, and in addition to events changing there is the possibility for enduring entities to change. Here’s perhaps the simplest way to make the story go. The universe is an enduring entity that continually gets older. My talk, then, recedes into the past in virtue of the universe ever becoming older than it was when I gave the talk. (If one is skeptical, as I am, that there is such an entity as the universe, one can give a more complex story about a succession of substances becoming older and older.)

Can one run any version of the McTaggart argument against a theory on which fundamental change consists in a substance’s changing rather than in the change of events? I am not sure, but at the moment I don’t see how. If a person changes from young to old, we have two events: their youth A and their old age B. But we can now say that neither A nor B changes fundamentally: A recedes into the past because of the person’s (or the universe’s) growing old.

If this line of thought is right, then we do learn something from McTaggart: an A-theorist should not locate fundamental change in events, but in enduring objects.

Four philosophy / adjacent jobs at Baylor

We have four jobs in philosophy or closely adjacent areas at Baylor, with most of the deadlines coming in mid-October:

• Tenure-track open-area (with some preferences) position in the  Philosophy Department
• Tenure-track position for a philosopher in the Great Texts Department
• Tenure-track position in social psychology or history and philosophy of science in the Baylor Interdisciplinary Core
• Senior position in bioethics in the Religion Department.

Special treatment of humans

Sometimes one talks of humans as having a higher value than other animals, and hence it being appropriate to treat them better. While humans do have a higher value, I don't think this is what justifies favoring them. For to treat something well is to bestow value on them. But it is far from clear why the fact that x has more value than y justifies bestowing additional value on x rather than on y. It seems at least as reasonable to spread value around, and preferentially treat y.

A confusing factor is that we do have reason to preferentially treat those who have more desert, and desert is a value. But the reason here is specific to desert, and does not in any obvious way generalize to other values.

I don't deny that we should treat humans preferentially over other animals, nor that humans are more valuable. But these two facts should not be confused. Perhaps we should treat humans preferentially over other animals because humans are persons and other animals are not--but this is a point about personhood rather than about value. I am inclined to think we shouldn't argue: humans are persons, personhood is very valuable, so we should treat humans preferentially. Rather, I suspect we should directly argue: humans are persons, so we should treat humans preferentially, skipping the value step. (To put it in Kantian terms, beings with dignity are valuable, but what makes them have dignity isn't just that they are valuable.)

Laws and mathematical complexity

Over the last couple of days I have realized that the laws of physics are rather more complex than they seem. The lovely equations like G = 8πT and F = Gmm′/r² (with a different G in the two equations) seem to be an iceberg most of which is submerged in the icy waters of the foundations of mathematics where the foundational concepts of real analysis and arithmetic are defined in terms of axioms.

This has a curious consequence. We might think that F = Gmm′/r² is much simpler than F = Gmm′/r² + Hmm′/r³ (where H is presumably very, very small). But if we fill out each proposal with the foundational mathematical structure, the percentage difference in complexity will be slight, as almost all of the contribution to complexity will be in such things as the construction of real numbers (say, via Dedekind cuts).

Perhaps, though, the above line of thought is reason to think that real analysis and arithmetic are actually fundamental?

Moral conversion and Hume on freedom

According to Hume, for one to be responsible for an action, the action must flow from one’s character. But the actions that we praise people for the most include cases where someone breaks free from a corrupt character and changes for the good. These cases are not merely cases of slight responsibility, but are central cases of responsibility.

A Humean can, of course, say that there was some hidden determining cause in the convert’s character that triggered the action—perhaps some inconsistency in the corruption. But given determinism, why should we think that this hidden determining cause was indeed in the agent’s character, rather than being some cause outside of the character—some glitch in the brain, say? That the hidden determining cause was in the character is an empirical thesis for which we have very little evidence. So on the Humean view, we ought to be quite skeptical that the person who radically changes from bad to good is praiseworthy. We definitely should not take such cases to be among paradigm cases of praiseworthiness.

Humeanism and knowledge of fundamental laws

On a "Humean" Best System Account (BSA) of laws of nature, the fundamental laws are the axioms of the system of laws that best combines brevity and informativeness.

An interesting consequence of this is that, very likely, no amount of advances in physics will
suffice to tell us what the fundamental laws are: significant advances in mathematics will also be needed. For suppose that after a lot of extra physics, propositions formulated in sentences p₁, ..., p_n are the physicist’s best proposal for the fundamental laws. They are simple, informative and fit the empirical data really well.

But we would still need some very serious mathematics. For we would need to know there isn’t a simpler collection of sentences {q₁, ..., q_m} that is logically equivalent to {p₁, ..., p_n} but simpler. To do that would require us to have a method for solving the following type of mathematical problem:

1. Given a sentence s in some formal language, find a simplest sentence s′ that is logically equivalent to s,

in the case of significantly non-trivial sentences s.

We might be able to solve (1) for some very simple sentences. Maybe there is no simpler way of saying that there is only one thing in existence than ∃x∀y(x=y). But it is very plausible that any serious proposal for the laws of physics will be much more complicated than that.

