From properties to sets

If we have abundant properties in our ontology, do we need to posit a second kind of entities, the sets?

Properties are kind of like sets. If P is a property, write x∈P if and only if x has P. A whole bunch of the Zermelo-Fraenkel axioms then are quite plausible. But not all. The most glaring failure is extensionality. The property of being human and the property of being a member of a globally dominant primate species have the same instances, but are not the same property.

We can get extensionality by a little trick and an axiom. Assume the following Axiom of Choice for Properties:

1. If R is any symmetric and transitive relation, then there is a property P such that (a) if x has P, then x stands in R to itself, and (b) for all x if x stands in R to itself, there exists a unique y such that x stands in R to y and y has P.

Like the ordinary Axiom of Choice, this is a kind of principle of plenitude. Apply (1) to the relation C of coextensionality that holds between two properties if and only if they have the same instances. This generates a property S₁ that is had only by properties and is such that for any property P there exists exactly one property Q such that P and Q are coextensive and Q has S₁. In other words, S₁ selects a unique property coextensive with a given property.

To a first approximation, then, we can think of those entities that have S₁ as sets. Then every set is a property, but not every property is a set. We certainly have extensionality, with the usual restriction to allow for urelements (i.e., extensionality only applies to sets). All the other axioms of Zermelo-Fraenkel with urelements minus Separation, Foundation and Choice are pretty plausibly true (they follow from plausible analogues for properties on an abundant view of properties). We get Choice for sets for free from (1).

Unfortunately, we cannot have Separation, however. For the property S₁ is coextensive to some set U by our assumptions. And the members of U will just be the instances of S₁, i.e., all the sets. And so we have a universal set, and of course a universal set plus Separation implies Comprehension, and hence the Russell Paradox.

So matters aren't so easy. The Axiom of Foundation is also not so clear. Might there not be a self-instancing property?

Thus the above simple approach gives us too many sets. But there is a solution to this problem, and this is simply to postulate the following second axiom about properties:

2. There is a property S₂ of properties such that (a) concreteness has S₂, and (b) all the axioms of Zermelo-Fraenkel Set Theory with Urelements minus Extensionality are satisfied when we stipulate that (i) a set is anything that has S₂ and (ii) A∈B if and only if A is an instance of B.

This axiom is fairly plausible, I think.

Now suppose that S₁ is as before, and let S₂ be any property satisfying (2). Then let S be the conjunction of S₁ and S₂. It is easy to see that if we take our sets to be those properties that have S, we will have all of Zermelo-Fraenkel with Choice and Urelements (ZFCU). Or at least so it seems to me—I haven't written out formal proofs, and maybe I need some further plausible assumptions about what abundant properties are like.

Of course, we cannot expect S₁ and S₂ to be unique. So there will be multiple candidates for sets. That's fine with me.

The big question is whether (1) and (2) are true. But if the theoretical utility of positing sets is a reason to think sets exist, then theoretical utility plus parsimony plus the reasons to believe in properties are a reason to think (1) and (2) are true.

A disconnect between lay and philosophical pro-choicers

Without having done any scientific survey, I get the impression that philosophical pro-choicers tend to agree with philosophical pro-lifers on positive answers to the questions:

1. Does a fetus have basically the same intrinsic moral standing as a normal newborn baby?
2. Does the life of members of the human biological species begins at or around conception?

(In some cases, (1) will need to be qualified to: "fetus with brain states", and then the following discussion will need to be restricted to somewhat later abortions.) Of course, the pro-choice and pro-life philosophers disagree on the implications of these positive answers. Thus, pro-choice philosophers who give a positive answer to (1) will either say that killing a normal newborn is permissible or that it is wrong for reasons other than its intrinsic moral standing (e.g., the hurt to adults in our society). And pro-choice philosophers working on abortion tend to distinguish between us and members of our biological species, holding that we are constituted by and not identical with members of our biological species.

