The importance of the future

It would be bad for me to permanently cease to exist in five minutes. But why? Suppose first a metaphysics of time on which there is no future, namely Growing Block or Presentism. On such a metaphysics there is no such thing as my future life, so how could it be bad for there to be a cessation of it?

Since the only tenable alternative to Growing Block and Presentism is Eternalism, the view that the past and future are real (oddly, there are no Futurists who think the future is real but deny the reality of the past), Eternalism is true.

Now, given Eternalism, we have a choice for three visions of our persistence through time. On one vision, Exdurantism, we are instantaneous stages that do not persist through time at all—at most we have temporal counterparts at other times. This does not fit with the intuition of my radical incompleteness should I cease to exist in five minutes. The second vision is Endurantism: I am wholly present at each time at which I exist. But then if the present moment is real, and eternally will be real, and I wholly exist at this present moment, then the intuition about the deep incompleteness I would have were my existence to permanently end in five minutes is undercut. So that can't be right either.

What remains is a family of views on which we are strung out four-dimensionally. The most common member of the family is Perdurantism: I am four-dimensional but have three-dimensional stages localized at times. A less common view is that I am four-dimensional, but not divided up into stages. Both of these views do justice to the idea that my existence is deeply incomplete, in something like the way it would be if I were missing an arm, should I cease to exist in five minutes.

As far back as I thought much about time (probably going back to age 10) I was an Eternalist. Until a couple of years ago, I was an Endurantist. Then I started being unsure whether Endurantism or a stageless four-dimensional view is right. The above argument strongly pushes me towards a four-dimensional view, and since I don't believe in stages, a stageless one.

Moreover, the above may help with a puzzle I used to have, which was how a B-Theorist should think about the badness of impending evils (especially death). How can a B-Theorist make sense of the badness of being closer and closer to something bad? But that may primarily be a problem for the Endurantist, since the Endurantist thinks we are three-dimensional beings wholly located in the here and now (as well as in the there and later, of course).

Hope

If there are ten lottery tickets, and I hold one, I shouldn't hope to win, but I should simply assign probability 1/10 to my winning. Anything beyond the probabilities in the way of hope would be irrational. Likewise, if I have probability 9/10 of winning. Then I can have confidence, but this confidence should no more be a hope than in the former case. It's just a confidence of 9/10.

But if my friend has fallen morally many times but promises to do better, I shouldn't simply calculate the probability of his doing better using the best inductive logic and leave it at that. I should hope he will do better.

What makes for the difference? In the case of the friend, he should do better. But it is, of course, false that I should win the lottery. Indeed, the outcome of my winning the lottery is in no way normatively picked out. I can appropriately hope that the lottery will be run fairly, but that's that.

If this is right then it seems hope is of what should be. Well, that's not quite right. For if I have done something so terrible that my friend is under no obligation to forgive me, I can still hope for her supererogatory forgiveness. So, perhaps, hope is of what should be or what goes over and beyond a should.

If this is right, then this neatly dovetails with my account of trust or faith. Faith has as its proper object a present state of affairs that should be, such as a testifier's honesty and reliability, or perhaps—I now add—a present state of affairs that goes beyond a should. Hope has as its proper object a future state of affairs that should be or goes beyond a should. Both of these flow from love.

If this is right, then in order for there to be appropriate hope in things beyond human power—such as a hope that an asteroid won't wipe out all life on earth—there must be shoulds, or beyond-shoulds, that go beyond human power. This requires an Aristotelian teleology or theism.

Despair

1. If there is no hope of an afterlife, all is hopeless.
2. If all is hopeless, ultimate despair is the right attitude.
3. Ultimate despair is not the right attitude.
4. So there is hope of an afterlife.

I will argue for 1 and 3. If there is no hope of an afterlife, any redeeming value we might hope for is overshadowed by the ultimate evil of both the end of our individual lives and of the human race. But despair is not the right attitude, since despair makes it impossible for us to live our moral lives, both in terms of the motivation to pursue the good and in our duty to comfort others. Despair saps our motivation. And faced with ultimate hopelessness, any comfort we might offer to others is insincere and dishonest. In despair at ultimate hopelessness, we could only live the good human life through self-deceit and the deceit of others. But that is not right. So ultimate despair cannot be the right attitude as it makes the good life impossible.

So there is hope.

Deep Thoughts XXXV

Meeting the minimum requirements is always good enough.

[This is a variant on XXXIV. There are times when we say things like "The minimum is not good enough." When we do that, what I think happens is that we have a context shift. "The minimum" is understood relative to one set of ends or norms while the "good enough" is understood relative to another. One kind of a case is where you're competing for a job. Meeting the minimum required qualifications is good enough for being hirable in principle (if it's not, the minimum requirements were incorrectly stated), but is not good enough for beating the competition. Another kind of case is where someone is being evaluated in a number of areas (or with respect to a multiplicity of assignments). In each area, there is a minimum requirement. Meeting that requirement is good enough for not failing according to that requirement. But there may be a second, meta requirement to exceed the minimum in most of the areas. (There cannot coherently be a requirement to exceed the minimum in all areas. For if there were such, then the "minimum" in each area would not be a minimum requirement but a maximum disqualifier.) In any case, when we keep the context constant (and as a rule in natural language context in short sentences stays constant), "The minimum is not good enough" is a self-contradiction.]

A simple consequence argument

Say that p and q are nomically equivalent provided that the laws of nature entail that p holds if and only if q does.

Assume:

1. If q is not up to you, and p is nomically equivalent to q, then p is not up to you.

Suppose determinism. Let L be the laws. Let t₀ be 1000 years ago. Let p be a proposition reporting something you do. Let q be the disjunction of all the nomically possible states of the universe at t₀ that evolve under L in such a way as to make p true. Then, plausibly:

2. p and q are nomically equivalent.

For given the deterministic laws, if p is true, then a thousand years ago the universe must have been such as to have to evolve to make p true.[note 1] And conversely, the laws entail that if it was such, then p is true.

Finally, observe that events a thousand years ago aren't up to you:

3. q is not up to you.

We conclude that p is not up to you. So no actions are up to you if determinism holds.

Substance causation, agent causation and time

Aristotelians about causation think all causation is substance causation. Events are causes only derivatively. What does the real causing are substances. This should make Aristotelians very sympathetic to the use of agent causation in the theory of free will. And insofar as the theory of agent causation is just that the agent is the cause of free actions, the Aristotelian who believes that we are substances[note 1] is surely going to agree that we are the causes of our free actions, and we are both agents and substances, so the agent is the cause of her free actions.

So far so good. But there is more to agent causation in regard to free will. Typically, agent causalists invoke agent causation to solve problems such as the randomness problem for libertarians. Agent causation is what makes an action be genuinely one's own action rather than a random blip. But the Aristotelian's embrace of substance causation is too broad. For not only does the Aristotelian think that her free actions are caused by her, she also thinks her non-free actions are caused by her, and even things like the circulation of the blood, which isn't an action at all, are caused by her. Moreover, since she is an agent, she thus thinks all of these things are caused by an agent. But if agent causation metaphysically lumps free actions with non-free ones, and doesn't distinguish them metaphysically from the circulation of the blood, then agent causation can't do the job it's designed for. The Aristotelian believes in agent causation, of course, and may do so with good metaphysical reason, but this agent causation cannot be used to solve the problems that the free will theorists want it to solve.

This line of thought might lead some Aristotelians about causation to accept a version of Cartesian dualism on which we are souls. For then one might hold, contrary to Aristotle, that only our free actions are caused by us and that the circulation of the blood and so on is not caused by us, because we are immaterial beings whose only direct effects are in whatever the equivalent of the pineal gland on this theory will be. This is not in the Aristotelian spirit, though, and it leads to unhappy ethical conclusions (bodies as akin to property).

I think there is something else one should say here. One shouldn't say that agent causation just is causation by an agent. Rather, agent causation is causation by an agent qua agent. You cause your free actions qua agent and you circulate your blood qua mammal, though of course you are both agent and mammal. It is a bit odd to say that you don't perform your non-free actions qua agent, though. After all, how can there be an action without an agent? Aren't all actions, free or not, the actions of an agent qua agent? Maybe. But maybe the distinction is still of some help, for maybe the kinds of mere randomness we want to rule out with the distinction isn't an action at all when looked at more closely.

There is another issue around here. There needs to be more to substance causation than the simple structure substance x causes event E. For paradigmatic substances persist over a long time, but many of their effects happen only at particular times in their existence. And there is an explanation of why the substance causes an effect at one time or another. For instance, I caused oatmeal to be assimilated to me earlier to day because I was hungry. The explanation includes not just the substance, but a state of a substance (and maybe other substances—but that's for another day). I caused the assimilation of my breakfast not only qua agent, but qua hungry agent. Likewise, I circulate the blood not just qua mammal, but qua mammal with a brain stem that sends such-and-such electrical signals to the heart.

Apart from considerations of free will, then, Aristotelians should say that the structure of substance causation is something like: substance x qua in state S causes event E. If we have an Aristotelian constituent ontology, then the state will be a mode (an essence, a necessary accident or a contingent accident) of the substance, and the causation relation will be a ternary relation between the substance, the mode and the event.

