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 wihout using any version of the Axiom of Choice I think 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.

Here, I say that X<Y iff X≤Y but not Y≤X. 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 possibiltiy 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.)