I also suspect, again without any scientific survey, that lay pro-choicers by and large answer (1) and (2) with "no". Moreover, I suspect that many of them think that (1) and (2) are crucial disputed questions in the discussion of abortion. In fact, it may be that quite a number of them think that abortion is permissible because the answers to (1) and (2) are negative and even accept the conditional:

3. If the answers to (1) and (2) are positive, then abortion is at least typically impermissible.

If so, then the position of lay pro-choicers is apt to be unstable. It is predicated at least in part on negative answers to (1) and (2), whereas the relevant experts—philosophers working on abortion, whether pro-choice or pro-life—tend to agree that the answers are positive.

Like I said, these are just anecdotal impressions. It would be valuable to have research on both lay and philosophical pro-choicers to see if these impressions are correct or not. Suppose it turns out that my anecdotal impressions turn out to be correct. Then the disconnect between lay and philosophical pro-choicers suggests that even if the philosophical debate is at a stalemate, there are ways for the social debate to move.

A funny discrete view of time

A number of my posts are exercises in philosophical imagination rather than serious philosophical theories. These exercises can have several benefits, including: (a) they're fun, (b) they expand the range of possibilities to think about and thus might contribute to a new and actually promising approach, and (c) they potentially contribute to philosophical humility by making us question whether the views that we take more seriously are actually better supported than these. This is one of those posts.

Suppose that time is discrete and made up of instants. However instead of saying that always some instant is present, we now allow for two possibilities. Sometimes an instant is present. But sometimes presently we are between instants. When an instant is present, there is a present moment. When an instant is not present, when we are between instants, there is a present interval, bounded by the last past instant and the first future instant.

Why posit that sometimes we are between instants? Because this lets us get out of Zeno's paradox of the arrow. Zeno notes that at no instant is the arrow moving, because at no instant does it occupy two places, and so the arrow never moves. But now that we have two possibilities, that of an instant being present and of an interval being present, we see that Zeno's inference from

1. At no instant is the arrow moving

to

2. The arrow never moves

uses the implicit assumption that we are always at an instant. But if sometimes instead of being at an instant we are between them, we are at an interval, then the inference fails. And indeed when instead of a present moment we have a present interval, we can say that the arrow really is moving in the present—it is in two places in the present, in one place in the last past instant and in another in the first future instant.

So we have positions when an instant is present and velocities when an interval is present.

Of course there are other ways out of the Zeno paradox of the arrow, the best of which is to adopt the at-at theory of motion. But it's nice to have other solutions besides the usual ones.

The traveling minds interpretation of indeterministic theories

I'm going to start by offering a simple way—likely not original, but even if so, not very widely discussed—of turning an indeterministic physical theory into a deterministic physical theory with an indeterministic dualist metaphysics. While I do not claim, and indeed rather doubt, that the result correctly describes our world, the availability of this theory has some rather interesting implications for the mind-body and free will and determinism debates.

Start with any indeterministic theory that we can diagram as a branching structure. The first diagram illustrates such a theory. The fat red line is how things go. The thin black dotted lines are how things might have gone but didn't. At each node, things might go one way or another, and presumably the theory specifies the transition probabilities—the chances of going into the different branches. The distinction between the selected branches and the unselected branches is that between the actual and the merely possible.

The Everett many-worlds interpretation of Quantum Mechanics then provides us with a way of making an indeterministic theory deterministic. We simply suppose that all the branches are selected. When we get to a node, the world splits, and so do we its observers. All the lines are now fat and red: they are all taken. There are some rather serious probabilistic problems with the Everett interpretation—it works best if the probabilities of each branch coming out of a node are equal, but in general we would not expect this to be true. Also, there are serious ethics problems, since we don't get to affect the overall lot of humankind—no matter which branch we ourselves take, there will be misery on some equally real branches and happiness on others, and we can do nothing about that.

To solve the probabilistic problems, people introduce the many-minds interpretation of the many-worlds interpretation. Each person has infinitely many minds. When we get to a branch point, each mind indeterministically "chooses" (i.e., is selected to) an outgoing branch according to the probabilities in the physics. Since there are infinitely many of these minds, at least in the case where there are finitely many branches coming out of a node we will expect each outgoing branch to get infinitely many of the minds going along it. So we're still splitting, and we still have the ethics problems since we don't get to affect the overall lot of humankind—or even of ourselves (no matter which branch we go on, infinitely many of our minds will be miserable and infinitely many will be happy).

