Should the A-theorist talk of tensed worlds?

For this post, suppose that an A-theory of time is true, so there is an absolute present. If we think of possible worlds as fully encoding how things can be so that:

1. A proposition p is possible if and only if p holds at some world,

then we live in different possible worlds at different times. For today a Friday is absolutely present and tomorrow a Saturday is absolutely present, and so how things are is different between today and tomorrow (or, in terms of propositions, that it’s Saturday is false but possible, so there must be a world where it’s true). In other words, given (1), the A-theorist is forced to think of worlds as tensed, as centered on a time.

But there is something a little counterintuitive about us living in different worlds at different times.

However, the A-theorist can avoid the counterintuitive conclusion by limiting truth at worlds to propositions that cannot change their truth value. The most straightforward way of doing that is to say:

2. Only propositions whose truth value cannot change hold at worlds

and restrict (1) to such propositions.

This, however, requires the rejection of the following plausible claim:

3. If (p or q) is true at a world w then p is true at w or q is true at w.

For the disjunction that it’s Friday or it’s not Friday is true at some world, since it’s a proposition that can’t change truth value, but neither disjunct can be true at a world by (2).

Alternately, we might limit the propositions true at a world to those expressible in B-language. But if our A-theorist is a presentist, then this still leads to a rejection of (3). For on presentism, the fundamental quantifiers quantify over present things, and the quantifiers of B-language are defined in terms of them. In particular, the B-language statement “There exist (tenselessly) dinosaurs” is to be understood as the disjunction “There existed, exist or will exist dinosaurs.” But if we have (3), then worlds will have to be tensed, because different disjuncts of “There existed, exist or will exist dinosaurs” will hold at different times. A similar issue comes up for growing block.

So on the most popular A-theories (presentism and growing block), we have to either allow that we inhabit different worlds at different times or deny (3). I think the better move is to allow that we inhabit different worlds at different times.

A pre-established harmony with genuine mind-world causation

Molinists have the ability to give a distinctive pre-established harmony account of how the exceptionless truth of deterministic laws of nature could be made compatible with libertarianism.

Here is the story. God considers possible worlds where dualistic agents have the causal power to miraculously contradict the physical laws in their free choices, but where outside of exercises of free will, there are elegant deterministic mathematical laws governing the world. Call such worlds Candidate Worlds.

God then narrows his consideration to Finalist Worlds, which are Candidate Worlds that are feasible—i.e., compatible with the Molinist conditionals—and where as it happens the agents’ free choices accord with the elegant deterministic mathematical laws that govern the rest of the world.

And then God wisely and prudently chooses one of the Finalist Worlds for actualization.

On this story, there are elegant deterministic mathematical laws of nature which are true even of the agents’ choices, but they are true of the agents’ choices because the agents freely chose as they did. The agents had the causal power to violate, say, the conservation of momentum, but in fact freely did not do so.

There is an ambiguity in the concept of “exceptionless laws”. “Exceptionless laws” could mean: laws that allow no exception (they are pushy laws that are so strong as to make no exception possible) and laws that in fact have no exception. The deterministic laws in this story are exceptionless in the sense of having no exception, which is why I am talking of their exceptionless truth rather than their exceptionless power.

In this story, there is a dual explanation of the agents’ choices. On the one hand, there is a standard libertarian story about the agents’ free causality. On the other hand, the laws have explanatory power, because God chose the Finalist Worlds because they are worlds where the laws have exceptionless truth.

The big difficulty with the above story is Molinism. Also, it is worth noting that it is metaphysically possible that there turn out not to be any (feasible) Finalist Worlds: in that scenario, God wouldn’t be able to create a world where there is freedom and elegant deterministic mathematical laws holding exceptionlessly. But for reasons similar to why many people think Transworld Depravity is unlikely to be true, I think it is unlikely that there would be no Finalist Worlds.

It is also interesting to note that there are two more views that could be plugged into the story that can do the same job: Thomism and compatibilism.

On Thomism, God can use primary causation to make agents freely choose as he desires. Then we can suppose that God surveys the same Candidate Worlds as on the Molinist story. Then he chooses Finalist Worlds as those Candidate Worlds where the agents’ free choices in fact do not contradict the mathematical laws. And then God uses primary causation to actualize one of the Finalist Worlds. Again, the agents have the power to contradict the laws, but freely choose not to exercise it.

Finally we have straightforward compatibilism. A dualist can just as easily be a compatibilist as a materialist. On this story, we skip the Candidate Worlds, and the Finalist Worlds are worlds with compatibilist agents, with a deterministic non-physical mental life, who have the power of contradict the physical laws of the world but who are mentally determined, in a way compatible with freedom, never to exercise such a power. And then God chooses one of the Finalist Worlds. The agents then are as free as any compatibilist agents.

