Four problems and a unified solution

A similar problem occurs in at least four different areas.

1. Physics: What explains the values of the constants in the laws of nature?

2. Ethics: What explains parameters in moral laws, such as the degree to which we should favor benefits to our parents over benefits to strangers?

3. Epistemology: What explains parameters in epistemic principles, such as the parameters in how quickly we should take our evidence to justify inductive generalizations, or how much epistemic weight we should put on simplicity?

4. Semantics: What explains where the lines are drawn for the extensions of our words?

There are some solutions that have a hope of working in some but not all the areas. For instance, a view on which there is a universe-spawning mechanism that induces random value of constants in laws of nature solves the physics problem, but does little for the other three.

On the other hand, vagueness solutions to 2-4 have little hope of helping in the physics case. Actually, though, vagueness doesn’t help much in 2-4, because there will still be the question of explaining why the vague regions are where they are and why they are fuzzy in the way there are—we just shift the parameter question.

In some areas, there might be some hope of having a theory on which there are no objective parameters. For instance, Bayesianism holds that the parameters are set by the priors, and subjective Bayesianism then says that there are no objective priors. Non-realist ethical theories do something similar. But such a move in the case of physics is implausible.

In each area, there might be some hope that there are simple and elegant principles that of necessity give rise to and explainingthe values of the parameters. But that hope has yet to be born out in any of the four cases.

In each area, one can opt for a brute necessity. But that should be a last resort.

In each area, there are things that can be said that simply shift the question about parameters to a similar question about other parameters. For instance, objective Bayesianism shifts the question of about how much epistemic weight we should put on simplicity to the question of priors.

When the questons are so similar, there is significant value in giving a uniform solution. The theist can do that. She does so by opting for these views:

1. Physics: God makes the universe have the fundamental laws of nature it does.

2. Ethics: God institutes the fundamental moral principles.

3. Epistemology: God institutes the fundamental epistemic principles for us.

4. Semantics: God institutes some fundamental level of our language.

In each of the four cases there is a question of how God does this. And in each there is a “divine command” style answer and a “natural law” style answer, and likely others.

In physics, the “divine command” style answer is occasionalism; in ethics and epistemology it just is “divine command”; and in semantics it is a view on which God is the first speaker and his meanings for fundamental linguistic structs are normative. None of these appeal very much to me, and for the same reason: they all make the relevant features extrinsic to us.

In physics, the “natural law” answer is theistic Aristotelianism: laws supervene on the natures of things, and God chooses which natures to instantiate; theistic natural law is a well-developed ethical theory, and there are analogues in epistemology and semantics, albeit not very popular ones.

Theistic Natural Law and the Euthyphro Problem

Theistic Natural Law (TNL) theory seems to be subject to the Euthyphro problem much as divine command theory (DCT) is. On DCT, the Euthyphro problem takes the form of the question:

1. Why did God command what he commanded rather than commanding otherwise?

On TNL, the Euthyphro problem takes the form of the question:

2. Why did God create beings with the natures he did rather than creating beings with other natures?

In both cases, one can respond by talking of the essential goodness of God, by virtue of which he makes a good choice as to how to fittingly match the non-normative with the normative features of creatures. In the DCT case, God makes the match by benevolently choosing what sorts of creatures to create and what sorts of commands to give them. In the TNL case, God makes the match by benevolently choosing the non-deontic and deontic features of natures and then creating creatures with these natures. Thus, in the DCT case, God has reason to coordinate the sociality of creatures with the command to cooperate, while in the TNL case God has reason to actualize natures that either both include sociality and the duty to cooperate or to actualize natures that include neither.

So in what way is TNL better off than DCT with regard to the Euthyphro problem? The one thing I can think of in the vicinity is this: TNL allows for there to be deontic features that necessarily every natural includes, and it allows for there to be some deontic features of creatures that are entailed by the non-deontic features. For instance, perhaps every possible nature of an agent includes a prohibition against pointless imposition of torture, and every possible nature of a linguistic agent includes a prohibition against lying. But I am not sure this difference is really relevant to the Euthyphro problem.

I do prefer TNL to DCT, but not because of the Euthyphro problem. My reason for the preference is that many moral obligations appear to be intrinsic features of us.

Of course, the above arguments presuppose a particular picture of how natural law works. But I like that picture.

Captain Proton's Ray Gun

The kids and I are big Star Trek fans (well, the 5-year-old is just a fan, not a big fan, as yet), and my son wanted to have Captain Proton's Ray Gun. Captain Proton is a cheesy character in a fictional series of 1950s movies in Star Trek Voyager. So, I guess, he's a fictional fictional character. I found some photos of a prop, traced the images in Inkscape, exported to OpenSCAD, and made 3D printable files, which are here. I printed it (it prints in two halves that join together), but we have yet to paint it (may not paint it right away, as in silver and gray it will look too much like a real gun at a distance to use outside the house).

