All-false open futurism

On All-False Open Futurism (AFOF), any future tensed statement about a future contingent must be false. It is false that there will be a sea battle tomorrow, for instance.

Suppose now I realize that due to a bug, tomorrow I will be able to transfer ten million dollars from a client’s account to mine, and then retire to a country that won’t extradite me. A little angel says to me:

1. Your freely taking your client’s money without permission tomorrow entails your being a thief tomorrow.

I don’t want to be a thief, tomorrow or ever, so I am about to decide not to do it. But now a little devil convinces me of AFOF and says that while (1) is true, so is:

2. Your freely taking your client’s money without permission tomorrow entails your being a saint tomorrow.

Perhaps I am not very good at modal logic and the devil needs to explain. Given AFOF, it is necessarily false that I will freely take my client’s money without permission tomorrow, and a necessary falsehood entails everything. So, the devil adds, I might as well buy my plane tickets now.

The angel, however, grants AFOF for the sake of argument, but says that notwithstanding (2), the following holds:

3. Tomorrow it will be the case your taking your client’s money without permission entails your being a thief.

For the entailment holds always.

At this point, we have an interesting question. Given AFOF, should I guide my actions by the entailment between future-tensed claims in (2) or by the future-tensed entailment claim in (3)? The angel urges that the devil’s reasoning undercuts all rationality, while the angel’s reasoning does not, and hence is superior.

But the devil has one more trick up his sleeve. He notes that it is a contingent question whether there will be a tomorrow at all. For God might freely decide to end time before tomorrow. Thus, that there will be a tomorrow is false on AFOF. But (3) implies that there will be a tomorrow, and so (3) is false as well. I try to argue on the basis of Scripture that God has made promises that entail a future eternity, but the devil is a lot better at citing the Bible than I, and convinces me that God might transfer us to a timeless state or maybe eternal life is a supertask lasting from 8 to 9 pm tonight. And in any case, surely it should not depend on revelation whether the angel has a good argument not to take the client’s money. This is a problem for AFOF.

Maybe this is the way out. The angel could say this:

4. Necessarily, if there will be a tomorrow, then it will be true tomorrow that taking your client’s money without permission entails your being a thief.

But while this conditional is true on AFOF, if the devil has made his case that God hasn’t promised there will be a tomorrow, he can respond with:

5. Necessarily, if God hasn’t promised there will be a tomorrow and there will be a tomorrow, then it will be true tomorrow that taking your client’s money without permission entails your being a saint.

For the antecedent of the conditional here is necessarily false on AFOF, it being contingent that there will be a tomorrow absent a divine promise. And it seems that (5) is even more relevant to guiding action than (4), then.

Maybe the defender of AFOF can insist that the future must be infinite. But this does not seem plausible.

Possible futures

Given a time t and a world w, possible or not, say that w is t-possible if and only if there is a possible world w_t that matches w in all atemporal respects as well as with respect to all that happens up to and including time t. For instance, a world just like ours but where in 2027 a square circle appears is 2026-possible but not 2028-possible.

Here is an interesting and initially plausible metaphysical thesis:

1. The world w is possible iff it is t-possible for every finite time t.

But (1) seems false. For imagine this:

2. On the first day of creation God creates you and promises you that on some future
   day a butterfly will be created ex nihilo. God never makes any other promises. God never makes butterflies. And nothing else relevant happens.

I assume God’s promises are unbreakable. The world described by (2) seems to be t-possible for every finite time t. For the fact that no butterfly has come into existence by time t does not falsify God’s promise that one day a butterfly will be created. But of course the world described by (2) is impossible.

(It’s interesting that I can’t think of a non-theistic counterexample to (1).)

So what? Well, here is one applicaiton. Amy Seymour in a nice paper responding to an argument of mine writes about the following proposition about situation where there are infinitely many coin tosses in heaven, one per day:

3. After every heads result, there is another heads result.

She says: “The open futurist can affirm that this propositional content has a nearly certain general probability because almost every possible future is one in which this occurs.” But in doing so, Seymour is helping herself to the idea of a “possible future”, and that is a problematic idea for an open futurist. Intuitively:

4. A possible future is one such that it is possible that it is true that it obtains.

But the open futurist cannot say that, since in the case of contingent futures, there can be no truth about its obtaining. The next attempt at accounting for a possible future may be to say:

5. A future is possible provided it will be true that it is possible that it obtains.

But that doesn’t work, either, since any future with infinitely many coin tosses (spaced out one per day) is such that at any time in the future, it is still not true that it is possible that it obtains, since its obtaining still depends on the then-still-future coin tosses. The last option I can think of is:

6. A future is possible provided that for every future time t it is t-possible.

But that fails for exactly the same reason that the t-possibility of worlds story fails.