The compatibilist version of this story is close to Leibniz’s pre-established harmony, except that it has real mind-world causation, which is a big improvement.

Of course, the non-theist can’t make any of these moves. And, alas, neither can I, since my mere foreknowledge view denies Molinism, Thomism (about free will) and compatibilism.

If open futurism is true, then there are possible worlds that can't ever be actual

Assume open futurism, so that, necessarily, undetermined future tensed “will” statements are either all false or all lack truth value. Then there are possible worlds containing me such that it is impossible for it to be true that I am ever in that world. What do I mean?

Consider possible worlds where I flip an indeterministic fair coin on infinitely many days, starting with day 1. Among these worlds, there is a possible world w_H where the coin always comes up heads. But it is impossible for it to be true that I am in that world. For that I ever am in that world entails that infinitely many future indeterministic fair coin tosses will be heads. But a proposition reporting future indeterministic events cannot be true given an open future. So, likewise, it cannot be true that I am ever in that world.

But isn’t it absurd that there be a possible world with me such that it is impossible that it be true that I am in it?

My presence is an unnecessary part of the above argument. The point can also be put this way. If open futurism is true, there are possible worlds (such as w_H) that can’t possibly ever be actual.

A heuristic argument for the Brouwer axiom

Suppose that:

1. We cannot make sense of impossible worlds, but only of possible ones, so the only worlds there are are possible ones.

2. Necessarily, a possible worlds semantics for alethic modality is correct.

3. Worlds are necessary beings, and it is essential to them that they are worlds.

Now, suppose the Brouwer axiom, that if something is true then it’s necessarily possible, is not right. Then the following proposition is true at the actual world but not at all worlds:

4. Every world is possible.

(For if Brouwer is false at w₁, then there is a world w₂ such that w₂ is possible at w₁ but w₁ is not possible at w₂. Since w₁ is still a world at w₂, at w₂ it is the case that there is an impossible world.)

Say that the “extent of possibility” at a world w is the collection of all the worlds that are possible at w. Thus, given 1-3, if Brouwer fails, the actual world is a world that maximizes the extent of possibility, by making all the worlds be possible at it. But it seems intuitively unlikely that if worlds differ in the extent of possibility, the actual world should be so lucky as to be among the maximizers of the extent of possibility.

So, given 1-3, we have some reason to accept Brouwer.

Saying with possible worlds what can't be said with box and diamond

The literature contains a number of examples of a modal claim that can be made with possible worlds language but not in box-diamond language. Here is one that occurred to me that is simpler than any of the examples I’ve seen:

• Reality could have been different.

Very simple in possible worlds language: There is a non-actual world. (Note: This doesn’t work on the version of Lewis’s modal realism that allows for duplicate worlds. All the worse for that version.) But no box-diamond statement expresses (*). One can, of course, say that there aren’t any unicorns but could be, which implies (*), but that’s not the same as saying (*).

Universal countable numerosity: A hypothesis worth taking seriously?

Here’s a curious tale about sets and possible worlds: What sets there are varies between metaphysically possible worlds and for any possible world w₁, the sets at w₁ satisfy the full ZFC axioms and there is also a possible world w₂ at which there exists a set S such that:

1. At w₂, there is a bijection of S onto the natural numbers (i.e., a function that is one-to-one and whose range is all of the natural numbers).

2. The members of S are precisely the sets that exist at w₁.

Suppose that this tale is true. Then assume S5 and this further principle:

3. If two sets A and B are such that possibly there is a bijection between them, then they have the same numerosity.

(Here I distinguish between “numerosity” and “cardinality”: to have the same cardinality, they need to actually have a bijection.) Then:

4. Necessarily, all infinite sets have the same numerosity, and in particular necessarily all infinite sets have the same numerosity as the set of natural numbers.

For if A and B are infinite sets in w₁, then at w₂ they are subsets of the countable-at-w₂ set S, and hence at w₂ they have a bijection with the naturals, and so by (3) they have the same numerosity.

Given the tale, there is then an intuitive sense in which all infinite sets are the same size. But it gets more fun than that. Add this principle:

5. If two pluralities are such that possibly there is a bijection between them, then the two pluralities have the same numerosity.

(Here, a bijection between the xs and the ys is a binary relation R such that each of the xs stands in R to a unique one of the ys, and vice versa.) Then:

6. Necessarily, the plurality of sets has the same numerosity as the plurality of natural numbers.

For if the xs are the plurality of sets of w₁, then there will be a world w₂ and a countable-at-w₂ set S such that the xs are all and only the members of S. Hence, there will be a bijection between the xs and the natural numbers at w₂, and hence at w₁ they will have the same numerosity by (5).