Infinite grounding regresses

Suppose, as seems possible, that every day for eternity you will toss a coin and get heads.

Then that you will get heads on every future day seems to be grounded in:

1. You will get heads on day 1, and you will get heads on every day starting with day 2.

And the second conjunct of (1) seems to be grounded in:

2. You will get heads on day 2, and you will get heads on every day starting with day 3.

And the second conjunct of (2) seems to be grounded in:

3. You will get heads on day 3, and you will get heads on every day starting with day 4.

And so on.

So, it seems, infinite propositional grounding regresses are possible.

I suspect that infinite existential grounding regresses are not possible, though.

Counseling the lesser evil

A controversial principle in Catholic moral theology is the principle of “counseling the lesser evil”, sometimes confusingly (or confusedly) presented as the “principle of the lesser evil”. The principle is one that the Church has not pronounced on. (For a survey of major historical points, see this piece by Fr. Flannery.)

First, a clarification. Nobody in the debate thinks it is ever permissible to do the lesser evil. The lesser evil is still an evil, and it is never permissible to do evil, no matter what might result from it. The debate is very specifically the following. Suppose someone is determined to do an evil, and cannot be dissuaded from doing some evil or other. Is it permissible to counsel a lesser evil in order to redirect the person from a greater evil? For instance, if someone is about to murder you, and cannot be dissuaded from an evil course of action, are you permitted to counsel theft instead, as on some interpretations the ten men in Jeremiah 41:8 do? (But see quotations in Flannery for other interpretations.)

There is no question that if the potential murderer is redirected to theft, the theft will still be wrong, indeed quite possibly a mortal sin (depending on the amount stolen). The moral question about “the lesser evil” is not about the primary evildoer but about the counselor. On the one hand, it appears that if the counselor’s counsel is sincere, the counselor is wrongfully endorsing an evil—albeit less evil—course of action. Indeed, it seems that the counselor is even intending the evil, albeit as an alternative to a greater evil.

On the other hand, a number of people will have very strong intuitions that it is not wrong to say to a potential murderer “Don’t kill me: here, take my laptop!” (Note: I assume the coerced circumstances do not render this a valid gift, so the potential murderer will indeed be a thief by taking the laptop.)

Let me add that the argument I will give leaves open the question of the advisability of counseling the lesser evil. Often it may be better to inspire the evildoer to do the good thing rather than the lesser of the evils. Moreover, one needs to be extremely wary of any public counseling of the lesser evil, because it is apt to encourage people who are not determined on evil to do the lesser evil. I think it is unlikely that such counseling is often advisable.

So, here’s the argument. Start with this thought. Agents deliberate about options. As they do so, they come to favor some options over others. Eventually, as they narrow in on the decision, they favor one option over all the others. Moreover:

1. If a deliberating agent in the end favors B over C, typically the agent will not choose C as a result of this deliberation.

There are at least two reasons for the “typically”. First, maybe the agent is irrational. Second, maybe there can be cases of circular favoring structures, so that the agent favors B over C, favors A over B, and favors C over A, so that she ends up choosing C anyway.

Next observe this:

2. If option B is better than option C, then it is good for a deliberating agent to favor B over C.

This is true regardless of whether B and C are both good options, or B is good and C is bad, or both B and C are bad. It is simply a good thing to favor the better over the worse.

With (1) and (2) in mind, consider a case where the agent has three options: a good A (e.g., going away), a lesser evil B (e.g., theft) and a greater evil C (e.g., murder). By (2) it is good if the agent to favors B over C. Suppose the counselor strives to lead the agent who is determined on evil to favor B over C (e.g., by emphasizing the resale value of the laptop, or the likelihood that the police will investigate a murder more thoroughly than a theft, or the greater sinfulness of murder, depending on what is more likely to impress the particular agent). Then the conditions for the Principle of Double Effect can be satisfied on the side of the counselor.

3. The counselor is pursuing a good end, the agent’s not choosing C.

4. The counselor’s chosen means to the good end is the agent’s favoring B over C. By (1), such favoring is likely to be effective in fulfilling the counselor’s good end (namely, the agent’s not choosing C) and by (2), such favoring is good.

5. There is a foreseen but not intended evil of the agent opting for B. It is not intended, because the counselor’s plan of action will be successful whether or not the agent opts for B (as foreseen) or for A (an unexpected bonus).