But now I want to offer a traveling minds interpretation of the indeterministic theory. On the physical side, this interpretation is just like the many-worlds interpretation. It is a dualist interpretation like the many-minds one: we each have a non-physical mind. But there is only one mind per person, as per common sense, and minds never split. Moreover our minds are all stuck together: they always travel together. When we come to a branching point, the physical world splits just as on the many-worlds interpretation. But the minds now collectively travel together on one of the outgoing branches, with the probability of the minds taking a branch being given by the indeterministic theory.

In the diagram, the red lines indicate physical reality. So unlike in the original indeterministic theory, and like in the many worlds interpretation, all the branches are physically real. But the thick red lines and the filled-in nodes, indicate the observed branches, the ones with the minds. (Of course, if God exists, he observes all the branches, but here I am only talking of the embodied observers.) On the many-worlds interpretation, all the relevant branches were not only physically real, but also observed. Presumably, many of the unobserved branches have zombies: they have an underlying physical reality that is very much like the physical reality we observe, but there are no minds.

The traveling minds interpretation solves the probability problems. The minds can travel precisely according to the probabilities given by the physics. Traveling minds as generated in the above way will have exactly the same empirical predictions as the original indeterministic theory. (In particular, one can build traveling minds from a Copenhagen-style consciousness-causes-collapse interpretation of Quantum Mechanics, or a GRW-style interpretation.)

Traveling Minds helps a lot with the ethics problem that many-worlds and many-minds faced. For although physical reality is deterministically set, it is not set which part of physical reality is connected with the minds. We cannot affect what physical reality is like, but we can affect which part of physical reality we collectively experience. And that's all we need. Note that "we" here will include all the conscious animals as well: their minds are traveling as well. In fact, as a Thomist, I would be inclined to more generally make this a "traveling forms" theory. Thus the unselected branches not only have zombies, but they have physical arrangements like those of a tree, but it's not a tree but just an arrangement of fields or particles because it lacks metaphysical form. But in the following I won't assume this enhanced version of the theory.

Now while I don't endorse this theory or interpretation—I don't know if it can be made to fit with hylomorphic metaphysics—I do want to note that it opens an area of logical space that I think a lot of people haven't thought about.

Traveling minds is an epiphenomenalist theory (no mind-to-physics causation) with physical determinism, and is as compatible with the causal closure of the physical as any physicalist theory (it may be that physicalist theories themselves require a First Cause; if so, then so will the traveling minds theory). Nonetheless, it is a theory that allows for fairly robust alternate possibilities freedom. While you cannot affect what physical reality is like, you can affect what part of physical reality we collectively inhabit, and that's almost as good. We have a solution to the mind-to-world causation problem for dualism (not that I think it's an important problem metaphysically speaking).

I expect that I and other philosophers have incautiously said many things about things like epiphenomenalism, determinism and causal closure that the traveling minds theory provides a counterexample to. For instance, while traveling minds is a version of epiphenomenalism, it is largely untouched by the standard objections to epiphenomenalism. For instance, one of the major arguments against epiphenomenalism is that if minds make no causal difference, then I have no reason to think you have a mind, since your mind makes no impact on my observations. But this argument fails because it assumes incorrectly that the only way for your mind to make an impact on my observations is by affecting physical reality. But your mind can also make an impact on my observations by leaving physical reality unchanged, and simply affecting which part of physical reality we are all collectively hooked up to.

Particles

I used to worry for Aristotelian reasons about the particles making up my body. The worry went something like this: Elementary particles are fundamental entities. Fundamental entities are substances. But no substance has substances as parts. The last is, of course, a very controversial bit. However there are good Aristotelian reasons for it.

But I shouldn't have worried much. Elementary particles are not all that likely to be fundamental entities. Quantum mechanics, after all, allows all sorts of superpositions between different particles. But substances either simply exist or simply don't. In the superposition case, they don't simply exist. So they simply don't. But I would expect that the superposition case is more the rule than the exception (if only with small coefficients for all but one one state). I guess we could think that when the wavefunction is in a pure state with respect to the existence of a particle, the particle then pops into existence, and when the state becomes mixed, it pops right out. But notice that the physics behaves in much the same way when we have a pure state and when we have a mixed state that is to a very high approximation pure. So whatever explanatory role the particles play when they pop into existence can be played, it seems, by the wavefunction itself when the particles aren't around. This suggests that the wavefunction is the more explanatorily fundamental entity, not the particles. Of course, the above relies on denying the Bohmian interpretation of quantum mechanics. But it's enough, nonetheless, to establish that elementary particles aren't all that likely to be fundamental entities. And hence they aren't all that likely to be substances.