But now that we have all of this detail in place, we can go back and ask whether we still get benefits of an agent causal theory in regard to free will. That's not so clear. For instance, when I qua hungry agent caused my breakfast to be assimilated to me, the work distinguishing this from non-actions like circulation is being done by my state of hungry agency. It is because this state is involved in the causation—and not just involved, but involved in the right way (my being a hungry agent could cause me to grow a tail if I was rigged the wrong way)—that I am acting. But event causalists can say something exactly parallel. What distinguishes my eating the breakfast from my circulating the blood is that the former is caused by the event of my being a hungry agent qua my being a hungry agent.

So I do not know that Aristotelian agent causalists can claim to do better than event causalists. In fact, for certain ends they might well want to join cause with the event causalists.

Pascal's Wager in a social context

One of our graduate students, Matt Wilson, suggested an analogy between Pascal's Wager and the question about whether to promote or fight theistic beliefs in a social context (and he let me cite this here).

This made me think. (I don't know what of the following would be endorsed by Wilson.) The main objections to Pascal's Wager are:

1. Difficulties in dealing with infinite utilities. That's merely technical (I say).
2. Many gods.
3. Practical difficulties in convincing oneself to sincerely believe what one has no evidence for.
4. The lack of epistemic integrity in believing without evidence.
5. Would God reward someone who believes on such mercenary grounds?
6. The argument just seems too mercenary!

Do these hold in the social context, where I am trying to decide whether to promote theism among others? If theistic belief non-infinitesimally increases the chance of other people getting infinite benefits, without any corresponding increase in the probability of infinite harms, then that should yield very good moral reason to promote theistic belief. Indeed, given utilitarianism, it seems to yield a duty to promote theism.

But suppose that instead of asking what I should do to get myself to believe the question is what I should try to get others to believe. Then there are straightforward answers to the analogue of (3): I can offer arguments for and refute arguments against theism, and help promote a culture in which theistic belief is normative. How far I can do this is, of course, dependent on my particular skills and social position, but most of us can do at least a little, either to help others to come to believe or at least to maintain their belief.

Moreover, objection (4) works differently. For the Wager now isn't an argument for believing theism, but an argument for increasing the number of people who believe. Still, there is force to an analogue to (4). It seems that there is a lack of integrity in promoting a belief that one does not hold. One is withholding evidence from others and presenting what one takes to be a slanted position (for if one thought that the balance of the evidence favored theism, then one wouldn't need any such Wager). So (4) has significant force, maybe even more force than in the individual case. Though of course if utilitarianism is true, that force disappears.

Objections (5) and (6) disappear completely, though. For there need be nothing mercenary about the believers any more, and the promoter of theistic beliefs is being unselfish rather than mercenary. The social Pascal's Wager is very much a morally-based argument.

Objections (1) and (2) may not be changed very much. Though note that in the social context there is a hedging-of-the-bets strategy available for (2). Instead of promoting a particular brand of theism, one might instead fight atheism, leaving it to others to figure out which kind of theist they want to be. Hopefully at least some theists get right the brand of theism—while surely no atheist does.

I think the integrity objection is the most serious one. But that one largely disappears when instead of considering the argument for promoting theism, one considers the argument against promoting atheism. For while it could well be a lack of moral integrity to promote one-sided arguments, there is no lack of integrity in refraining from promoting one's beliefs when one thinks the promotion of these beliefs is too risky. For instance, suppose I am 99.99% sure that my new nuclear reactor design is safe. But 99.9999% is just not good enough for a nuclear reactor design! I therefore might choose not promote my belief about the safety of the design, even with the 99.9999% qualifier, because politicians and reporters who aren't good in reasoning about expected utilities might erroneously conclude not just that it's probably safe (which it probably is), but that it should be implemented. And the harms of that would be too great. Prudence might well require me to be silent about evidence in cases where the risks are asymmetrical, as in the nuclear reactor case where the harm of people coming to believe that it's safe when it's unsafe so greatly outweighs the harm of people coming to believe that it's unsafe when it's safe. But the case of theism is quite parallel.

Thus, consistent utilitarian atheists will promote theism. (Yes, I think that's a reductio of utilitarianism!) But even apart from utilitarianism, no atheist should promote atheism.

Conditional probability and probability comparisons

Suppose we want to say that event B is more likely than another A. What does that mean? A natural thing to say is that P(B)>P(A). But that doesn't fit our intuitions. For all measure zero sets then end up being equally likely. We can get a sharper comparison if we instead of starting with unconditional probabilities, we work with primitive conditional probabilities. A natural way that I've considered in the past is to say that A≤B if and only if P(A|A∪B)≤P(B|A∪B). This lets you compare tiny sets, like a single point to two points. But this approach has the intuitive disadvantage that then if we use uniform measure on [0,1], then [0,1]≤[0,1).

But there is a better way to generate a comparison from a conditional probability. Say that A≤B if and only if the two sets are identical or P(A−B|(A−B)∪(B−A))≤P(B−A|(A−B)∪(B−A)). It's not that hard (unless one is as sleepy as I am this morning) to show that this relation is reflexive, transitive and total—i.e., a weak order. Moreover, this weak order has the property that if A is a proper subset of B, then we're guaranteed to have strict inequality: A<B.

Update: A. Paul Pedersen informs me that De Finetti gives a definition equivalent to this on page 367 of the second volume of his probability book.

There is no regular approximately invariant finitely additive probability measure on all subsets of a cube or ball

For a totally ordered field K, say a hyperreal one, write x≈y (and say that they are approximately equal) provided that x−y is 0 or infinitesimal. A K-valued probability P defined for all subsets of Ω is said to be regular provided that P(A)>0 whenever A is non-empty. It is approximately rigid motion invariant provided that P(A)≈P(gA) for every rigid motion g and set A such that A∪gA⊆Ω. The following can be proved in Zermelo-Fraenkel (ZF) set theory without any Axiom of Choice:

Theorem 1. There is no totally ordered field K and a regular K-valued approximately rigid motion invariant finitely additive probability on all subsets of a ball or cube Ω.

If we delete "approximately", this follows from this.

The result follows from this post. Given such a regular measure we can define a preorder ≤ by letting A≤B if and only if P(A)≤P(B). By the Theorem from that post, it follows in ZF that Banach-Tarski is true. But Banach-Tarski implies that there is no approximately rigid motion invariant finitely additive probability on all subsets of a ball or cube.

(Why ball or cube? This saves me from having to worry about some edge effects given our definition of invariance.)

Another result, proved by similar methods:

Theorem 2. Let Ω be a subset of three-dimensional Euclidean space invariant under rotations about the origin 0. If K is a totally ordered field and P is a regular K-valued finitely additive probability on all subsets of Ω approximately invariant under rotations about the origin, then P({0})≈1.

Suppose now that we have a particle undergoing Brownian motion released at time t₀ at the origin, and then observed at time t₁. The probability of its being in some set at t₁ should be at least approximately invariant under rotations, and of course it is unacceptable to say that the probability that it is at the origin is approximately one—on the contrary, with approximately unit probability it is going to be away from the origin.

Update: Similar things hold for full-conditional probabilities, where approximate invariance is replaced with
invariance conditionally on the whole space (but there is no requirement of invariance conditionally on subsets of the space).

Interesting riddle concerning the Axiom of Choice

There is an interesting riddle over at MathOverflow concerning the Axiom of Choice.

Love of God is needed, not just love of neighbor

Suppose Jim believes that he has a long-lost elder brother, and in order to become the next Marquess of Winchester he strives to hunt down and murder that brother. The above is incompatible with Jim's being virtuous, but it is logically compatible with (though psychologically unlikely to coexist with) Jim's loving every human being, since Jim's belief might be false, and he might thus have no long-lost brother, and hence no failure of love for that brother. Thus, loving each human being does not entail being virtuous.

But loving God—that's a different matter. For if one loves God, then one is thereby disposed to love all that God has made. Thus, Jim, while he does not fail in love of any particular neighbor, does fail in love for God.

Theologians who say "God doesn't exist"

Some theologians say that God doesn't exist--God is beyond being. Here is one way to make sense of their claim: Non-relational claims with "God" as the subject term have the word "God" functioning like the "It" in "It's raining" (we should think of "It's raining" on this reading as a nullary predicate). In other words, what we are really expressing are subjectless claims. When the vulgar believer says "God doesn't exist", that's not literally true. What is literally true is something like: "It's Godding". And when the vulgar believer says "God is wise", what's literally true is: "It's Godding wisely" just as "The rain is intense" really means that it's raining intensely. Relational claims can be similarly handled. "The rain falls on me" is more precisely expressed as "It's raining on me", and "God creates the earth" is better said as "It's Godding creatively with respect to the earth." A theologian of this stripe can then talk with the vulgar, but she has a preferred explication of what is being said.

I think this story fails when we talk about love between God and humans. For love is essentially relational. An account of love that eliminates either the subject term or the object term is automatically not an account of love in the proper sense. (There is an extended sense in which someone might be said love a non-existent person. But I think it's more proper to say that she seems to love. If presentism and no-afterlife are true, then in this example, Sally does not love Fred—she only thinks she does.)