6. The good of the agent’s not choosing C is proportionate to the foreseen evil of the agent’s choosing B, and there is, we may suppose, no better way of achieving the good.

In particular, there is no intention that the agent choose B, or even choose B over C. The intention is that the agent favor B over C, which is all that is typically needed, given (1), for the agent not to choose C.

Note 1: This provides a defense of pretty strong cases of counseling the lesser evil. The argument works even in cases where the agent being counseled wouldn’t have thought of evil B prior to the counseling (that is the case in Jeremiah 41:8). It might even work where B is impossible prior to the counseling. For instance you might unlock your safe in order to make it easier for the agent to steal your money in place of killing you. In so doing, your end is still that C not be done, and the means is that B is favored over C.

Note 2: This solves the problem of bribes.

Note 3: I am not very confident of any of the above.

Bribes and conditional intentions

You are trying to get a permit that you are both morally and legally entitled to, but an official requires a bribe to give you the permit. Are you permitted to pay the bribe?

I always thought: Of course!

But now I think this is more difficult than it has seemed to me. Initially, it seems that your action plan is very simple:

1. Give the bribe in order that the official give you the permit.

But suppose that you pay the bribe and the official never notices the money slipped onto her desk, though when you lean over her desk, from that angle you look just like her nephew, so she gives you the permit out of nepotism. In that case, while you got what you wanted, you didn’t fulfill your plan–your bribery was not a success. That shows that (1) is only a part of your action plan. More fully, your plan is:

2. Give the bribe in order that the official be motivated by it (in the usual way bribes motivate) to give you the permit.

But now it seems to be a moral evil that an official be motivated by a bribe to do something, even if the thing she is motivated to do is the right thing. So in setting oneself on plan (2), it seems one intends something immoral.

I wonder if this isn’t a case similar to asking a murderer: “If you are going to kill me, kill me painlessly” (which one might even put in the simple phrase “Kill me painlessly”, with everybody understanding that the request is conditional). In that case, your intention is not that the murderer kill you painlessly, but that:

3. If the murderer kills you, she kills you painlessly.

And that conditional isn’t a bad thing.

One makes the request of the murderer on the expectation–but certainly neither intention nor hope!–that the the antecedent of the conditional will turn out to be true. Nonetheless, one does not intend the consequent.

Perhaps in the bribery case one has a similar intention:

4. If the official isn’t going to be motivated by duty, she will be motivated by the bribe.

One then gives the bribe on the expectation–but neither intention nor hope–that the official will be unmotivated by duty.

But things aren’t quite that simple. Suppose that I prefer Coca Cola to cocaine, and in a really shady restaurant I place this order:

5. I’ll have a Coca Cola, but if you can’t do that, then I’ll have some cocaine.

Here I’ve done something wrong: I’ve conditionally procured illegal drugs. But how to distinguish (5) from (3) and (4)?

One psychological difference is that in (5), presumably I desire the cocaine, just not as much as I desire the Coca Cola. But in (3) and (4), I don’t desire the painless killing or the taking of the bribe. (Compare this case: Malefactors will forcibly give you Coca Cola, cocaine or cyanide. You say “I’ll have a Coca Cola, but if you can’t do that, I’ll have some cocaine.” Here, I presume, you don’t desire the cocaine, but it’s better than the Coca Cola. That’s more like (3) and (4) than like the restaurant version of (5).)

But I don’t really want to rest the relevant moral distinctions on desires.

Here’s what I’d like to say, but I have a hard time making it work out. In (5)–the restaurant coke/cocaine order–when the antecedent of the conditional is met, your will stands behind the consequent. In (3)–the killing case–your will doesn’t stand behind the consequent even when the antecedent of the conditional is met. Even when it is inevitable that you will be killed, you don’t intend to die, but only not to die painfully. But I worry about this. Suppose then you die painlessly. Isn’t your intention not to painfully die satisfied by the painless death, and hence the painless death was the means to avoiding the painful death? And in the bribery case you intend not to have your request denied, but wasn’t the taking of the bribe the means to the request?

Perhaps there is something much simpler, though, that doesn’t involve intentions so much. Perhaps it’s not morally wrong for the official to give the permit because of the bribe. What is wrong is for the official to give the permit solely because of the bribe. But you needn’t intend that. On the contrary, you might have emphasized to the official that you are morally and legally entitled to the permit. There are many ways the bribe can work. It might be the sole motive. But it might also be a partial motive. Or it might be a defeater for a defeater: "It's a lot of trouble to give permits, so I won't bother. But if I get a bribe, then the trouble is worth it." Of course, that still leaves the probably purely hypothetical case where you know that the only way the bribe will work is by being the sole motive. But now it's not so clear that it's permissible to give it.