Here is one way out: Deny classical theism, say that God is in time, and insist that God has to act at t in order to create something ex nihilo at t. But God, being perfect, can’t make a promise unless he has a way of ensuring the promise to come true. But how can God make sure that he will one day create the butterfly? After all, on any future day, God is free not to create it then. Now, if God promised to create a butterfly by some specific date, then God could be sure that he would follow through, since if he hadn’t done so prior to the specified date, he would be morally obligated to do so on that day, and being perfect he would do so. So since God can’t ensure the promise will come true, he can’t make the promise. (Couldn’t God resolve to create the butterfly on some specific day? On non-classical theism, maybe yes, but the act of resolving violates the clause “nothing else relevant happens” in (2).)

This way out doesn’t work for classical theism, where God is timeless and simple. For given timelessness, God can timelessly issue the promise and “simultaneously” timelessly make a butterfly appear on (say) day 18, without God being intrinsically any different for it. So I think the classical theist has reason to deny (1), and hence has no account of “possible futures” that is compatible with open futurism, and thus probably has to deny open futurism. Which is unsurprising—most classical theists do deny open futurism.

Logical consequence

There are two main accounts of ψ being a logical consequence of ϕ:

• Inferentialist: there is a proof from ϕ to ψ

• Model theoretic: every model of ϕ is a model of ψ.

Both suffer from a related problem.

On inferentialism, the problem is that there are many different concepts of proof all of which yield an equivalent relation of between ϕ and ψ. First, we have a distinction as to how the structure of a proof is indicated: is a tree, a sequence of statements set off by subproof indentation, or something else. Second, we have a distinction as to the choice of primitive rules. Do we, for instance, have only pure rules like disjunction-introduction or do we allow mixed rules like De Morgan? Do we allow conveniences like ternary conjunction-elimination, or idempotent? Which truth-functional symbols do we take as undefined primitives and which ones do we take as abbreviations for others (e.g., maybe we just have a Sheffer stroke)?

It is tempting to say that it doesn’t matter: any reasonable answers to these questions make exactly the same ψ be logical consequence of the same ϕ.

Yes, of course! But that’s the point. All of these proof systems have something in common which makes them "reasonable"; other proof systems, like ones including the rule of arbitrary statement introduction, are not reasonable. What makes them reasonable is that the proofs they yield capture logical consequence: they have a proof from ϕ to ψ precisely when ψ logically follows from ϕ. The concept of logical consequence is thus something that goes beyond them.

None of these are the definition of proof. This is just like the point we learn from Benacerraf that none of the set-theoretic “constructions of the natural numbers” like 3 = {0, 1, 2} or 3 = {{{0}}} gives the definition of the natural numbers. The set theoretic constructions give a model of the natural numbers, but our interest is in the structure they all have in common. Likewise with proof.

The problem becomes even worse if we take a nominalist approach to proof like Goodman and Quine do, where proofs are concrete inscriptions. For then what counts as a proof depends on our latitude with regard to the choice of font!

The model theoretic approach has a similar issue. A model, on the modern understanding, is a triple (M,R,I) where M is a set of objects, R is a set of relations and I is an interpretation. We immediately have the Benacerraf problem that there are many set-theoretic ways to define triples, relations and interpretations. And, besides that, why should sets be the only allowed models?

One alternative is to take logical consequence to be primitive.

Another is not to worry, but to take the important and fundamental relation to be metaphysical consequence, and be happy with logical consequence being relative to a particular logical system rather than something absolute. We can still insist that not everything goes for logical consequence: some logical systems are good and some are bad. The good ones are the ones with the property that if ψ follows from ϕ in the system, then it is metaphysically necessary that if ϕ then ψ.

Modal details in Unger's argument against his existence

Unger famously argues that he doesn’t exist, by claiming a contradiction between three claims (I am quoting (1) and (2) verbatim, but simplifying (3)):

1. I exist.

2. If I exist, then I consist of many cells, but a finite number.

3. If I exist and I consist of many but a finite number of cells, then removal of the least important cell does not affect whether I exist.

Unger then says:

these three propositions form an inconsistent set. They have it that I am still here with no cells at all, even while my existence depends on cells. … One cell, more or less, will not make any difference between my being there and not. So, take one away, and I am still there. Take another away: again, no problem. But after a while there are no cells at all.

But taken literally this is logically invalid. Premise (2) says that I consist of many but a finite number of cells. But to continue applying premise (3), Unger needs that premise (2) would still be true no matter how many cells were taken away. But premise (2) does not say anything about hypothetical situations. It says that either I don’t exist, or I consist of a large but finite number of cells. In particular, there are no modal operators in (2).

Now, no doubt this is an uncharitable objection. Presumably (2) is not just supposed to be true in the actual situation, but in the hypothetical situations that come from repeated cell-removals. At the same time, we don’t want (2) to be ad hoc designed for this argument. So, probably, what is going on is that there is an implied necessity operator in (2), so that we have:

4. Necessarily, if I exist, then I consist of many cells, but a finite number.

The same issue applies to (3), since (3) needs to be applied over and over in hypothetical situations. Another issue with (3) is that to apply it over and over, we need to be told that removal of the cell is possible. So now we should say:

5. Necessarily, if I exist and I consist of many but a finite number of cells, then removal of the least important cell is possible and does not affect whether I exist.