Of course, it may be that the things that are fundamental physical entities will turn out to be just as problematic for the Aristotelian as the particles were...

The simplest way to run an infinite fair lottery?

I've posted two ways to run an infinite fair lottery (this and this). There is also a very simple way. Just take infinitely many people and have them each independently toss an indeterministic fair coin. If you're lucky enough that exactly one person rolls heads, that's the winner. Otherwise, the lottery counts as a failure. The probability of failure is high—it's one—but nonetheless success should be causally possible. And if you succeed, you've got what is intuitively an infinite fair lottery.

My earlier thought experiments requires a version of the Axiom of Choice. This version doesn't, but the earlier ones has the merit of working always or almost always. However, for the purposes of generating paradoxes and supporting causal finitism this version might be good enough.

A note to fellow mathematicians: Any mathematician reading this and some of my other posts on infinite fair lotteries is apt to be frustrated. There is a lot that isn't rigorous here. But I'm not doing mathematics. One can perhaps best think of what I'm doing as a physicsy thought experiment. When I think of independent indeterministic coin flips, take these as actual causally-independent physical processes, e.g., each indeterministic coin flip happening in a different island universe of an infinite multiverse. I am fully aware, for instance, that the stuff I say in this post isn't fully modeled by the standard Kolmogorovian probability theory. For instance, an infinite sequence of i.i.d.r.v.'s X_n with P(X_n=1)=P(X_n=0)=1/2 need not have any possible state such that exactly one of the variables is 1, depending on how the i.i.d.r.v.'s are constructed. That's an artifact of the fact that probabilistic independence as normally defined is not a sufficient model of genuine causal independence (see here). I am also assuming that permutation symmetries in the space of coin flips persist even when we consider nonmeasurable or null sets. Again that's going beyond the mathematics, but justified as a physicsy thought experiment. If we put each coin flip in a relevantly similar separate universe of a multiverse, then of course everything should be intuitively invariant under permutations of the coins. Probabilities understood vaguely as measures of rational believability go beyond the mathematical theory of probability.

Antipresentism

Presentists think that the past and future are unreal but the present is real. I was going to do a tongue-in-cheek post about an opposed view where we have the past and future but no present. But as I thought about it, the position grew a little on me philosophically, at some expense of the tongueincheekness. Still, please take all I say below in good fun. If you get a plausible philosophical view out of it, that's great, but it's really just an exercise in philosophical imagination.

One way to think about antipresentism is to imagine the eternalist's four-dimensional universe, but then to remove one slice from it. Thus, we might have 1:59 pm and 2:01 pm, but no 2:00 pm. Put that way, the view isn't particularly attractive. Still, I do wonder why it would be more unattractive to remove just one time slice than to remove everything but that one time slice as the presentist does. It would, of course, be weird for the antipresentist to say that events first exist in the future, then pop out of existence just as one would have thought that they would come to be present, and then pop back into existence in the past. But perhaps no weirder than events coming out of nothing and going back into nothing, as on presentism. This way to think about antipresentism makes it a species of the A-theory.

But the antipresentisms I want to think about are ones that might be compatible with the B-theory. Start with the famous puzzles of Zeno and Augustine about the now. Augustine worried about the infinite thinness of the now. Zeno on the other hand worried about the fact that there are no processes in the now; there is no change in the now since within a single moment all is still.

One way of taking these ideas seriously is to see the present as an imaginary dividing line between the past and the future. There is in fact no dividing line: there is just the past and the future. (I think Joseph Diekemper's work inspired this thought.)