And in the case of the murder, you are trying to dissuade the murder from killing you painfully by drawing her attention to the argument that option C is bad because there is a better–albeit still bad–option B? She might then go for option B or she might go for the good option A. Either way, she refrains from doing C. There is, in fact, a way in which the murder case is easier than the bribe case, because your being killed painlessly is not a means to your avoiding the painful death–it is what occurs in its place. If I am offered coffee or water and I go for the water, my drinking water isn’t a means to avoiding coffee, though it happens in its place.

A two-stage view of proportionality in the Principle of Double Effect

A question about Double Effect that hasn’t been sufficiently handled is in what way, if any, the good effects of bad effects are screened off when judging proportionality.

It seems that some sort of screening off is needed. Consider this case. An evildoer says that he’ll free five innocents unless you kill one innocent; otherwise, she’ll kill them. So you shoot at the innocent’s shirt covering his chest, intending to learn how the fabric is rent by the bullet (knowledge is a good thing!), while foreseeing without intending that the innocent should die, and also foreseeing without intending that the evildoer will free the five.

This is clearly a travesty of double effect reasoning. But the only condition that isn’t obviously satisfied is the proportionality condition. So let’s think about proportionality. Here are two ways to think here:

1. All good and bad effects count for proportionality. Thus, both the death of the one and the saving of the five count, as does the trivial good of knowing how the shirt rips. Thus proportionality is satisfied: the goods are proportionate.

2. The good effects that are causally downstream of the bad effects of one’s action don’t count. On this view, it is the intended effect that must be proportionate to the unintended bad effects. Thus, the death of the one counts, and the trivial good of knowing about how the fabric rips counts, but the saving of the five does not count, as it is not intended (if it were intended, the act would be impermissible, of course). But of course the good of knowing how the fabric rips is not proportionate to the death of the one innocent.

Option 2 fits better with the intuition that the initial case was a travesty of double effect reasoning.

But option 2 doesn’t seem to be the right one in all other cases. Suppose I am guarding five innocents sentenced to death by an evil dictator. If I free them, I will be killed. I also know that unless the innocents leave the country, they will be recaptured soon. The innocents are planning to bribe the border officials, which is quite likely to work. But it will be wrong for the border officials to let them escape, because the border officials will have the false belief that these people are justly sentenced, but are venal.

It seems permissible to free the innocents. Here, the unintended but foreseen bad effect is my own death. The good effect is the innocents’ being allowed out of prison. But it seems that if we don’t get to consider effects downstream of bad stuff, we don’t get to consider the fact that the innocents will escape the country, as that’s downstream of the border officials’ venal acceptance of bribes.

Here’s one theory I developed today in conversation with a graduate student. Proportionality is very complex. Perhaps there are two stages.

Stage I: Are the intended good effect and the foreseen bad effects are in the same ballpark? This is a very loose proportionality consideration. One life and ten lives are in the same ballpark, but knowing how the fabric rips is far out of that ballpart. If the intended good effect is so much less than the foreseen bad effects that they are not in the same ballpark, proportionality is not met. Here, the good effects that are downstream of the bad effects don’t count.

If the Stage I proportionality condition is violated, the act is wrong. If it’s met, I proceed to Stage II.

Stage II: Now I get to do a proportionality calculation taking into account all the foreseen goods and bads, regardless of how they are connected to intentions.

The proportionality condition now requires a positive evaluation by means of both stages.

On this two stage theory, shooting the innocent’s shirt in the initial case is wrong, as proportionality is violated at Stage I. On the other hand, the release of the prisoners may be permissible. For the freedom of the innocents is in the same ballpark as my life—it’s a big ballpark—even if they are going to be recaptured. It’s not a trivial good, like the taste of a mint.

I am not happy with this. It’s too complicated!

Certamen practice machine

This summer, the big kids and I built a practice machine for the Junior Classical League's Certamen competition, based on an Arduino (clone) Mega 2560. It's a "practice machine" as it's not officially approved for tournament use (and perhaps can't be without the clicker shape being changed). Cost is about $80 (including filament), as compared to $400+ for the official version.

Build instructions are here. Code is here.

Existence, causation and individuation

Suppose a cause C produces horses, in the following way:

• When C produces a horse, a horse instantly comes into existence made out of some mass of non-equine matter M.

• The genetic makeup of the resulting horse is randomly distributed over all DNA compatible with being a horse.

(Imagine lightning striking a bog and randomly turning the bog mass into a horse.)