Loving without knowing

Fred is lost in the desert and dies. Without knowing this, Sally, his loving wife who is a presentist and thinks there is no afterlife spends weeks searching for Fred in the desert, in uncertainty whether Fred is still alive, despite great hardship and danger to her own life.

Observe that a presentist who disbelieves in an afterlife thinks that the dead are simply nonexistent. So Sally is not only uncertain whether Fred is alive, but she is uncertain whether Fred exists. Yet she acts out of love. Hence:

• It is possible to love someone while being unsure whether he exists.

(Of course, one might also think that this case points to there being something defective in believing in presentism or in doubting an afterlife. )

Better than perfect

A necessary and sufficient condition for a student to have perfect performance on a calculus exam is to correctly, perfectly clearly and with perfect elegance answer every question within the time allotted. But what if the student also includes her proof of the Riemann Zeta Conjecture on the last page? Hasn't the student done better than perfect?

Well, the student hasn't done something more perfect. But the student has taken her answers above and beyond the nature of a calculus exam. So, yes, while one cannot be more than perfect, one can go above nature. Perfection is not the same as maximality of value.

How did we come to justifiedly believe that there are three dimensions?

I think a really interesting philosophy of science project would be to ask how it is that we came to have the then-justified belief that there are three dimensions of space? (I don't know that that belief is still justified now given the serious possibility that String Theory is true.) Do any of my readers know anything on the intellectual history of the tridimensionality of space? I don't even know when it was first proposed. Euclid would be my guess.

A nominalist reduction

Suppose that there were only four possible properties: heat, cold, dryness and moistness. Then the Platonic-sounding sentences that trouble nominalists could have their Platonic commitments reduced away. For instance, van Inwagen set the challenge of how to get rid of the commitment to properties (or features) in:

1. Spiders and insects have a feature in common.

On our hypothesis of four properties, this is easy. We just replace the existential quantification by a disjunction over the four properties:

2. Spiders and insects are both hot, or spiders and insects are both cold, or spiders and insects are both dry, or spiders and insects are both moist.

And other sentences are handled similarly. Some, of course, turn into a mess. For instance,

3. All but one property are instantiated

becomes:

4. Something is hot and something is cold and something is dry but nothing is moist, or something is hot and something is cold and something is moist but nothing is dry, or something is hot and something is dry and something is moist but nothing is cold, or something is cold and something is dry and something is moist but nothing is hot.

Of course, this wouldn't satisfy Deep Platonists in the sense of this post, but that post gives reason not to be a Deep Platonist.

And of course there are more than four properties. But as long as there is a finite list of all the possible properties, the above solution works. But in fact the solution works even if the list is infinite, as long as (a) we can form infinite conjunctions (or infinite disjunctions—they are interdefinable by de Morgan) and (b) the list of properties does not vary between possible worlds. Fortunately in regard to (b), the default view among Platonists seems to be that properties are necessary beings.

Reverse Frankfurt cases

On standard Frankfurt cases, there is a counterfactual intervener who does nothing in the actual world, but who would prevent the action if one willed otherwise. I've been musing about reverse counterfactual interveners who do nothing in the actual world, but who would enable the action if one willed otherwise. For instance:

• Fred is sitting on the sofa watching The Good Guys. Unbeknownst to him, freak cosmic rays have just severed the nerve connections between his brain and his leg muscles. Fred knows the baby needs a change, but decides not to get up, and keeps on watching the show.

If that's the whole story, then:

1. Fred can't get up.
2. Fred is responsible for not trying to change the baby.
3. Fred is not responsible for his baby not being changed by him (since he can't change the baby).

But now add to the story:

• An alien monitoring Fred's thoughts would instantly reconnect the nerve connections as soon as Fred started trying to go change the baby.

The alien doesn't affect (2), of course. But does she affect (1) and (3)?

I have a hard time deciding whether Fred can get up with the alien in place. Consider:

• I don't try to run as fast as possible. But an alien is monitoring my thoughts, and were I to try to run as fast as possible, he would supercharge my muscles and the grippiness of my shoes and I'd run at Mach 3.

Can I run at Mach 3? There is something that I can do such that were I to do it, I would run at Mach 3. But maybe this doesn't make it be the case that I can run at Mach 3. Rather, maybe this just makes it be the case that I can do something that would make me able to run at Mach 3. After all, consider a very different case.

• I don't try to run as fast as possible. But an alien is monitoring my thoughts, and were I to try to run as fast as possible, he would make me able to speak Cantonese.

In this case, clearly I can't speak Cantonese, though there is something I can do such that were I to do it, I would become able to speak Cantonese. If this is like the Mach 3 case—and I am not completely sure of that—then in that case, I too can't run at Mach 3. And that suggests that even with the alien in place, Fred can't get up—though, again, I am not completely sure the Mach 3 case is like the Fred-and-alien case.

But perhaps the ability and responsibility don't line up. For I find it plausible that Fred is responsible for the baby not being changed by him in the case of the alien. After all, such double prevention things are not that unusual. To adapt Locke's example, you're at a party, and the host for security reasons locks the door but installs a doorman who will unlock the door who will open the door for anyone who wants to leave. It sure seems clear that if you stay at the party, you are responsible for that.

Maybe what happens is this. Assessment of outcome responsibility ("Is Fred responsible for the baby being unchanged by him?") tracks something like counterfactuals, while ability ("Can Fred get up?") tracks "internal features". The line between the two ways of tracking may not always be clear (I have some scepticism about how precisely defined counterfactuals are), but perhaps nothing of great moral significance rides on either one. For what matters morally for guilt and praiseworthiness is not what outcomes you are responsible for, but only what choices, what acts of will or failures to will, what tryings and failures to try, you are responsible for. Outcome responsibility does matter for the court system, especially but not only in civil cases, but that's mainly a matter of policy.

If that's right, then it does something interesting to some of the dialectics about alternate possibilities. For instance, Peter van Inwagen has argued that determinism and Frankfurt-style interveners would take away one's responsibility for certain outcomes. The compatibilist can embrace this conclusion. For the morally important question is about responsibility for one's will, not for outcomes. It could in principle be that we are responsible for no outcomes (if only because it could be that our acts of will have no outcomes), but we are responsible for our will.

But I don't know that this gets the compatibilist off the hook entirely. For something like an ability to try is important to assessing responsibility for a failure to try. And it is not clear that compatibilists have very good accounts of the ability to try.

Ordering subsets of the line and the Banach-Tarski paradox

Suppose we want to rank the subsets of the interval [0,1] by size. Consider this:

1. There is a total, transitive and reflexive ordering (i.e., a total preorder) ≤ of subsets of the interval [0,1] such that (a) if A⊆B, then A≤B and (b) if A and B are disjoint non-empty sets with A≤B, then A<A∪B,

where X<Y means that X≤Y but not conversely.

Condition (a) is intuitively correct for any notion of size comparison. Condition (b), then, is like a regularity condition in probability theory. Any regular probability is going to yield a comparison like in (1).

One can get such an ordering by applying Szpilrajn's Theorem to get a total ordering of the subsets of [0,1] that extends subset inclusion. Note, however, that Szpilrajn's Theorem uses a version of the Axiom of Choice. It's an interesting question whether one can have (1) without any Axiom of Choice.

Now, the Banach-Tarski Paradox—that a solid ball can be disassembled into a finite number of subsets that can be reassembled into two solid balls each of the same size as the original—also uses a version of the Axiom of Choice. It would be nice to have (1) while avoiding the Banach-Tarski Paradox. One might even think Bayesian epistemology requires that one be able to hold on to (1) while avoiding the Banach-Tarski Paradox, since the latter spells doom for rigid-motion invariant probabilities in a three-dimensional region. But without using any version of the Axiom of Choice one can prove:

Theorem. If (1) is true, then the Banach-Tarski Paradox holds.

This essentially follows from Note 1 in Pawlikowski's paper, together with the fact that the solid ball has the same cardinality as [0,1] (so an order on the subsets of [0,1] as in (1) transfers to an order on the subsets of the ball that satisfies (1)), and the following pleasant observation:

2. If ≤ is as in (1) and A₁,A₂,A₃,A₄ are non-empty sets, then if B=A₁∪A₂∪A₃∪A₄, then at most one of the A_i satisfies the condition B−A_i<A_i and at least one of the A_i satisfies the condition A_i<B−A_i.

Say that X~Y iff X≤Y and Y≤X. To prove (2), suppose first that A_i>B−A_i and j≠i. Then A_j⊆B−A_i. Moreover, A_i⊆B−A_j. Thus, B−A_j≥A_i>B−A_i≥A_j, and so we do not have A_j>B−A_j. Next, suppose A_i does not satisfy A_i≤B−A_i. Thus, by totality of ≤, we have A_i>B−A_i. But this can happen for at most one i. So there will be three distinct i such that A_i≤B−A_i. To obtain a contradiction, suppose all three inequalities are non-strict and suppose i and j are two of the three indices. Then A_i~B−A_i and A_j~B−A_j. Observe that A_j is a subset of B−A_i, and hence A_j≤A_i. By the same argument, A_i≤A_j, and so A_j~A_i. The same will go for the third index, so there are three indices i, j and k such that A_i~B−A_i~A_j~B−A_j~A_k~B−A_k. But B−A_k contains the union of A_i and A_j and so by (1) we have A_i<B−A_k, which is a contradiction.