We might, for instance, instead of thinking of times as instants think of the basic entities as temporally extended events or time intervals, not made out of instantaneous events or moments. An event or interval might be past, or it might be future, or—like the writing of this post—it might be both past and future. (Thus, "past" and "future" is taken weakly: "at least partly past" and "at least partly future".) Some events or time intervals have the special property of being both past and future. We can stipulate that those events or time intervals are present. But they aren't real because they are present. They're just lucky enough to have two holds on reality: they are past and they are present. (In this framework, the presentist's claim that only present events are real sounds very strange. For why should reality require both pastness and futurity—why wouldn't one be enough?) There are no events or time intervals that are solely present.

There is a natural weakly-earlier-than relation e on events. If we had instants of time, we would say that EeF if and only if some time at which E happens is earlier than some time at which F happens. But that's just to aid intuition. Because there are no instantaneous events, every event is weakly earlier than itself: e is reflexive. It is not transitive, however. The antipresentist theory I am sketching takes e to be primitive. There is also a symmetric temporal overlap relation o that can be defined in terms of e: EoF if and only if EeF and FeE.

If we like, we can now introduce abstract times. Maybe we can say that an abstract time is a maximal pairwise overlapping set of time intervals (or of events, if we prefer). We can say that t₁ is earlier than t₂ provided that some element of t₁ is strictly earlier than some element of t₂ (where E is strictly earlier than F provided EeF but not FeE). I haven't checked what formal properties this satisfies—I need to get ready for class now (!).

How to make an infinite fair lottery out of infinitely many coin flips

This is a technical post arising from a question Rob Koons asked me.

An infinite sequence of fair and independent coin flips determines a sequence of zeroes and ones (e.g., zero = tails, one = heads). Let Ω be the set of all infinite sequences of zero/one sequences, equipped with the probability measure P corresponding to the fair and independent coin flips.

Notice an invariance property capturing at least part of the independence and fairness assumption. If ρ_n is the operation of flipping the nth element in the sequence, and ρ_nA for a subset A of Ω is the set obtained by applying ρ_n to every sequence in A, then P(ρ_nA)=P(A) whenever A is measurable. Moreover, intuition extends this idea beyond the measurable sets: A and ρ_nA are always going to be probabilistically on par.

Let Ω₀ be the subset of Ω consisting of those sequences that have only finitely many ones in them. There is a natural one-to-one correspondence between Ω₀ and the natural numbers N. Suppose a=(a₀,a₁,...,a_k,0,0,0,...) is a member of Ω₀. Then let N(a) be the natural number whose binary digits are a_k...a₁a₀. Conversely, given a natural number n with binary digits a_k...a₁a₀, let n* be the sequence (a₀,a₁,...,a_k,0,0,0,...) in Ω₀. Thus, we can interpret the members of Ω₀ as binary numbers written least significant digit first.

For any members a and b of Ω, write a#b for the sequence whose nth element is the sum modulo 2 (xor) of the nth elements of a and b. For a subset B of Ω, let a#B = { a#b : b∈B }. We can think of a#B as a twist of B by a. If a is in Ω₀, I will call it a finite twist. Any finite twist can be written as a finite sequence of flips ρ_n, where the positions n correspond to the non-zero digits in the sequence we twist by. Thus, if A is measurable, a finite twist of it will have the same probability as A does, and even if A is not measurable, a finite twist will be intuitively equivalent to A.

Say that a~b if and only if a and b differ in only finitely many places. Thus, a~b if and only if a#b is a member of Ω₀. This is an equivalence relation. By the Axiom of Choice, there is a set A₀ such that for every b in Ω, there is a unique a in A₀ with a~b. (Thus, A₀ contains exactly one member of each equivalence class.) For any natural number n other than 0, let A_n=n*#A₀ and it's easy to check that this equation holds for n=0 as well.

It's easy to see that the A_n are disjoint and their union is all of Ω. They are disjoint because if a is in n*#A₀ and m*#A₀, then a=n*#b and a=m*#c for b and c in A₀. It follows that b~c. But A₀ contains only one member from each equivalence class, so b=c, and so n*#b=m*#b, from which it obviously follows that n*=m* and so n=m. Their union is all of Ω, because if b is in Ω, and a is the unique member of A₀ such that a~b, then N(a#b)*#a=(a#b)#a=b (by obvious properties of addition modulo 2), and so b is a member of A_N(a#b).