So if my curious tale is true, not only does each infinite set have the same numerosity, but the plurality of sets has the same numerosity as each of these infinite sets.

We can now say that a set or plurality has countable numerosity provided that it is either finite or has the same numerosity as the naturals. Then the conclusion of the tale is that each set (finite and infinite), as well as the plurality of sets, has countable numerosity. I.e., universal countable numerosity.

But hasn’t Cantor proved this is all false? Not at all. Cantor proved that this is false if we put “cardinality” in place of “numerosity”, where cardinality is defined in terms of actual bijections while numerosity is defined in terms of possible bijections. And I think that possible bijections are a better way to get at the intuitive concept of the count of members.

Still, is my curious tale mathematically consistent? I think nobody knows. Will Brian, a colleague in the Mathematics Department, sent me a nice proof which, assuming my interpretation of its claims is correct, shows that if ZFC + “there is an inaccessible cardinal” is consistent, then so is my tale. And we have no reason to doubt that ZFC + “there is an inaccessible cardinal” is consistent. So we have no reason to doubt the consistency of the tale.

As for its truth, that's a different matter. One philosophically deep question is whether there could in fact be so much variation as to what the sets are in different metaphysically possible worlds.

Multi-Bohm

I am exploring what seem to me to be under-explored parts of the logical space of interpretations of Quantum Mechanics. I may be wasting my time: there may be good reasons why those parts of logical space are not explored much. But I am also hoping that such exploration will broaden my mind.

So, here’s a curious interpretation: multi-Bohm. Assume no collapse as in Everett. At any given time t, there is the set S_t of all particle position assignments compatible with the value of the wavefunction ψ(t) (we can extend to spin and other things in the same way that Bohm gets extended to spin and other things). Typically, this set will include every possible position assignment, and will have continuum cardinality.

Now on Bohm’s interpretation, one member of S_t is privileged: it is the actual positions of the particles. But drop that privileging. Suppose instead that all the assignments of S_t are on par. Then S_t gives us a synchronic decomposition of the Everettian multiverse into "branches". Now stitch the synchronic decomposition into trajectories using the guiding equation: a position assignment s_t ∈ S_t is part of the same trajectory as a position assignment s_t′ ∈ S_t′ if and only if the guiding equation evolves s_t into s_t′ over the time span from t to t′ given the actual wavefunction ψ.

We can think of the above as a story with infinitely many (continuum many) parallel Bohmian universes. But that bloats the ontology by including infinitely many ensembles of particles. Since the wavefunction fully determines the sets S_t of position assignments (or so I assume—there are some worries about null-measure stuff that I am not perfectly sure of), we can stop thinking about real particles and just as a way of speaking superimposed on top of the many-worlds interpretation.

This means that we can interpret the many-worlds interpretation not as a branching-worlds story, but as a deterministic parallel worlds reading. For given the two-way (I assume) determinism in the guiding equation, the trajectories never meet: the branches always stay separate and parallel. Moreover, the probability problem of the many-worlds interpretation is unsolved, and so we cannot say that the story fits better with one set of experimental results rather than another.

This isn’t very attractive…

Cardinality and worlds

For every initial ordinal k, there is a possible world with exactly k photons. But there is no set of all initial ordinals (proof: suppose there is such a set; the union of the members of any set of ordinals is an ordinal; so the union of the initial ordinals is an ordinal; it must have the same cardinality as some initial ordinal in the set; but for any ordinal in the set, there is a larger one in the set). So there is no set of all possible worlds.

This argument doesn't use the Axiom of Choice and hence it improves on the argument I gave here.

An Axiom of Choice strong enough to puzzle

All the main puzzles that follow from the Axiom of Choice (AC)--nonmeasurable sets, Banach-Taski and guessing future coin tosses--need only a weaker version of AC. One weaker version that suffices is this:

(*) There is a choice function for any partition of the interval (0,1) into non-empty countable sets.

Now imagine worlds with point-sized particles that never move, but can perish and come into existence. The world starts at time 0. Each particle has a lifetime between 0 and 1, exclusive. Some locations in the world are never occupied by a particle. Call these "vacant". At all other locations, a particle comes into existence at time 0. Two particles never occupy the same location at the same time. Call such worlds p-worlds.
For each non-vacant location x in a p-world w, there is an associated set L(w,x) of numbers in (0,1), where a number y is in L(w,x) iff some particle at x has lifetime of length y. I now need a crucial metaphysical plenitude assumption:

(**) For any set S such that (a) every member of S is a countable non-empty collection of members of (0,1) and (b) the cardinality of S is at most that of the continuum, there is a p-world w such for each A in S there is a unique location x in w such that L(w,x)=A.