Corollary 1. The claim that every partial order extends to a total order implies the Banach-Tarski Paradox.

And since it's well known that the Banach-Tarski Paradox cannot be proved in Zermelo-Fraenkel (ZF) set theory unless ZF is inconsistent:

Corollary 2. Claim (1) cannot be proved in ZF, unless ZF is inconsistent.

I have some thoughts on how one might perhaps be able to improve Corollary 1 to show the result under the weaker assumption that every set has a total order. (The existence of Lebesgue nonmeasurable sets can be proved from that.)

Philosophical implications: I think standard Bayesianism requires both (1) and the falsity of Banach-Tarski. So standard Bayesianism fails. Moreover, we get an argument for the possibility of incommensurability in decision theory. For if (1) is false, then there can be incommensurability (denying (1) requires affirming that there are some partially preordered sets that have no total preorder whose strict order relation extends the strict order relation of the original preorder), and if Banach-Tarski is true, there can be incommensurability (incommensurability in probabilities implies incommensurability in choices).

Introducing doppelgangers

Yesterday, I mentioned that one might reinterpret the quantifier symbols in a language so as to introduce doppelgangers: extra quasi-entities that just don't exist, but get to be talked about using standard quantifier inference rules. Here I want to give a bit more detail, and then offer a curious application to the philosophy of mind: an account of how a materialist could use a doppelganged reinterpretation of language to talk like a hard-core dualist. The application shows that it is philosophically crucial that we have a way to distinguish between real quantifiers and mere quasi-quantifiers (in the terminology of the previous post) if we want to distinguish between materialism and dualism.

Onward! Let L be a first-order language with identity. A model M for L will be a pair (D,P), where D is a non-empty set ("domain") and P is a set of subsets ("properties") of D. A doppel-interpretation of L is a pair (I,s) where I is a function from the names and predicates other than identity to D and P respectively and s is a function from names to the set {0,1} ("signature"). The signature function s tells us which name is attached to an ordinary object (0) and which to a doppelganger (1).

Now a substitution vector for a doppel-interpretation (I,s) will be a partial function v from the names and variables of L to D such that v(a)=I(a) whenever a is a name. A signature vector is a partial function f from the names and variables of L to D such that f(a)=s(a) whenever a is a name. If x is a variable and u is in D, then I will write v(x/u) for the substitution vector that agrees with v except that v(x/u)(x)=u. I.e., v(x/u) takes v and adds or changes the substitution of u for x. Likewise, if f is a signature vector, then s(x/n) agrees with s except that s(x/n)(x)=n for n in {0,1}.

We can now define the notion of a substitution and signature vector pair (v,f) doppel-satisfying a formula F in L under a doppel-interpretation (I,s). We begin with the normal Tarskian inductive stuff for truth-functional connectives:

• (v,f) doppel-satisfies F or G iff (v,f) doppel-satisfies F or (v,f) doppel-satisfies G
• (v,f) doppel-satisfies F and G iff (v,f) doppel-satisfies F and (v,f) doppel-satisfies G
• (v,f) doppel-satisfies ~F iff (v,f) doesn't doppel-satisfies F.

Now we need our quasi-quantifiers:

• (v,f) doppel-satisfies ∃xF iff for some u in D, (v(x/u),f(x/0)) doppel-satisfies F or for some u in D, (v(x/u),f(x/1)) doppel-satisfies F
• (v,f) doppel-satisfies ∀xF iff for every u in D, (v(x/u),f(x/0)) doppel-satisfies F and for every u in D, (v(x/u),f(x/1)) doppel-satisfies F.

Then we need atomic formulae:

• (v,f) doppel-satisfies h=k (where h and k are variables-or-names) iff v(h)=v(k) and f(h)=f(k)
• (v,f) doppel-satisfies Q(h₁,...,h_n) where Q is other than identity iff (I(h₁),...,I(h_n))∈I(Q).

And finally we can define doppel-truth: a sentence S of L is doppel-true provided that it is doppel-satisfied by every substitution and signature vector pair.

It is easy to see that if s_m is the usual sentence that asserts that there are m objects, and if D has n objects, then s_m is doppel-true if and only if m=2n. It is also easy to see that all the first order rules for quantifiers are valid for our quasi-quantifiers ∃ and ∀.

Now on to our fake dualism. We need a more complex doppelganging. Specifically, we need to divide our stock of predicates into the mental and non-mental predicates. Start with a materialist's first order language L that includes mental predicates (the materialist may think they are in some sense reducible). Add a predicate SoulOf(x,y) (we won't need to specify whether it's mental or not) which will count for us as neither mental nor non-mental. I will assume for simplicity that the mental predicates are all unary (e.g., "thinks that the sky is blue")—things get more complicated otherwise, but one can still produce the fake dualism. Now, instead of doppelganging all the objects, we only doppelgang the minded objects. Thus, our models will be triples (D,D_m,P) where D_m is a subset of D (the minded objects, in the intended interpretation). We say that a substitution and signature vector pair (v,f) is licit if and only if f(a)=1 implies v(a)∈D_m ("only members of D_m have doppelgangers"), and in all our definitions we only work with licit pairs. Moreover, our reinterpretations do not need to give any extension to the predicate SoulOf(x,y): that's handled in the semantics.

Finally, we modify doppel-satisfaction for quantifiers and predicates:

• (v,f) doppel-satisfies ∃xF iff for some u in D, (v(x/u),f(x/0)) doppel-satisfies F or for some u in D_m, (v(x/u),f(x/1)) doppel-satisfies F
• (v,f) doppel-satisfies ∀xF iff for every u in D, (v(x/u),f(x/0)) doppel-satisfies F and for every u in D_m, (v(x/u),f(x/1)) doppel-satisfies F.
• (v,f) doppel-satisfies h=k (where h and k are variables-or-names) iff v(h)=v(k) and f(h)=f(k)
• (v,f) doppel-satisfies Q(h₁,...,h_n) where Q is non-mental and other than identity iff (I(h₁),...,I(h_n))∈I(Q) and f(h₁)=...=f(h_n)=0
• (v,f) doppel-satisfies Q(h) where Q is mental iff I(h)∈I(Q) and f(h)=1
• (v,f) doppel-satisfies SoulOf(h,k) iff v(h)=v(k), f(h)=1 and f(k)=0.

And then we say we have doppel-truth of a sentence when every licit pair is satisfied.

The intended materialist doppel-interpretation (I,f) consists of the usual materialist interpretation I of the names and predicates other than Soul(x) and that gives to Soul(x) the extension of all the minded objects, sets D_m to be the set of minded objects, plus has a signature f such that f(n)=1 where n is a name of a minded object and otherwise f(n)=0.

Now let's speak an informal version of our doppelganged materialist language. Let M be any mental predicate and Q any non-mental one. Say that a soul is any x such that ∃y(SoulOf(x,y)). Suppose "Jill" is the name of a minded object. The following sentences will be doppel-true:

• Some objects have souls.
• Every object that has a soul is a non-soul.
• Only souls satisfy M.
• No soul satisfies Q.
• Jill is a soul.
• Jill does not satisfy Q.

The doppel-language is strongly dualist. Only the souls have mental properties predicated of them and only the non-souls have non-mental properties (or stand in non-mental relations). A materialist community could stipulate that henceforth their language bears this kind of doppel-interpretation. They could then talk like dualists. But they wouldn't be dualists.. Hence the doppelganged ∃ and ∀ aren't really quantifiers.

Assume materialism. If there are n material objects, including m minded objects, then in the doppelganged language it will be true to say something like "There are n+m objects." For every one of the minded objects has a doppelganger.

My favorite Aquinas quote

Hence we must say that the distinction and multitude of things come from the intention of the first agent, who is God. For He brought things into being in order that His goodness might be communicated to creatures, and be represented by them; and because His goodness could not be adequately represented by one creature alone, He produced many and diverse creatures, that what was wanting to one in the representation of the divine goodness might be supplied by another. For goodness, which in God is simple and uniform, in creatures is manifold and divided and hence the whole universe together participates the divine goodness more perfectly, and represents it better than any single creature whatever. (S.Th. I.47.1)

Quasi-quantifiers

In First Order Logic (FOL), there are three aspects to a quantifier:

• grammar: a quantifier attaches to a formula and generates a new formula binding one variable
• inference: we have the FOL universal and existential introduction and elimination rules
• semantics: the Tarskian definition of truth in a model treats quantifiers in a particular way with respect to a domain.

The first two aspects are normally lumped together under "syntactic aspects", but I think keeping them separate is important.