So now suppose that in world w₁, a female Arabian, Green Lightning, comes into existence as a result of C, while in w₂, a male Exmoor pony, Tigger, comes into existence as a result of C.

Presumably, Green Lightning and Tigger are numerically distinct horses. Why are they distinct? Presumably because they are qualitatively different—specifically, because their DNA is different. If in w₁ and w₂, C respectively produced horses that were exactly alike out of M, those horses would have to have been numerically identical. (Haecceitists will disagree.)

But now we have a puzzle for Aristotelians. Both Green Lightning and Tigger are of the same species. (If you think that breeds or sexes make for different metaphysical species, modify the example and make them be of the same sex and breed, but still very different from each other.) Let the Fs be the qualitative features that Green Lightning and Tigger initially differ in.

1. The Fs in are accidents in the Aristotelian sense: they are accidental to their horsehood, which is their form.

(They may not be accidents in the contemporary modal sense. It may be that it is impossible for a horse to be of another sex than it is.)

But:

2. The Fs make Green Lightning be distinct from Tigger.

3. If what makes Green Lightning be distinct from Tigger are the Fs, then the Fs help make Green Lightning be Green Lightning.

4. Nothing that helps make x be x can be explanatorily posterior to x.

5. So, the Fs are not explanatorily posterior to Green Lightning. (2-4)

6. The accidents of x are explanatorily posterior to x.

7. So, the Fs are explanatorily posterior to Green Lightning. (1,6)

8. Contradiction! (5,7)

The case where C makes a horse come into existence from non-equine matter makes the above argument a bit more vivid. In the ordinary case of equine reproduction, a sperm and egg contribute their DNA and give rise to the DNA of the offspring. There it could be argued that the relevant thing that helps make the resulting horse be the horse it is is the DNA in the sperm and the DNA in the egg.

One could conclude that a horse can’t come into existence from matter that doesn’t already contain implicit in it the DNA of the horse. But that is implausible, especially since God could create a horse even without any matter.

This puzzle worries me a lot. I initially thought it was a special puzzle for four-dimensionalist temporal-parts Aristotelianism, because it showed that the first temporal part of the horse was explanatorily prior to the whole, whereas Aristotelianism forbids parts to be prior to wholes. But then I realizes that the same point could be made about accidents without reference to four-dimensionalism.

Here is my best solution. There is something about Green Lightning that is prior to her being Green Lightning. It is her being caused by C to exist with the Fs (i.e., her being caused by C to exist as a female Arabian, etc.). Admittedly, that sounds just as much as an accident of Green Lightning as the Fs are. It’s not Green Lightning’s form, so what else could it be but an accident? There is no answer in Aristotle, but there is a potential answer in Aquinas: this could be Green Lightning’s act of being, her esse. And it is not crazy to take Green Lightning’s esse to be something that (a) is prior to Green Lightning, (b) Green Lightning could not exist without, and (c) an individuator of Green Lightning.

This reminds me of a line of thought in the Principle of Sufficient Reason book where I argued that the esse of a contingent being is its being caused. If my present solution is correct, that was only a partial description of the esse of a contingent being. And I think there may well be an argument for the principle that ex nihilo nihil fit in the vicinity, just as in the PSR book—for it is absurd to think that anything contingent could be prior to x if x has no cause, while this esse is something contingent.

Murder by slowdown?

Zeno wants Alice dead and he has the following plan. He slows down Alice’s functioning—say, by cooling her or by sending her around the earth on a spaceship so fast that relativistic time dilation does the job—so much that each second of Alice’s internal time takes a billion years of external time. In six seconds of Alice’s internal time, she’s dead, because the sun runs out of hydrogen and turns into a red giant.

Did Zeno kill Alice or did the sun kill Alice? Both: Zeno kills Alice by shifting her future life into a spatiotemporal position where that life would be destroyed by the sun. This is akin to sending Alice now into the sun on a speeding rocket.

(I am not a lawyer, but I expect Zeno could only be convicted of attempted murder, since a conviction for murder requires the victim to be dead; similarly, I assume that an 80-year-old person who gives someone a poison that takes forty years to work can only be convicted of attempted murder, because by the time the poison does its work, the murderer will be dead.)

But now imagine that Zeno lives in a universe where the earth will be habitable forever. He sets up an automated system that slows down Alice’s internal time to such a degree that in the first year of external time, Alice’s internal time moves ahead only 3 seconds; in the next external year, it moves ahead by 1.5 seconds; in the next year, it moves ahead by 0.75 seconds; and so on. What happens? Well, Alice still cannot have more than six seconds of life ahead of him. In n years of external time, she will have had 6 − 6/2ⁿ seconds of internal time.