Here is one reason to think that any credible proposal for fundamental laws is going to be pretty complicated. Past experience gives us good reason to think the proposal will involve arithmetical operations on real numbers. Thus, a full statement of the laws will require including a definition of the arithmetical operations as well as of the real numbers. To give a simplest formulation of such laws will, thus, require us to solve the problem of finding a simplest axiomatization of the portions of arithmetic and real analysis that are needed for the laws. While we have multiple axiomatizations, I doubt we are at all close to solving the problem of finding an optimal such axiomatization.

Perhaps the Humean could more modestly hope that we will at least know a part of the fundamental laws—namely the part that doesn’t include the mathematical axiomatization. But I suspect that even this is going to be very difficult, because different arithmetical formulations are apt to need different portions of arithmetic and real analysis.

Chanceability

Say that a function P : F → [0,1] where F is a σ-algebra of subsets of Ω is chanceable provided that it is metaphysically possible to have a concrete (physical or not) stochastic process with a state space of the same cardinality as Ω and such that P coincides with the chances of that process under some isomorphism between Ω and the state space.

Here are some hypotheses ones might consider:

1. If P is chanceable, P is a finitely additive probability.

2. If P is chanceable, P is a countably additive probability.

3. If P is a finitely additive probability, P is chanceable.

4. If P is a countably additive probability, P is chanceable.

5. A product of chanceable countably additive probabilities is chanceable.

It would be nice if (2) and (4) were both true; or if (1) and (3) were.

I am inclined to think (5) is true, since if the P_i are chanceable, they could be implemented as chances of stochastic processes of causally isolated universes in a multiverse, and the result would have chances isomorphic to the product of the P_i.

I think (3) is true in the special case where Ω is finite.

I am skeptical of (4) (and hence of (3)). My skepticism comes from the following line of thought. Let Ω = ℵ₁. Let F be the σ-algebra of countable and co-countable subsets (A is co-countable provided that Ω − A is countable). Define P(A) = 1 for the co-countable subsets and P(A) = 0 for the countable ones. This is a countably additive probability. Now let < be the ordinal ordering on ℵ₁. Then if P is chanceable, it can be used to yield paradoxes very similar to those of a countably infinite fair lottery.

For instance, consider a two-person game (this will require the product of P with itself to be chanceable, not just P; but I think (5) is true) where each player independently gets an ordinal according to a chancy isomorph of P, and the one who gets the larger ordinal wins a dollar. Then each player will think the probability that the other player has the bigger ordinal is 1, and will pay an arbitrarily high fee to swap ordinals with them!

Culpability incompatibilism

Here are three plausible theses:

1. You’re only culpable for a morally wrong choice determined by a relevantly abnormal mental state if you are culpable for that mental state.

2. A mental state that determines a morally wrong choice is relevantly abnormal.

3. You are not culpable for anything that is prior to the first choice you are culpable for.

Given these theses and some technical assumptions, it follows that:

4. If determinism holds, you are not culpable for any morally wrong choice.

For suppose that you are blameworthy for some choice and determinism holds. Let t₁ be the time of the first choice you are culpable for. Choices flow from mental states, and if determinism holds, these mental states determine the choice. So there is a time t₀ at which you have a mental state that determines your culpable choice at t₁. That mental state is abnormal by (2). Hence by (1) you must be culpable for it given that it determines a wrong choice. But this contradicts (3).

The intuition behind (1) is that abnormal mental states remove responsibility, unless either the abnormality is not relevant to the choice, or one has responsibility for the mental state. This is something even a compatibilist should find plausible.

Moreover, the responsibility for the mental state has to have the same valence as the responsibility for the choice: to be culpable for the choice, you must be culpable for the abnormal state; to be praiseworthy for the choice, you must be praiseworthy for the abnormal state. (Imagine this case. To save your friends from a horrific fate, you had to swallow a potion which had a side-effect of making you a kleptomaniac. You are then responsible for your kleptomania, but in a praiseworthy way: you sacrificed your sanity to save your friends. But now the thefts that come from the kleptomania you are not blameworthy for.)

Premise (2) is compatible with there being normal mental states that determine morally good choices, as well as with there being normal mental states that non-deterministically cause morally wrong choices (e.g., a desire for self-preservation can non-deterministically cause an act of cowardice).

What I find interesting about this argument is that it doesn’t have any obvious analogue for praiseworthiness. The conclusion of the argument is a thesis we might call culpability incompatibilism.

The combination of culpability incompatibilism with praiseworthiness compatibilism (the doctrine that praiseworthiness is compatible with determinism) has some attractiveness. Leibniz cites with approval St Augustine’s idea that the best kind of freedom is choosing the best action for the best reasons. Culpability incompatibilist who are praiseworthiness compatibilists can endorse that thesis. Moreover, they can endorse the idea that God is praiseworthy despite being logically incapable of doing wrong. Interestingly, though, praiseworthiness compatibilism makes it difficult to run free will based defenses for the problem of evil.