A quasi-quantifier, then, is something has the grammar and inferential structure of a quantifier, but may have different semantics. Every quantifier is also a quasi-quantifier. A quasi-quantifier that isn't a quantifier—i.e., that has aberrant semantics—will be a quantifier. Quasi-quantifiers can be of types, like "existential" or "universal", that correspond to those of quantifiers. One can have formal languages with existential and universal quasi-quantifiers. In fact, to an approximation English is a language with quasi-quantifiers: "there is" is a mere quasi-quantifier. I will argue for the possibility of mere quasi-quantifiers, connect the issue with fundamentality and then make my suggestion about English.

For any natural number n and quantifier E, let s_n(E) be the analogue of the FOL sentence using ∃ that asserts that there are n objects. For instance, s₂(E) is the sentence

• ExEy(x≠y&~Ez(z≠x&z≠y)).

A sufficient condition for E to be an existential mere quasi-quantifier is that E has the grammar and inferential rules of ∃ but is interpreted in such a way that s_n(E) is false in some model with a domain with n objects.

An uninteresting way to get an existential mere quasi-quantifier is by domain restriction. Restrict interpretations in such a way that names must all be in a subdomain of the model and quantifiers are restricted to the subdomain. A non-trivial quasi-quantifier is a mere quasi-quantifier that isn't just a restricted quantifier.

A sufficient condition for E to be an existential non-trivial quasi-quantifier is that E has the grammar and inferential rules of ∃ but is interpreted in such a way that s_n(E) is true in some model with a domain with fewer than n objects.

It isn't hard to generate languages with interpretations that make them have non-trivial quasi-quantifiers, though we will have to reinterpret the identity as well. For instance, it's not hard to generate a pair of existential and universal "doppelganging quantifiers"[note 1], that have the same inferential rules as the existential and universal quantifiers, but a sentence gets interpreted in a model as if each item in the model had a doppelganger, where a doppelganger of x stands in the same relations as x, except for identity (x=x but x's doppelganger isn't identical with x), and yet without adding any objects to the domain.[note 2]

Whether a quasi-quantifier is a quantifier depends on how that quasi-quantifier is treated in a Tarski-style definition of truth. Now, when we quasi-quantify also over non-fundamental objects, like holes and shadows, I think the Tarski-style definition of truth will give the truth conditions in terms of how the fundamental objects are (say, perforate or shadowing). This is going to be controversial, but, hey, this is only a blog post.

It follows immediately that when we quasi-quantify also over non-fundamental objects, we have a mere quasi-quantifier. Moreover, it's not going to be a restricted quantifier, so it's a non-trivial quasi-quantifier.

Now the English "there is" to an approximation is a quasi-quantifier. (It's not quite a quasi-quantifier, as the rules of inference for it will not quite match that of ∃ due to vagueness.) Moreover, it quasi-quantifies also over things like holes and defects and chairs, which are non-fundamental. Therefore, it is a mere quasi-quantifier. Nor is it just a restriction of a quantifier, so it is a non-trivial quasi-quantifier.

Once we see this, temptations to quantifier pluralism should be decreased. Of course, we have quasi-quantifier pluralism: There are quantifiers, there are doppelganging quasi-quantifiers, there are English quasi-quantifiers, there are mereological quasi-quantifiers, and so on. But only the first of these are quantifiers.

Now, in the formal examples, like of my doppelganging quantifiers, one can give a paraphrase of the quasi-quantifiers in terms of quantifiers: one just writes out the Tarski definition of truth for each sentence. But in natural language examples, the Tarski definition of truth is not going to be formally statable (at least not in any way tractable to us). And so there won't be a paraphrase of the quasi-quantifier sentences in quantified sentences. Quine won't like that. And what I said above about the Tarski definition when I characterized quasi-quantifiers won't be easy to say in the natural language case. There is much more work to be done here.

And of course just as there is no entity without identity, there is no quasi-quantifier without quasi-identity.

Two kinds of Platonism

There are two kinds of Platonism. Both hold that there are properties. But they differ as to the grounding relation that holds between predication and property possession (I will also assume that what goes for properties goes for relations, but sometimes formulate things just in terms of properties for simplicity). Both agree that if there is a property of Fness (there might not be if F is gerrymandered or negative, on sparse Platonisms), then x is F if and only if x instantiates Fness. Deep Platonism further affirms:

1. If there is a property of Fness, then the fact that x is F is grounded in the fact that x instantiates Fness.

Shallow Platonism denies (1). It is likely to instead affirm:

2. If there is a property of Fness, then the fact that x is F partly grounds or explains the fact that x instantiates Fness.

Deep Platonism faces two problems. The first is the Regress Problem. For if Deep Platonism is true, then "instantiates" seems non-gerrymandered and positive, and so it should correspond to a Platonic entity, the relation of instantiation. Then, the fact that x instantiates Fness will be grounded in the fact that x and Fness instantiate instantiation. But this leads to a vicious regress where each instantiation relation is grounded in the next.

The second is the Creation Problem. Everything that exists and is distinct from God is created by God. If the properties are all distinct from each other, then at most one is identical with God, and hence all but at most one property are created by God. But explanatorily prior to creating anything, will have multiple properties such as that he is able to do something and that he knows something. But how can he have those properties when there is at most one property at this point in the explanatory story?

Both problems have Deep Platonist solutions. For instance, one might say that "instantiates" is the unique non-gerrymandered and positive predicate that has no Platonic correspondent, or one might say that (1) has an exception in the case of instantiation. And one might say that God has at most one property, say divinity, and he is identical to that property. (But this, too, seems to lead to exceptions for (1), or perhaps an implausible view of what predicates correspond to properties. For there sure seem to be many other non-gerrymandered and positive predicates, like "is wise" and "is powerful", that apply to God.)

But Shallow Platonism has a particularly neat solution to both problems. There either is no regress, or if there is a regress, it is an unproblematic forward regress: because x is F, x and Fness instantiate instantiation, and because of that x, Fness and instantiation instantiate instantiation, and so on. Forward regresses are not at all problematic. And while it may be explanatorily prior to the creation of properties (or of all but one property) that God is wise, it is not explanatorily prior to the creation of properties that God instantiates wisdom.

Dominating reasons

Some things just aren't reasons for a choice. For instance, the fact that a portion of ice cream has an odd number of carbon atoms is by itself not a reason at all for eating the ice cream, and the fact that I find hot chocolate unpleasant is by itself not a reason to choose the hot chocolate. (The "by itself" qualifier is needed. I might have some instrumental reason for consuming an odd number of carbon atoms, and I might be ascetically training myself to consume what is unpleasant.)

Sometimes, however, something can be a reason for A without being a reason for A rather than B. For instance, that I enjoy hot chocolate to degree 100 is a reason to have hot chocolate. But if I enjoy ice cream to degree 150 on the very same scale, then my enjoying hot chocolate to degree 100 is not by itself a reason to have hot chocolate rather than ice cream. In the absence of other reasons, it would then make no rational sense to choose hot chocolate over ice cream, since my reason for hot chocolate is strictly dominated by my reason for ice cream.

At least roughly speaking:

• Reason R (not necessarily strictly) dominates reason S if and only if S is not at all a reason for choosing an action supported by S over an action supported by R.
• Reason R strictly dominates reason S if and only if R dominates S and S does not dominate R.

And of course reasons can be replaced by sets of reasons here. Then, Buridan's Ass cases are ones where the reasons for each action non-strictly dominate the reasons for the other.

Rational choice between A and B occurs only when one has reason to choose A over B and reason to choose B over A. Thus, rational choice between A and B occurs only when the reasons for neither option dominate the reasons for the other.

Definition: Reasons R and S are incommensurable if and only if neither dominates the other.

Thus, rational choice is possible only given sets of reasons that are incommensurable.

Hour of Code

I think computer programming should be taught from early grades, both in order to expand the mind and to be able to use the computing tools around us more effectively (it's no harder than cursive handwriting, and so much more useful!). And then I came across Hour of Code, which is an attempt to introduce kids to programming in an hour during Computer Science Education Week (Dec. 9-15). I hope to run an afterschool Hour of Code event at my kids' school for grades five and six.

I tested the "Write your first computer program" tutorial on my 11-year-old daughter, and she completed it in half an hour, so it seems right for children her age, though she wasn't deeply excited about it. (But it did frustrate my 8-year-old son.)

My daughter then went for the "Create a holiday card" activity with Scratch, and made an animated Christmas card. Scratch is an event-driven graphical programming environment for kids that reminds me a lot of Hypercard (which I still have a full version of on our Powerbook 190 laptop). Scratch has her hooked. I tried Scratch with her about two years ago, but the computer we were running it on was a bit too old and we didn't see the nice little intro they now have for Hour of Code, so it was frustrating. It helps, of course, that she's done some Mindstorms programming before.

Both of the activities were web-based. They're still in beta, but I highly recommend them for kids. There are lots of other activities there. And it's not too late to volunteer at your kids' school to run an Hour of Code activity for them.