So just as in the first scenario, Zeno has ensured that Alice has less than six seconds of internal time left. It sure sounds like murder. But wait! In the second scenario, it seems that Alice never dies: she is alive this year, just sluggish; she will be alive next year, though even more sluggish; and so on.

But Alice will be dead in exactly six seconds of internal time. So what will be the cause of death? The unfortunate misalignment between Alice’s internal time and the external time of the universe, together with the universe running out of time “once year ω rolls around”? Maybe. I am not sure. This is paradoxical.

There is a way of getting out of this paradox. Suppose internal time must be discrete. Then to slow down Alice’s time means to space out the discrete ticks of her time. Suppose for simplicity that Alice has a hundred ticks per internal second. Then in the next year, she will have 300 ticks. Some time in year ten, the 599th tick of Alice’s future life happens. And the 600th tick will never happen. So, the gradual slowdown story is is impossible. The speed his zero after the tenth year. The best (or worst?) Zeno can do is ensure that the 599th tick of Alice’s life is the last one. But if that’s what he does, then he causes her death by ensuring that the 600th tick never happens. But if that’s what he does, there is no gradual slowdown paradox.

Why my present existence can't depend on future events

I find very persuasive arguments like this:

1. If theory T is true, then whether I exist now depends on some future events.

2. Facts about what exists now do not depend on future events.

3. So, theory T is not true.

For instance, some four-dimensionalist solutions to problems of fission according to which the number of people there are now depends on whether fission will are subject to this criticism.

But I’ve had a nagging worry about arguments like this, that in accepting (2), I am not being faithful to my eternalist four-dimensionalist convictions: why should the present aspects of the four-dimensional me have this sort of priority? Moreover, I didn’t really have an argument for (2). Until today.

Here is an argument for (2). Start with this.

4. If facts about my present existence depend on future events, then facts about my present existence depend on future events that happen to me.

For instance, suppose that whether I exist now depends on whether some surgeon cuts my brain in half tomorrow. Well, then, some of the events that my present existence depends on will be events that happen entirely to someone else—for instance, whether the surgeon gets to work on time. But other events, such as the cutting or non-cutting of the brain, will happen to me. It would be absurd to think that facts about my present existence or identity depend on future events that happen entirely to something other than me.

Then add:

5. Any events that happen to me in the future depend on my present existence.

For, such events presuppose my future existence, and my future existence is caused by my present existence.

6. Circular dependence is impossible.

7. So, facts about my present existence do not depend on future events.

Note that (6) is a very strong premise, and is one place the argument can get attacked. Many people think that you can have circular dependence when the dependence in the two directions is of a different sort. In the case at hand, facts about my present existence might depend constitutively on future events, while the future events depend causally on my present existence. Nonetheless, I think (6) is true, even if the dependence in the two directions is of a different sort.

Another move is to describe the future events on which my existence depends without reference to me. Don’t describe what the surgeon does as the splitting of my brain, but as the splitting of brain x. Then we could say that the future event of the surgeon’s splitting my brain does depend on my present existence, but my present existence doesn’t depend on that event. Instead, it depends on the future event of the surgeon’s splitting brain x. This objection denies (5): while the splitting of my brain depends on my present existence, the splitting of brain x does not, and yet it happens to me.

I think this is mistaken. The splitting of brain x depends on the future existence of that brain, and that brain depends on me, because parts depend on wholes—that is a deep Aristotelian premise I accept. Thus I think (5) is true. An event that happens to me is an event that involves at least a part of me, and none of my parts could exist without me. Granted, a brain like mine could exist without me. But token events are individuated in part by the things caught up in them. A splitting of a brain merely like mine would be a different event from the splitting of this particular brain. And it is a token event that my present existence is supposed to depend on.

The above argument won’t move non-Aristotelians who think that wholes depend on parts rather than parts depending on wholes. But it works for me. And hence it assuages the worry that in accepting (2), I am being unfaithful to my views about time.

All that said, I don’t really want to affirm (2) in an exceptionless way. If I am a time-traveller born in the year 2200, then my present existence does depend on what will happen in the future. But it only depends on what will happen in the external-time future not on what will happen in my internal-time future. And, crucially, I think time-travel is only possible when it doesn’t result in causal loops. So even if I am a time-traveller from the future, I cannot affect anything that is causally relevant to whether I will be born, etc. This probably means that if time-travel is possible, it is possible only in very carefully limited settings.