But all the A_n are going to be intuitively probabilistically on par: they are each a finite twist of A₀.

Our lottery is now obvious. Given a random sequence of coin flips, we take its representation a in Ω and choose the unique number n such that a is in A_n.

This is really the Vitali-set construction applied directly to sequences of coin flips. Note that along the way we basically showed that Ω has nonmeasurable subsets. For the sets A_n cannot be measurable with respect to P, since they would all have equal probability, and so by countable additivity they would have to have probability zero, which would violate the total probability axiom.

The construction in this post is more complicated than the one here, I guess, but it has the advantage that it always works, while that construction only worked with probability 1.

A divine command and an open future

I'm piling on to the argument here.

Suppose God creates Adam and Eve, and gives them eternal life. He then commands them that:

1. They freely pray for at least a minute on each of the infinitely many Sabbaths starting with day t₇ (the day after their creation).

This seems a reasonable command. But it is unreasonable to command something that the agent cannot ever make true. And on open future views, it is impossible for (1) ever to be true. For at any time, (1) depends on future free choices. So on open future views, the command (1) is unreasonable. And that's a problem for open future views.

Yet another infinite population problem

There are infinitely many people in existence, unable to communicate with one another. An angel makes it known to all that if, and only if, infinitely many of them make some minor sacrifice, he will give them all a great benefit far outweighing the sacrifice. (Maybe the minor sacrifice is the payment of a dollar and the great benefit is eternal bliss for all of them.) You are one of the people.

It seems you can reason: We are making our decisions independently. Either infinitely many people other than me make the sacrifice or not. If they do, then there is no gain for anyone to my making it—we get the benefit anyway, and I unnecessarily make the sacrifice. If they don't, then there is no gain for anyone to my making it—we don't get the benefit even if I do, so why should I make the sacrifice?

If consequentialism is right, this reasoning seems exactly right. Yet one better hope that it's not the case that everyone reasons like this.

The case reminds me of both the Newcomb paradox—though without the need for prediction—and the Prisoner's Dilemma. Like in the case of the Prisoner's Dilemma, it sounds like the problem is with selfishness and freeriding. But perhaps unlike in the case of the Prisoner's Dilemma, the problem really isn't about selfishness.

For suppose that the infinitely many people each occupy a different room of Hilbert's Hotel (numbered 1,2,3,...). Instead of being asked to make a sacrifice oneself, however, one is asked to agree to the imposition of a small inconvenience on the person in the next room. It seems quite unselfish to reason: My decision doesn't affect anyone else's (I so suppose—so the inconveniences are only imposed after all the decisions have been made). Either infinitely many people other than me will agree or not. If so, then we get the benefit, and it is pointless to impose the inconvenience on my neighbor. If not, then we don't get the benefit, and it is pointless to add to this loss the inconvenience to my neighbor.

Perhaps, though, the right way to think is this: If I agree—either in the original or the modified case—then my action partly constitutes the a good collective (though not joint) action. If I don't agree, then my action runs a risk of partly constituting a bad collective (though not joint) action. And I have good reason to be on the side of the angels. But the paradoxicality doesn't evaporate.

I suspect this case, or one very close to it, is in the literature.

Aristotelian propositions, promises and an open future

Aristotelian propositions are "tensed propositions" that are supposed to be able to change their truth value. If I say "It is sunny", this is supposed to express an Aristotelian proposition p such that p is true today, but p was false on cloudy days.

Now, a necessary condition for me to have fulfilled a promise is that

1. the proposition that was the object of the promise is true.

Suppose yesterday—i.e., on Sunday—I promised:

2. Tomorrow, I will do a blog post on Aristotelian propositions.

And I do make such a post today, i.e., on Monday, but I won't make another one on Tuesday. If the propositions expressed by tensed sentences are Aristotelian, then I have not fulfilled my promise. For the tensed proposition expressed by (2) is not true.

So tensed sentences don't express Aristotelian propositions, it seems. Rather, the proposition that yesterday I expressed with (2) is different from the proposition that would have been expressed with (2) today. The proposition that I expressed with (3) yesterday is "tenseless".