Necessary coincidences

On standard naturalist views, neither the objective facts of mathematics and morality nor their grounds (e.g., Platonic entities, etc.) have any influence on how matter behaves and hence on how we think. This seems to imply that if our mathematical or moral beliefs happen to be true, that's just a coincidence. But merely coincidentally true belief isn't knowledge (maybe it's Gettiered knowledge). Now consider a response on which:

1. Of biological necessity, we have evolved through unguided natural selection to have mathematical or moral beliefs of type N.
2. Of metaphysical necessity, most mathematical or moral beliefs of type N are true.
3. Therefore, there is no coincidence here and nothing that calls out for further explanation.

(Erik Wielenberg offered a response of roughly this sort last week here.) The response presupposes that there cannot be coincidences between necessary truths that call out for further explanation.

But there can. There are two real numbers, x and y, between 0 and 1 with the following property. If you write them out in binary, divide up the bits into groups of eight, and then put the bits into ASCII code, then you actually find a lot of comprehensible text in each. In particular:

4. In x, there are infinitely many occurrences of "Consider the following proposition:", and each of them is followed by a well-formed arithmetical sentence (say, written in TeX) and a period. In fact, all possible arithmetical sentence thus occur in x.
5. In y, at exactly the same point as each "Consider the following proposition:" string occurs, there instead occurs "That's true" or "That's false."
6. Moreover, "That's true" occurs in y precisely when the proposition given in that place in x is true, and "That's false" occurs when the proposition is false.

But of course, it is necessary that x and y have the binary expansion they do.

Now, if we're given two such numbers x and y, the above is an apparent coincidence that calls out for explanation. And maybe an explanation can be given, say in terms of a selection effect: Perhaps the reason we're considering these two numbers is because a logically omniscient being exhibited them to us, and the being chose the two numbers for these remarkable properties. No surprise then!

But what if turned out that x=π and y=e satisfy (4)-(6)? Then we would consider the above coincidence truly remarkable. We would search for some deep mathematical reason for it. But suppose this search fizzled out and we came to conclude that although, of course, it is necessary that π and e have the properties (4)-(6), e.g., it being necessary that Fermat's Last Theorem occur at location 12848994949494888 in π (I assume it doesn't) and "That's true" in e at the same location and so on, mathematically this is just an incredibly unlikely coincidence. That would be a highly intellectually unsatisfying position. So unsatisfying that we would reach for a metaphysical explanation like Descartes' story about God having designed mathematics or a science fictional one like Carl Sagan's novel about aliens having embedded a message in π. We would have good reason to accept such an explanation if it were offered, and if we rejected there being such an explanation, we would have to say we have just a coincidence.

Thus, we can imagine cases of agreement between necessary mathematical facts which genuinely call out for explanation. And we can imagine concluding that although they call out for explanation, there is none, and hence we have a coincidence. Thus we can imagine a coincidence in the realm of necessary truth.

The Axiom of Choice in some claims about probabilities

I spent the last week trying to get clear on the logical interconnections between a number of results about probabilities that are relevant to formal epistemology and that use a version of the Axiom of Choice in proof, such as:

1. For every non-empty set Ω, there is an ordered field K and a K-valued probability function that assigns non-zero finitely additive probability to every non-empty subset of Ω.
2. For every non-empty set Ω, there is a full finitely additive conditional probability on Ω (i.e., a Popper function with all non-empty subsets normal).
3. The Banach-Tarski Paradox holds: one can decompose a three-dimensional ball into a finite number of pieces that can be moved around and made into two balls of the same size.
4. There are Lebesgue non-measurable sets in the unit interval [0,1].

All of these results require some version of the Axiom of Choice. It turns out that there is a very simple map of their logical interconnections in Zermelo-Fraenkel (ZF) set theory:

• BPI→(1)→(2)→(3)→(4),

where BPI is the Boolean Prime Ideal theorem, a weaker version of the Axiom of Choice.

The proof from BPI to (1) is standard--just let K be an ultrapower of the reals with an appropriate ultrafilter. That from (1) to (2) is almost immediate: just define the conditional probabilities via the ratio formula and take the standard part. Pawlikowski's proof of Banach-Tarski easily adapts to use (2) (officially, he uses Hahn-Banach). Finally, Foreman and Wehrung show in ZF that every subset of Rⁿ is Lebesgue measurable iff every subset of [0,1] is. But it follows from (2) that not every subset of R³ is Lebesgue measurable.

This has important consequences. Without the Axiom of Choice, one can prove that either (a) there are sets that have no regular probabilities no matter what ordered field is chosen for the values and no full conditional probabilities, or (b) the Banach-Tarski Paradox holds and hence there are no rigid-motion-invariant probabilities on regions of three-dimensional space big enough to hold a ball. And in either case, Bayesianism has a problem.

Various challenges for evolutionary psychology

I took my kids to an event tonight with Eric Wielenberg, and my 11-year-old daughter found herself puzzling about how our emotions could evolve. I tried to convince her that some aspects of our emotional lives that have plausible evolutionary explanations. But she came up with a number of challenges for evolutionary explanations that withstood my critical scrutiny. They are an interesting bunch:

1. The desire to do what is wrong
2. Happiness, in the sense of contentment—think of a cat lying down while being stroked (my example)
3. The drive to achieve things the hard way even when one can get them without effort—wanting the achievement of getting a meal by hunting even if one can get an equally delicious meal without much effort (her example)
4. Mercy towards weaker animals, even ones that we could eat or that could eventually harm us.

Note that while occasional pleasure can definitely contribute to fitness by rewarding behavior, the kind of contentment in (2) might actually be deleterious, by making us less active. One might think of (3) as a form of practicing, but that seems a stretch in a lot of cases. And (4) is particularly puzzling—it's not so hard to come with stories about a lot cases of mercy towards members of our own species, but mercy to other species seems a lot harder. One might try a malfunction story for (1), but my daughter thinks (1) is too widespread to be anything like a disease, and we surely don't want to say that (2)-(4) are faults. I suppose one could try to find a spandrel story about some of these, but I am not sure how convincing these will be and whether they will fit well with the fact that (2)-(4) seem to be components of our flourishing.

Of course (1) is puzzling on a theistic view (whether evolutionary or not), though perhaps the theist can give a view on which it's a distortion of a desire to imitate God, by desiring to be ultimately in charge. On the other hand, (2) and (4) have very nice theistic explanations.

The Axiom of Choice gives and takes away probabilities

Suppose you have assigned coherent (i.e., finitely additive) probabilities to a collection of options, but then you come upon a refinement of this collection of options, including more fine-grained ones. For instance, previously you had assigned probabilities to propositions about which individuals in a population had brown or non-brown eyes. But now you realize you should refine the non-brown-eye group into the blue, green and none of the above groups, as well as considering people's hair-color. It is intuitively plausible that if your initial probabilities about brown versus non-brown eyes were coherent, you should be able to come up with a coherent assignment of probabilities to the refined cases, e.g., by equally dividing up the probabilities of the non-brown eye category between the three newly recognized suboptions. One way to state the above in full generality is this:

1. Whenever P is a finitely additive probability assignment on an algebra F of subsets of a space Ω, and F is a subalgebra of a finer algebra G of subsets of Ω, then P can be extended to G.

Is this true? Well, the Axiom of Choice implies it is and I think this was first proved by Tarski. The Axiom of Choice gives probabilities (or at least implies that they exist).

But the Axiom of Choice also takes away probabilities. One famous case is that of nonmeasurable subsets of an interval, but that's about countably additive stuff, while I am right now talking of finitely additive stuff. One way it does this is by implying the Banach-Tarski paradox:

2. A solid three-dimensional ball can be decomposed into a finite number of subsets which can be moved rigidly to produce two balls of the same size as the original. It follows that no region in three dimensions that has the room to contain a ball has a (finitely additive) probability assignment on all its subsets that is invariant under rigid motions.

Now, for those who, like Bayesians, think epistemology is basically probability theory, (1) is going to be attractive but (2) is going to be paradoxical and repugnant. These thinkers may be tempted to give up the Axiom of Choice in order to deny (2). But I think they are likely to still want (1). And since (1) is known not to actually be equivalent to the Axiom but weaker, there might seem to be some hope.

Question: Can we coherently deny (2) and accept (1) in Zermelo-Fraenkel Set Theory (ZF) without Choice?

It turns out that the hope is vain. For:

Theorem. Claim (1) implies claim (2) in ZF.

For Luxemburg (1969) proved that (1) is equivalent to the Hahn-Banach Theorem, while Pawlikowski (1991) proved that the Hahn-Banach Theorem implies the Banach-Tarski Paradox.

So we cannot get away from this (at least not without even more radical revision to set theory). If we want probability existence results like (1), we must accept probability nonexistence results like (2).

Moral and perfect freedom

Say that moral freedom is the ability to choose between a morally permissible and a morally wrong action. Perfect freedom, on the other hand, is a freedom to choose between morally permissible actions, but with a perfect and infallible directedness at the good of the sort that God and the saints in heaven are said to enjoy.

Morriston (and others before him, like Quentin Smith, but Morriston's piece is particularly well developed) basically offers this dilemma: Either moral freedom is better than than perfect freedom or not. If moral freedom is better, then God has the less valuable kind of freedom, which seems incompatible with God's perfection. If moral freedom is not better, then God should have created beings with perfect freedom, since this way all the evils flowing from our misuse of moral freedom would have been prevented.