In other words, any set S satisfying (a) and (b) is the set of sets of lifetime lengths for non-vacant locations in some p-world, without duplication.

Given the plenitude assumption, I get the version of AC needed for the paradoxes. For given a partition S of (0,1) into countable sets, there will be a p-world as in (**). Given a member A of S, there will be a unique location x such that L(w,x)=A. Let f(A) be the lifetime of the first particle at x in w. This is our choice function.

So the major paradoxes of AC follow from a plausible plenitude assumption about possible worlds.

Modal realism and God

According to David Lewis's modal realism, every possible world exists as a concrete universe, and a proposition is possible provided it holds at some universe. But this seems incompatible with theism. For necessarily God believes every truth, and we can now run the following argument.

Necessarily, if p is true, God believes p. So, if p is possible, possibly God believes p. Thus, possibly, God believes that there are no horses, since the proposition that there are no horses is possibly true. So there is a universe, say u₁, at which God believes that there are no horses. Now God either actually has this belief or not. If he actually has this belief, then he actually has conflicting beliefs, since he actually believes that there are horses. But God does not have conflicting beliefs. So we have to say that while at u₁ God believes there are no horses, actually God instead believes there are horses. Thus, what propositions God believes differs between universes. But how could that make any sense? Granted, perhaps our beliefs can be localized to brain hemispheres and then at a location in my left hemisphere I believe p and at another I don't. If that can be made sense of, then one could give a sense to the locution "believes p at x". But God's beliefs surely do not have any such localization. Wherever God is present, he is wholly present. He is not a material being to have partial presence of the sort that might allow for a spatial distribution of our beliefs.

Devotional use of fantasy and science fiction

The kind of fantasy and science fiction that I like describes a possible world as it were from the inside (i.e., the sentences are to be interpreted relative to that world, as if that world were actual, in the sense of two-dimensional semantics—this makes it possible for the stories to have alternate origins for "the human race" and so on). Some of these possible worlds are fairly close to ours (realistic kinds of science fiction) and some are quite far from ours. Besides the kinds of values that every kind of literature can have, such as giving us a richer picture of moral deliberation, imaginative fiction of the sort I like also performs a devotional service—it gives us a richer picture of the power of God. There perhaps are no hobbits, probably there are no vast plasma-based intelligent beings in the sun, perhaps we do not live in a multiverse, almost surely there are no vampire-like unconscious but sophisticatedly cognitive beings, and probably God did not become incarnate as a lion; but all these things might have been so, by the power of God.

That does not mean that the fiction has to be overtly theistic or by a theistic author. Any picture of a genuinely possible world is a picture of a world in which God would exist, since God exists necessarily, in all worlds (and in the case of "God", the two-dimensional intension is constant, so we don't need to distinguish between conceivability and possibility). If the story is not compatible with the existence of God—for instance, if it contains a story of the ultimate origination of the cosmos incompatible with theism, or if it contains innocents suffering for eternity, vel caetera—then the story fails to describe a possible world.

Personally, I am made uncomfortable by imaginative fiction that does not describe a possible world. Besides rare cases of stories that appear to be clearly incompatible with theism, an offender is time travel stories that often violate metaphysical strictures against causal loops and circular explanation. I was also made uncomfortable by a Greg Egan story where mathematics itself is changed by human activity. (I think I am also made a bit uncomfortable by stories that strongly imply that what is happening is in our world—this world we live in—whereas the content of the story is metaphysically incompatible with how things are up to now. For instance, stories that give an alternate account of how "we humans" came into existence. But that is easily taken care of by reinterpreting the story without the rigidity of "our world"—that's what two-dimensional semantics is for.)

Actuality, Possibility and Worlds is released

My Actuality, Possibility and Worlds book has now been released, in both hardcover and paperback. There is a table of contents and index here [PDF]. I have to say that Continuum did a very fine job producing this on a fast schedule.

The Principle of Sufficient Reason, now in paperback

Shame-free self-promotion: I was looking on Amazon, and it looks like my Principle of Sufficient Reason book is now available in paperback (oddly, rather less expensive than the Kindle edition), while Actuality, Possibility and Worlds can be preordered in hardback and paperback.

Actualities, Possibilities and Worlds

The book manuscript is all finished, submitted, and is about to move into production at Continuum.  Some of the material in the book was written this summer, some was written back when I was a graduate student, and some was written in between.  Memorably to me, some of the material on Spinoza was written in the hospital while my wife was asleep in labor with our first child.