The advocate of Aristotelian propositions does have a way out. She can modify the condition (1) for promise fulfillment to:

3. the proposition that was the object of the promise was true at the time of the promise.

Now, there is no difficulty. The Aristotelian proposition that would have been expressed by (2) was true yesterday (since today I do make such a blog post) but isn't true today (since tomorrow I won't—I hope!).

But note that the advocates of an open future cannot go for (3). For on their view, the proposition that was the object of the promise wasn't true when I made the promise. Thus, there is a tension between holding that tensed sentences express Aristotelian propositions and accepting an Open Future. But a lot of Open Futurists do just that.

This is not an insoluble difficulty. One can, for instance, suppose an operator B_y that acts on an Aristotelian proposition and "shifts it backward by y. Thus, B_{1 day} applied to the Aristotelian proposition that tomorrow I will do a blog post on Aristotelian propositions is the Aristotelian proposition that today I do such a blog post. Then we replace condition (1) with:

4. I fulfill at t₂ a promise I made at t₁ only if B_y(p) is true at t₂, where y=t₂−t₁ and p is the object of the promise I made at t₁.

Still, it's weird, isn't it, that I fulfill a promise by bringing about something other than what I promised?

Propositions that never become true but are probable

According to open future views, the proposition that in 2015 a fair and indeterministic coin lands heads has some probability but is not true. However, that proposition is apt to become true in 2015. So the probability of the proposition isn't the same as the probability of the proposition being true, since it's certainly not true now, but might well become true in 2015.

So far so good (or bad). Suppose God promises you that from 2015 onward, every year, a fair and indeterministic coin will be tossed. Now let Q be the proposition that there are infinitely many years after 2014 during each of which a fair and indeterministic coin lands heads [I screwed up in the original formulation of Q and wrote "that every year from 2015 onward, ad infinitum, a fair and indeterministic coin lands heads"; Alan Rhoda's response targets my screwup; see my response to him below]. Now note that on open future views Q can never possibly become true. For on any date, the proposition requires for its truth that there will be infinitely many fair and indeterministic heads results still past that date, and on open future views a proposition that requires an undetermined future event won't be true.

So, open future views have to say that it's impossible for Q to ever be true. But a proposition such that it's impossible for it ever to be true should get probability zero. But the probability that of the infinitely many coin tosses, infinitely many will be heads is 1 according to classical probability theory. So open future views should be rejected.

Here's another argument in the same vein. Suppose I know I will have an eternal afterlife, and I promise you that I will freely pray for you every day, ad infinitum, starting November 1, 2014. On open future views, the object of my promise is a proposition that can never be true. But it's clearly a bad thing to promise something that can never be true. Yet what I promised wasn't a bad thing to promise. So open future views are false.

One might even have the direct intuition that one could keep the promise. That intuition is incompatible with open future views.

Yet yet another probability paradox

Start with a set M of countably infinitely many people, and a set D of countably infinitely many fair dice. Suppose that there are no natural orderings on the set D, and that each person in M has exactly one of the dice in D assigned to her. (Or if you prefer, these are sets of unique names of people and coins respectively.) You are a person in M, and you know what all the members of D are but have no information whatsoever on which member of D is yours. Now all the dice are simultaneously and independently tossed. Obviously, your probability that your die showed sixes is 1/6.

Then the set of all the dice that landed sixes is revealed to you. Call the revealed set D₆.

Suppose—this will be no surprise, as it had probability one—that the set of six-landing dice is infinite and the set of non-six-landing dice is infinite as well. Before it was revealed to you which dice landed sixes, your probability that your die yielded a six was 1/6. Did that probability change after you learned which set was the set of dice that landed sixes?

There are three options:

1. No, it didn't change at all—it stayed at 1/6.
2. Yes, it changed to an undefined value.
3. Yes, it changed to some other defined value.

To choose between the options, observe first that your current probability that your die landed six must now be exactly the same as the probability that your die is a member of D₆. But the fact that D₆ is in fact the set of the six-showing dice carries no information as whether your die is in D₆. Since all the dice are independent and fair, learning which dice landed sixes is completely irrelevant to finding your die. So whatever probability you assign to your die being among the members of D₆ after the revelation must be the same as the probability you assigned to it before the revelation.