I want to make two points. First, the relevant question shouldn't be whether moral freedom is better than perfect freedom, but whether the action of creating beings with moral freedom is better than the action of creating beings with perfect freedom. An action can be better than another, even if its intended effect is no better. For instance, if I promised an editor a paper on modality, and I have the time for only one paper, the action of writing a paper on modality is better than the action of writing a paper on the Trinity, even if the effect of the latter action may be the better.

With this distinction in mind, one notices that there is a difference in value between God's creating a being that inevitably loves him back and his creating a being that gets to choose whether or not to love him back. Even if a being that inevitably loves him back is no better, God's action of inviting someone into communion with him very much has something very significant to be said for it that God's creating someone who will inevitably be in communion with him doesn't.

The second point is this. There is a value to loving someone by choice. Now when God and St Francis love each other, each loves the other by choice. Francis chooses to love God, while being able not to. But God likewise chooses to love Francis, while being able not to. Now you might say: "But doesn't God have to love everyone, given that he is love itself?" I agree (though I know some don't): necessarily, if Francis exists, God loves him. But Francis doesn't have to exist—Francis only exists because God chose to create him. Thus God has freedom whether to love Francis, a freedom he exhibits in choosing to create Francis, something he did not have to do.

Now there is a necessary asymmetry here. Since we cannot have a choice about whether God exists, and once God exists, there is the obligation to love him, our choice requires moral freedom: it is a choice between the good of love and the evil of not loving the supremely lovable God. But for God the choice whether to love Francis was at the same time a choice whether to create Francis. This choice does not require moral freedom, since it is not a choice between good and evil, but only good and good-or-neutral.

So on both sides, the relationship between God and Francis involves a freedom to love or not to love Francis. This freedom is valuable and God has it. But in Francis this freedom, of necessity, is moral freedom. So it is not that moral freedom is more valuable than perfect freedom. Rather, it is that in a creature, freedom whether to love God has to be an instance of moral freedom, while in God, freedom whether to love a creature is an instance of perfect freedom.

Objection: But doesn't Morriston's problem come back when we consider the doctrine of the Trinity? The Father and the Son do not choose to love each over not loving each other. The Son is not a creature, and so the Father does not choose to create the Son rather than the Father. Yet, surely, the intra-Trinitarian love is the most perfect kind of love. So wouldn't a creature that has to love God have a better kind of love than one that has a choice about it?

Response 1: A certain symmetry and equality in love are particularly valuable. In the Trinity, we have a symmetry: no Person of the Trinity has the freedom to fail to love another. But we automatically start off with God having a choice whether to be in a relationship of love with a creature, namely through his having a choice whether to create the creature. It makes for deeper equality and symmetry if the creature also has a choice about how to respond to God.

A love relationship that is chosen on one side but not on another is less valuable through the asymmetry. Imagine a woman who chose to have a baby had a drug that would ensure that the child would love her back. She had a choice, to some degree, whether to love the baby. But she refuses the child a choice about whether and how to reciprocate the relationship.

Response 2: To choose to love makes one intimately related to one's love. But in the Trinitarian case, there is an even deeper relation to love: God is identical with his love.

Manipulation, randomness and responsibility

Suppose you chose A over B, but that through minor changes in your circumstances, changes that at most slightly rationally affect the reasons for your decision and that do not intervene in your mental functioning, I could reliably control whether you chose A or whether you chose B. For instance, maybe I could reliably get you to choose B by being slightly louder in my request that you do A, and to choose A by being slightly quieter. In that case your choice is in effect random—the choice is controlled by features that from the point of view of your rational decision are random—and your responsibility slight.

Now suppose you are a friend of mine. To save my life, you would need to make a sacrifice. There is a spectrum of possible sacrifices. At the low end, you need to spend five minutes in my company (yes, it gets worse than that!). At the high end, you and everybody else you care about are tortured to death. With the required sacrifice being at the low end, of course you'd make the sacrifice for your friend. But with the required sacrifice being at the high end, of course you wouldn't. Now imagine a sequence of cases with greater and greater sacrifice. As the sacrifice gets too great, you wouldn't make it. Somewhere there is a critical point, a boundary between the levels of sacrifice you would undertake to save my life and ones you wouldn't. This critical point is where the reasons in favor of the sacrifice and those against it are balanced.

Speaking loosely, as the degree of required sacrifice increases, the probability of your making that sacrifice goes down. The "probability" here is something rough and frequentist, compatible with determinism. If determinism is true, however, in each precise setup around the critical point, there is a definite fact of the matter as to what you would do. And there are two possibilities about your character:

1. You have a neat and rational character, so that for all sacrifices below the critical level, you'd do it, and for all the sacrifices above the critical level, you wouldn't do it.
2. At around the critical value, whether you make the sacrifice or not comes to be determined not by the degree of sacrifice but by irrational factors—what shoes I'm wearing, how long ago you had lunch, etc.

I suspect that in most realistic cases we'd have (2). But on both options, we have effective randomness: your action can be controlled through minor changes in your environment that at most slightly affect your reasons. For instance, in option (1), where you are simply rationally going by the strength of the reasons, the slightest tipping of the scales will do the job—you'll undergo 747.848 minutes of torture but not 747.849. And in option (2), non-rational factors that have only a slight rational effect, or no rational effect, control your aciton. In both cases, your choice can be controlled. By the principle I started the post with, around the critical point you couldn't be very responsible.

But surely you would be very praiseworthy for undertaking a great sacrifice to save my life, especially around the critical point. That the sacrifice is so great that we're very near the point where the reasons are balanced does nothing to diminish your responsibility. If anything, it increases your praiseworthiness. Thus determinism is false.

This is not an argument for incompatibilism. I am not arguing here that responsible is incompatible with determinism. I am arguing that having full responsibility around the critical level is incompatible with determinism.

An interesting equivalent to the Hahn-Banach theorem

The Hahn-Banach Theorem (HB) cannot be proved without some version of the Axiom of Choice. (Technically, it's stronger than ZF but weaker than BPI.) A cool fact about HB is that it is sufficient for proving the existence of nonmeasurable sets and even for proving the Banach-Tarski paradox.

In 1969, Luxemburg proved that the Hahn-Banach theorem is equivalent to the claim that every boolean algebra has a (finitely additive) probability measure.

He also proved that the Hahn-Banach theorem is also equivalent to the following interesting claim:

1. For any set Ω and proper ideal N of subsets of Ω (i.e., N is closed under finite unions, any subset of a member of N is a member of N, but not every subset of Ω is in N), there is a (finitely additive) probability measure on all subsets of Ω that is zero on every member of N.

One direction can be proved by using the existence of a probability measure on the quotient boolean algebra 2^Ω/N via Hahn-Banach. The other direction follows (I don't know if Luxemburg did it this way) from this fact which can be proved without the axiom of choice:

2. Any boolean algebra is isomorphic to the quotient of a boolean algebra of sets.

Given the Axiom of Choice, this is a trivial consequence of the Stone Representation Theorem. Without the Axiom of Choice, (2) is a quick consequence of a theorem of Buskes, de Pagter and van Rooij. It can also be proved directly.[note 1]

(This is part of a general observation that some of the things that can be done with ultrafilters can also be done with filters, though the results may be weaker.)

Kaczor's review of One Body

Chris Kaczor has a review of my One Body over at Public Discourse.

Why faith in the testimony of others is loving: Notes towards a thoroughly ethical social epistemology

Loving someone has three aspects: the benevolent, the unitive and the appreciative. (I develop this early on in One Body.) Believing something and gaining knowledge on the testimony of another teaching involves all three aspects of love.

Appreciation: If I believe you on testimony, then I accept you as a person who speaks honestly and reasons well. It is a way of respecting your epistemic achievement. This does not mean that a failure to accept your testimony is always unappreciative. I may appreciate you, but have good reason to think that the information you have received is less complete than mine.

Union: Humans are social animals, and our sociality is partly constituted by our joint epistemic lives. To accept your testimony is to be united with you epistemically.

Benevolence: Excelling at our common life of learning from and teaching one another is a part of our flourishing. If I gain knowledge from you, you thereby flourish as my teacher. Thus by learning from you, I benefit not only myself as learner but I benefit you by making you a successful teacher.

We learn from John Paul II's philosophical anthropology that we are essentially givers and accepters of gifts. In giving, epistemically and otherwise, we are obviously benevolent, but also because it is the human nature to be givers, in grateful acceptance of a gift we benefit, unite with and affirm the giver, thereby expressing all three aspects of love.

A Moorean reason not to believe in an open future

Let A be the best valid argument that has been given for an open future. But I have really excellent reasons to think that it's true that my ears won't turn to diamond over the next hour, reasons clearly stronger than my reasons to think that all of A's premises are true. But if there is an open future, then it's not true that my ears won't turn into diamonds over the next hour (since that depends on indeterministic quantum phenomena). So I have on balance reason to think A is unsound.

Of course, there are other arguments for an open future. But I can pair each one with a fact about the future that I have reason to be a lot more confident in than the truth of the argument's premises.