Conciliationism is false or trivial

Suppose you and I are adding up a column of expenses, but our only interest is the last digit for some reason. You and I know that we are epistemic peers. We’ve both just calculated the last digit, and a Carl asks: Is the last digit a one? You and I speak up at the same time. I say: “Probably not; my credence that it’s a one is 0.27.” You say: “Very likely; my credence that it’s a one is 0.99.”

Concialiationists now seem to say that I should lower my credence and you should raise yours.

But now suppose that you determine the credence for the last digit as follows: You do the addition three times, each time knowing that you have an independent 1/10 chance of error. Then you assign your credence as the result of a Bayesian calculation with equal priors over all ten options for the last digit. And since I’m your epistemic peer, I do it the same way. Moreover, while we’re poor at adding digits, we’re really good at Bayesianism—maybe we’ve just memorized a lot of Bayes’ factor related tables. So we don’t make mistakes in Bayesian calculations, but we do at addition.

Now I can reverse engineer your answer. If you say your credence in a one is 0.27, then I know that of your three calculations, one of them must have been a one. For if none of your calculations was a one, your credence that the digit was a one would have been very low and if two of your calculations yielded a one, your credence would have been quite high. There are now two options: either you came up with three different answers, or you had a one and then two answers that were the same. In the latter case, it turns out that your credence in a one would have been fairly low, around 0.08. So it must be that your calculations yielded a one, and then two other numbers.

And you can reverse engineer my answer. The only way my credence could be as high as 0.99 is if all three of my calculations yielded a one. So now we both know that my calculations were 1, 1, 1 and yours were 1, x, y where 1, x, y are all distinct. So now you aggregate this data, and I do the same as your peer. We have six calculations yielding 1, 1, 1, 1, x, y. A Bayesian analysis, given the fact that the chance of error in each calculation is 0.9, yields a posterior probability of 0.997.

So, your credence did go up. But mine went up too. Thus we can have cases where the aggregation of a high credence with a low credence results in an even higher credence.

Of course, you may say that the case is a cheat. You and I are not epistemic peers, because we don’t have the same evidence: you have the evidence of your calculations and I have the evidence of mine. But if this counts as a difference of evidence, then the standard example conciliationists give, that of different people splitting a bill in a restaurant, is also not a case of epistemic peerhood. And if the results of internal calculations count as evidence for purposes of peerhood, then there just can’t be any peers who disagree, and conciliationism is trivial.

From the finite to the countable

Causal finitism lets you give a metaphysical definition of the finite. Here’s something I just noticed. This yields a metaphysical definition of the countable (phrased in terms of pluralities rather than sets):

1. The xs are countable provided that it is possible to have a total ordering on the xs such if a is any of the xs, then there are only finitely many xs smaller (in that ordering) than x.

Here’s an intuitive argument that this definition fits with the usual mathematical one if we have an independently adequate notion of nautral numbers. Let N be the natural numbers. Then if the xs are countable, for any a among the xs, define f(a) to be the number of xs smaller than a. Since all finite pluralities are numbered by the natural numbers, f(a) is a natural number. Moreover, f is one-to-one. For suppose that a ≠ b are both xs. By total ordering, either a is less than b or b is less than a. If a is less than b, there will be fewer things less than a than there are less than b, since (a) anything less than a is less than b but not conversely, and (b) if you take something away from a finite collection, you get a smaller collection. Thus, if a is less than b, then f(a)<f(b). Conversely, if b is less than a, then f(b)<f(a). In either case, f(a)≠f(b), and so f is one-to-one. Since there is a one-to-one map from the xs to the natural numbers, there are only countably many xs.

This means that if causal finitism can solve the problem of how to define the finite, we get a solution to the problem of defining the countable as a bonus.

One of the big picture things I’ve lately been thinking about is that, more generally, the concept of the finite is foundationally important and prior to mathematics. Descartes realized this, and he thought that we needed the concept of God to get the concept of the infinite in order to get the concept of the finite in turn. I am not sure we need the concept of God for this purpose.

Are there multiple models of the naturals that are "on par"?

Assuming the Peano Axioms of arithmetic are consistent, we know that there are infinitely many sets that satisfy them. Which of these infinitely many sets is the set of natural numbers?

A plausible tempting answer is: “It doesn’t matter—any one of them will do.”

But that’s not right. For the infinitely many sets each of which is a model of the Peano Axioms are not isomorphic. They disagree with each other on arithmetical questions. (Famously, one of the models “claims” that the Peano Axioms are consistent and another “claims” that they are inconsistent, where we know from Goedel that consistency is equivalent to an arithmetical question.)

So it seems that with regard to the Peano Axioms, the models are all on par, and yet they disagree.