So, if we choose option (1), then already before you found out that the double-six rollers were the members of D₆, you would have already assigned probability 1/6 to your die being in D₆. But there was no natural ordering on the set D of dice, so the set D₆ will be epistemically on par with its complement W−W₆. Both are simply countably infinite sets with countably infinite complements, and we can easily define an isomorphism of D onto itself that swaps the two sets. So if prior to learning the dice results you assigned 1/6 to your die being in D, you should have equally assigned 1/6 to your die being in D−D₆. But that's incoherent, since it's a given that the die is in D or D−D₆ but 1/6+1/6=1/3<1. So it seems that (1) is not an option.

That leaves (2) and (3). But those options are very strange. They imply that in such infinite die rolling scenarios, more data can always destroy your reasonable initial probability assignments.

Now, you might think that the above scenario only works when you don't know which die is yours, and that's kind of a strange scenario. But one can modify the scenario to work even when you do know which die is yours, but there is some other unique feature you don't know about your die, say, which of infinitely many (metaphysically) possible exotic particles is hidden inside the die, which of infinitely many angels has your die as a personal favorite, or what an independent sequence of rolls of the die yielded. Then the set D will be set of these unique features, and D₆ will be the set of these features among the dice that landed six.

Yet another probability paradox

You know for sure that infinitely many people, including yourself, each are independently tossing fair coins. You don't see your coin's result. But then you learn for sure something amazing: only finitely many of the coins came up heads. This is extremely unlikely—indeed, by the Law of Large Numbers it has zero probability—but it seems nonetheless possible. What probability should you now assign to your coin being heads?

Intuition: Very small, maybe zero, maybe infinitesimal.

Here's an argument, however, that you should stick to your guns and continue to assign 1/2. Let F be the proposition that only finitely many of the coins landed heads. Let G be the proposition that of the coins other than yours, only finitely many of the coins landed heads. Learning G does not affect your probability that your coin landed heads. The coins are all independent, so no information about the other tosses tells you about yours. But, now, necessarily (given the setup that you toss only one coin) F is true if and only if G is true. For your coin won't make the difference between infinitely and finitely many heads. So learning F does not affect your probability that your coin landed heads.

To make sticking to your guns even more amazing, note that this works for any infinity of people, even a very high uncountable infinity. Wow!

Scoring rules and epistemic rationality

Scoring rules measures the inaccuracy of one's credences. Roughly, when p is true, and one assigns credence r to p, then a scoring rule measures the distance between r and 1, while when p is false, the scoring rule measures the distance between r and 0. The smaller the score, the better.

Some scoring rules are better than others. Let's suppose some scoring rules are right. Then this thesis seems to be implicit in some applications of scoring rules (e.g., here):

1. If S is the right scoring rule, then a credence-assignment policy is epistemically rational only if following the policy minimizes expected total or average S-scores.

(And there will be a debate about whether we should have "total" or "average"—see link.)

But (1) is false. Here's a simple counterexample that works for most reasonable scoring rules. Consider a situation like this: A fair coin is flipped. If you assign credence 0.51 to heads, a mindreader who knows your credence assignments will immediately reveal to you how the coin landed. Otherwise, you will never have any information on how the coin landed.

Obviously, the epistemically rational thing to do is to assign 0.5 to heads. But this leads to higher expected total and average scores on most reasonable scoring rules. For if you assign 0.51, then once the mindreader tells you how the coin landed, you will update your credence to be very close to 0 or 1, and your score will be very low. And the only cost of this scenario is the slight inoptimality from briefly having score 0.51 instead of the optimal score of 0.5. So the epistemically rational policy for dealing with situations like this, namely assigning 0.5, does less well in expected scores than the epistemically irrational policy of assigning 0.51.

The case may seem farfetched. But there are real-life cases that may be similar. It may be that for psychological reasons when you are a bit more sure, or a bit less sure (depending on your character and the thesis), of a thesis than rationality calls for, you will be better able to investigate whether the thesis is true. Thus it may be better for your long term epistemic score that you do what is epistemically irrational.