Open theism and risk

We have many well-justified beliefs about how people will freely act. For instance, I have a well-justified belief that at most a minority of my readers will eat a whole unsweetened lemon today. Yet most of you can. (And maybe one or two of you will.) Notice that a fair amount of our historical knowledge is based on closely analogous judgments. When we engage in historical analysis we base ourselves on knowledge of how people freely act individually or en masse. We know that various historical events occurred because of what we know about how people who report historical events behave--given what we know about human character, we know the kinds of things they are likely to tell the truth about, the kinds of things they are likely to lie about and the kinds of things they are likely to be mistaken about. But it would be strange to claim knowledge about past human behavior and disclaim knowledge about future human behavior when exactly similar probabilistic regularities give us both.

But if open theism is true, then God cannot form such beliefs about the future. For open theists agree that God is essentially infallible in his beliefs: it is impossible for God to hold a false belief. But if God were in a habit of forming beliefs about how people will in fact act, then in at least some possible worlds, and probably in this one as well, God would have false beliefs—it may be 99.99% certain that I won't eat a whole unsweetened lemon today, but that just means that there is a 0.01% chance that I will.

So the open theist, in order to hold on to divine infallibility, must say that God keeps from having beliefs on evidence that does not guarantee truth. Why would God keep himself from having such beliefs, given that they seem so reasonable? Presumably to avoid the risk of being wrong about something.

But now notice that open theism has God take really great risks. According to open theism, in creating the world, God took the risk of all sorts of horrendous evils. The open theist God is not at all averse to taking great risks about creation. So why would he be so averse to taking risks with his beliefs?

The open theists who think that there are no facts about the future have an answer here. They will say that my belief that at most a minority of my readers will eat a whole unsweetened lemon today is certainly not true, since the fact alleged does not obtain, and hence that I shouldn't have this belief. Instead, I should have some probabilistic belief, like that present conditions have a strong tendency to result in the nonconsumption of these lemons. My argument here is not addressed to these revisionists.

The fixity of the past and laws

Suppose an ever-truthful dictator tells you that he is interested in the truth value of a proposition p solely about the laws of nature and how the world was like 1000 years ago. He tells you that in an hour he will infallibly find out whether p is true. If so, he will execute you. If not, he will let you go free.

Unless you have a time machine or a God who answers prayers before they are made is on your side, fatalism is surely the appropriate attitude. There is nothing you can do about whether you will be executed.

Next suppose that determinism is true. That shouldn't affect what was just said. Determinism does not create new supernatural powers to affect the past.

Now add that your identical twin is told by the dictator that he will be executed if and only if he scratches his head in an hour.

Finally suppose that your proposition p is the proposition that one thousand years ago the universe was such as to nomically determine that in an hour you will scratch your head, and suppose that the dictator's infallible method for finding out whether p is true is simply to see if you scratch your head. Then you are in the same boat as your twin! Each of you will be killed if and only if he scratches his head. Since fatalism is true for you, it's true for your twin. Thus he can't do anything about his execution, and in particular he has no freedom about scratching his head. And so compatibilism is false.

Powers and strivings

Neo-Aristotelian metaphysics of causation is centered around the notion of powers. I wonder, though, whether the term "power" isn't too passive to convey the notion. In my sexual ethics work, I have used the term "striving".

One's image of a power may be something like a match: it has the power to start a fire, but it is not doing anything until the triggering condition—heat from friction—is applied. On the other hand, one's imagine of a striving may be something like a bow pulled back, with the tension and compression in the bow's limbs actively exerting a force that is closely balanced by the tension in the archer's muscles. The striving in the bow is something active, as can be seen by the way the archer gets tired the longer the string is held back, and only needs the archer's resistance to end in order for it to be released. Or one's image of striving may be the archer slowly and effortfully (but perhaps with an appearance of effortlessness) pulling back the string.

It is clear that metaphysically the cases of the archer pulling back the string and the archer holding the string are alike. The only difference is that in the case of holding the strivings of the limbs and the archer are balanced while in the case of pulling back the strivings are imbalanced. I think that in the end the case of the match is also metaphysically alike. In my One Body book, I say in this connection that an army in readineness is not an idle army. Likewise, the match in readiness is not an idle match. The powers of the molecules in the match are in balance, holding each other back, actively striving against each other like the archer against the bow.

Even in God there is a constant striving. God intrinsically is pure act, says Aquinas, and the eternal Trinitarian relations of procession emphasize this. It is crucial not to think of the Trinitarian relations as something historical, in the way that human parenthood in non-ideal cases can be, but rather as continuing—the Father did not once and for all generate the Son at the beginning of time, but in the timeless eternity of God's dynamism always is begetting him, and the Holy Spirit eternally proceeds as the active love between the Father and the Son. And on the economic side, Leibniz talks of the divine ideas of the various worlds that God can create as competing with one another for God to actualize them. We shouldn't overemphasize that competitiveness, but some emphasis is helpful here. (And this competitiveness in the end fits better into an incommensurability picture of creation than Leibniz's optimalism.)

To a large degree everything I said here is metaphor. But good choice of metaphor guides good philosophizing.

Risk compensation

Rational decision theory predicts that decreasing the riskiness of a activity will tend to increase the prevalence and degree of risky activity. As paragliding is made safer, one expects more people to be engaged in paragliding and those who are engaged in it to do it more intensely. But of course increasing the prevalence and degree of behavior will tend to increase the prevalence of occurrences of the negative outcome that one was decreasing. This is the phenomenon of risk compensation: decreasing the risk of a negative outcome of an activity is to some degree—maybe sometimes completely—compensated for by an increase in the prevalence and degree of engagement in the risky activity. For instance, taxi drivers who have antilock brakes tend to follow the vehicle in front of them more closely.

Suppose that in some case the risk compensation to some safety measure is complete: i.e., the prevalence of the relevant negative outcome (say, crashes or fatalities) is unchanged, due to the compensating increase in the prevalence and degree of the risky behavior. One might think that at this point the safety measure was pointless.

Whether this conclusion is correct depends, however, on what values the risky behavior itself has when one brackets the risk in question. If the risky behavior has positive value when one brackets the risk, the safety measure does in fact achieve something good: an increase in prevalence and degree of valuable but risky behavior with no increase in negative outcome. Paragliding is (I assume) a pleasant way to enjoy the beauty of the earth and to stretch the limits of human ability. An increase in the prevalence of paragliding without an increase of negative outcomes is all to the good.

When the behavior is completely neutral, then the safety measure, however, is simply a waste given complete risk compensation.

Finally, if the risky behavior has negative value even when one brackets the particular risky outcome, then in the case where the risk compensation is complete, the safety measure is counterproductive. It does not decrease the negative outcome it is aimed at, but by increasing the prevalence of an otherwise unfortunate activity it on balance has a negative outcome. For instance, suppose bullfighting is an instance of immoral cruelty to animals. Then apart from the risks to the bullfighter, the activity has negative value: it harms the animal and damages the soul of the person. If a safety measure for prevention of goring then were compensated for by an increase in prevalence, the safety measure would have on balance a negative outcome: there would be no decrease in gorings but there would be an increase in immoral and harmful activity.

Moreover, in cases where the risky behavior has independently negative value, a safety measure can have on balance negative effect even when the risk compensation is quite modest. Suppose that (I am making up the numbers) gorings occur in 10% of bullfights and cruelty to bulls in 80%. Suppose, further, that cruelty to bulls is at least as bad as goring (since it not only harms the bull but more importantly it seriously damages the soul of the cruel person). Then a safety measure that decreases the probability of goring by a half but results in a modest 10% increase in the prevalence of bullfighting will have on balance negative effect. For suppose that previously there were 1000 bullfights, and hence 100 gorings and 800 instances of cruelty. Now there will be 1100 bullfights, and hence (at the new rate) 55 gorings and 880 instances of cruelty. We have prevented 45 gorings at the cost of 80 instances of cruelty, and that is not worth it.

Much of the public discussion of risk compensation and safety measures centers on sex, and particularly premarital sex. We should typically expect some behavioral change in the direction of risk compensation given a safety measure. If one thinks premarital sex to be itself typically valuable, then even given total risk compensation one will think the safety measure to be worthwhile. If one thinks premarital sex to be value-neutral, then as long as the risk compensation is incomplete (i.e., the decrease in the risks due to the safety measure is not balanced by the increase in prevalence), one will think the safety measure to be worthwhile (at least as long as the costs of the safety measure are not disproportionate). But if one thinks premarital sex to have negative moral value, then one may well think a safety measure to be counterproductive even if the risk compensation is incomplete—as in my imaginary bullfighting cases.

I think public discussion of things like condoms and sex education could be significantly improved if participants in the discussion were all open and clear about the fact that we should expect some degree of risk compensation—that's just decision theory[note 1]—and were mutually clear on what value they ascribe to the sexual activity itself, independently of the risks in question.

In these kinds of cases, it sounds very attractive to say: "Let's focus on what we all agree on. Being gored, getting AIDS and teen pregnancy are worth preventing." But a public policy focused successful at improving the outcomes we have consensus on can still be on balance harmful, as my (made up) bullfighting example shows.