The base for the manuscript was my dissertation, though a lot of new material was added, especially on a Spinozistic-Tractarian account of possibility that should be taken much more seriously than it is.  And this summer while revising I cut a number of passages, especially some technical ones, which looked to me like "the author is just trying to impress his dissertation committee."

Amusingly, as I was revising the manuscript I found that in it I had given and endorsed an argument against divine command theory that recently I criticized in print, where I attributed the argument to Wes Morriston.  When I wrote my critique, I completely forgot that I had once made the argument myself!  I actually find my own version of the argument fairly convincing, but not being able to decide if the argument or the critique was better, I cut the argument from the manuscript.

Objective probabilities

Suppose:

1. W is a set of possible worlds (or maybe situations?)
2. L is a first-order language suitable for talking about what is going on at a member of W, and with a finite symbol-set
3. S is the set of strings, of finite or countably infinite length, but with a starting point (i.e., "ababababab..." is acceptable, but "...ababababab" is not) in the symbol-set of L
4. e(s) is the proposition expressed by a sentence s of L
5. B_W(s) is the claim that s is a member of S such that e(s) is true at exactly one member of W
6. r is a random variable whose values range over the members of S and have the following property: P(r=s)=(n+1)^−(l(s)+1), where n is the number of symbols in the symbol-set of L and l(s) is the length of s; thus, r simply chooses a random string in S, letter by letter, with an equal likelihood of any particular letter or of ending the string there.

Then it seems we can define a probability of a first-order[note 1] proposition p relative to the worlds in W as follows: P_L,W(p)=P(e(r) entails p|B_W(r)).

If the language L is somehow natural for describing the members of W, then it makes sense to think of P_L,W as defining a natural probability measure for what goes on in members of W. If p is W-impossible, i.e., if p holds at no member of W, then P_L,W(p)=0.

What is particularly nice about P_L,W is that it favors worlds with simpler laws. Thus, it is a probability measure particularly well-suited to making scientific inferences.

A serious technical difficulty with the above definition is that P_L,W(p) will not be defined for all p, but only for those p for which the set of sentences r such that e(r) entails p is measurable. One can avoid this difficulty by restricting the ps for which P_L,W(p) is defined, or by replacing the Axiom of Choice with the axiom that all subsets of the reals are measurable.

A second technical difficulty is that P(B_W(r)) might be zero. This difficulty will be avoided if we have at least one finitely simple world, where a world w is finitely simple if and only if there is a finite sentence r such e(r) is true at w and only at w. I suspect (again, I haven't written out the proof) that in that case we get the following interesting theorem: With probability one, we are in a finitely simple world. This suggests that the measure P might be useful for inductive purposes—it seems to be a measure that prefers simpler worlds.

Distinguishing the A- and B-theories

It is not so easy to state a clear distinction between the A- and B-theories of time. One says things like: "According to the A-theory, there is an objective past, present and future". But although I've used formulations like this, I've cringed when doing so, because of various problems.

So, let me propose a different characterization: The A-theorist holds that at different times we inhabit different possible worlds, while the B-theorist denies this.

This formulation makes the A-theory seem implausible--according to the A-theory, interworld travel is possible, since if we wait, we will end up in a different world from the one we're in. Nonetheless, I think this formulation is right, follows from the "objective past, present and future" formulation insofar as we can understand it, and the A-theorist should agree with it. Consider that a world includes or encodes everything that is objectively the case, and if something is objectively the case at w₁ but not at w₂, then w₁ and w₂ are different worlds. According to the A-theorist, the difference between some event's being future or past is objective. So, if we consider the world we'll be in in ten minutes, there will be objective differences--some events that now are future will then be past. Hence, it's a different world.

Another reason the A-theorist should embrace this is that it allows one to neatly characterize the difference between A-theories with an open future and A-theories without an open future. On A-theories with an open future, for a future time t, there is a minimal set W(t) of worlds such that it is a fact that the world at t will be one of the members of W(t), and typically W(t) has more than one member. On A-theories without an open future, for any future time t, there is a unique world w_t such that it is a fact that the world at t will be w_t.

There is, however, an alternative to the above approach for the A-theorist. The argument for my characterization of the A-theory depended on the claim that possible worlds do not change. The A-theorist could deny that. But I think it would be an implausible denial. Possiible worlds are like propositions. They are abstracta, not spatiotemporal entities. They do not change. There is change in a possible world, just as there is change in a novel (i.e., in what the world or novel describes), but the world itself does not change, just as a novel itself need not change (it might, if it gets revised or destroyed, but that's a dififerent change from the change in the novel).