Here’s a point, however, that is known to specialists, but not widely recognized (e.g., I only recognized the point recently). When one says that some set M is a model of the Peano Axioms, one isn’t saying quite as much as the non-expert might think. Admittedly, one is saying that for every Peano Axiom A, A is true according to M (i.e., M⊨A). But one is not saying that according to M all the Peano Axioms are true. One must be careful with quantifiers. The statement:

1. For every Peano Axiom A, according to M, A is true.

is different from:

2. According to M, all the Peano Axioms are true.

The main technical reason there is such a difference is that (2) is actually nonsense, because the truth predicate in (2) is ineliminable and cannot be defined in M, while the truth predicate in (1) is eliminable; we are just saying that for any Peano Axiom A, M⊨A.

There is an important philosophical issue here. The Peano Axiomatization includes the Axiom Schema of Induction, which schema has infinitely many formulas as instances. Whether a given sequence of symbols is an instance of the Axiom Schema of Induction is a syntactic matter that can be defined arithmetically in terms of the Goedel encoding of the sequence. Thus, it makes sense to say that some sequence of symbols is a Peano Axiom according to a model M, i.e., that according to M, its Goedel number satisfies a certain arithmetical formula, I(x).

Now, non-standard models of the naturals—i.e., models other than our “normal” model—will contain infinite naturals. Some of these infinite naturals will intuitively correspond, via Goedel encoding, to infinite strings of symbols. In fact, given a non-standard model M of the naturals, there will be infinite strings of symbols that according to M are Peano Axioms—i.e., there will be an infinite string s of symbols such that its Goedel number g_s is such that I(g_s). But then we have no way to make sense of the statement: “s is true according to M” or M⊨s. For truth-in-a-model is defined only for finite strings of symbols.

Thus, there is an intuitive difference between the standard model of the naturals and non-standard models:

1. The standard model N is such that all the numbers that according to N satisfy I(x) correspond to formulas that are true in N.

2. A non-standard model M is not such that all the numbers that according to M satisfy I(x) correspond to formulas that are true in M.

The reason for this difference is that the notion of “true in M” is only defined for finite formulas, where “finite” is understood according to the standard model.

I do not know how exactly to rescue the idea of many inequivalent models of arithmetic that are all on par.

Approximate truth and the very recent past

Suppose I say that Jim yelled in delight at 12:31. But in fact he did so at 12:32. Then I said something false but approximately true.

Now, suppose that I hear Jim giving a loud yell of delight about 300 meters away. While I am listening to that yell, I think that Jim is yelling. But in the last second of my hearing, Jim is no longer yelling, but the sound waves are still traveling to me. No big deal. My belief that Jim is yelling is false, but approximately true. Or so I want to say.

And it’s important to say something like this, for it allows us to preserve the idea that our sense give us approximate truth. The case of sound from 300 meters away is particularly strong, but the point goes through in all our sensation, as none of it travels faster than the speed of light. Now, granted, often when we become aware of a stimulus, our sensory organs are still undergoing it. But nonetheless it is strictly speaking false to say that this very part of the stimulus that we are now aware of is in fact going on. So our senses seem to lead us slightly astray. But at most very slightly. It is approximately true that this part of the stimulus is going on now, because it is in fact going on a fraction of a second earlier. Or, perhaps, it is a part of our common sense knowledge of the world that the data of the senses is only meant as an approximation to the truth, and so there is no straying at all.

Now imagine that I say that Jim actually yelled in delight at 12:31, but he was actually completely silent all day, although in a very nearby possible world he did yell in delight at 12:31. Then what I said is not approximately true. In ordinary contexts, the modal difference between the actual and the merely possible vitiates approximate truth, no matter how nearby the merely possible world is.

So now on to one of my hobby horses: presentism. If presentism is true, then the difference between what is happening now and what happened earlier is relevantly like the difference between the actual and the possible. In both cases, it is a difference between a neat and clean predication and a predication in the scope of a modal operator, pastly or possibly, respectively. If this is right, then if presentism is true, I cannot say what I said about its being approximately true that Jim is yelling if Jim has actually stopped. That difference is a very deep modal difference. That the time when Jim is yelling is in a nearby past no more suffices for the approximate truth of “Jim is yelling now” than that Jim is yelling in a nearby possible world is enough for the approximate truth of “Jim is actually yelling”. The ontological gulf between the actual and the possible is vast; so would be the ontological gulf between the present and the past if presentism were true.

Thus, the presentist cannot say that the senses tend to deliver approximate truth.

Objection: We know to correct the data of the senses for the delay.

Response: We know. But that's a recent development.