Now, I guess, we can have a valid argument in S4.

Is this a merely technical issue here? I am not sure. I think that once we’ve inserted “Necessarily” into (4) and (5), our intuitions may start to shift. While (2) is very plausible if we grant the implied materialism, (4) makes us wonder whether there couldn’t be weird situations where I exist but don’t consist of many but a finite number of cells. First, it’s not obviously metaphysically impossible for me to grow an infinitely long tail? That, however, is a red herring. The argument can be retooled only to suppose that I necessarily have many cells and I actually have a finite number. But, second, and more seriously, is it really true that there is no possible world where I exist with only a few cells? In fact, perhaps, I once did exist with only a few cells in this world!

Similarly about (5). It’s clear that right now I can survive the loss of my least important cell. But it is far from clear that this is a necessary truth. It could well be metaphysically possible that I be reduced to some state of non-redundancy where every cell is necessary for my existence, where removal of any cell severs an organic pathway essential to life. I would be in a very different state in such a case than I am right now. But it’s far from clear that this is impossible.

Perhaps, though, the modality here isn’t metaphysical modality, but something like nomic modality. Maybe it’s nomically impossible for me to be in a state where every cell is non-redundant. Maybe, but even that’s not clear. And it’s also harder to say that the removal of the least important cell has to (in the nomic necessity sense) be nomically possible. Couldn’t it be that nomically the only way the least important cell could be removed would be by cutting into me in ways that would kill me?

Furthermore, once we’ve made our modal complications to the argument, it becomes clear that of the three contradictory premises (1), (4) and (5), premise (1) is by far the most probable. Premise (1) is a claim about my own existence, which seems pretty evident to me, and is only a claim about how things actually are now. Premises (4) and (5) depend on difficult modal details, on how things are in other worlds, and on metaphysical intuitions that are surely more fraught than those in the cogito.

(One of the things I’ve discovered by teaching metaphysics to undergraduates, with a focus on formulating logically valid arguments, is that sometimes numbered arguments in published work by smart people are actually quite some distance from validity, and it’s hard to see exactly how to make them valid without modal logic.)

Reducing de re to de dicto modality

In my previous post, I gave an initial defense of a theory of qualitative haecceities in terms of qualitative origins: qualitative haecceities encapsulate complete qualitative descriptions of an entity’s initial state and causal history. I noted that among the advantages of the theory is that it can allow for a reduction of de re modality to de dicto modality, without “the mystery of non-qualitative haecceities”.

I want to expand on this, and why qualitative-origin haecceities are superior to non-qualitative haecceities here. A haecceitistic account of de re modality proceeds in something like the following vein. First, introduce the predicate H(Q,x) which says that Q is a haecceity of x. Then we reduce de re claims as follows:

• x is essentially F ↔︎ ∀Q(H(Q,x)→□(∀y(Qy→Fx)))

• x is accidentally F ↔︎ ∃Q(H(Q,x)∧◊(∃y(Qy∧Fx))).

Granted, this involves de re modality for second-order variables like Q. But this de re modality is less problematic because we can suppose the Barcan and converse Barcan formulas to hold as axioms for the second-order quantifiers, and we can treat the second-order entities as necessary beings. De re modality is particularly difficult for contingent beings, so if we can reduce to a modal logic where only necessary beings are subject to de re modal claims, we have made genuine progress.

We will also need some axioms. Here are two that come to mind:

• ∀x∀Q(H(Q,x)→Qx) (things have their haecceities)

• ∀x∃Q(H(Q,x)) (everything has a haecceity).

Now, here is why I think that qualitative-origin haecceities are superior to non-qualitative haecceities. Given qualitative-origin haecceities, we can give an account of what H(Q,x) means without using de re modality. It just means that Qy attributes to y all of the actual qualitative causal origins of x, including x’s initial qualitative state. On the other hand, if we go for non-qualitative haecceities, we seem to have two options. We could take H(Q,x) to be primitive, which always should be a last resort, or we could try to define in some way like:

• H(Q,x) ↔︎ (□(Ex→Qx) ∧ □∀y(Qy→y=x))

where Ex says that x exists (it might be a primitive in a non-free logic, or it might just be an abbreviation for ∃y(y=x)). But this definition uses de re modality with respect to x, so it is not satisfactory in this context, and I can’t think of any way to do it without de re modality with respect to potentially contingent individuals like x.

Rooted and unrooted branching actualism

Branching actualist theories of modality say that metaphysical possibility is grounded in the powers of actual substances to bring about different states of affairs. There are two kinds of branching actualist theories: rooted and unrooted. On rooted theories, there are some necessarily existing items (e.g., God) whose causal powers “root” all the possibilities. On unrooted theories, we have an ungrounded infinite regress of earlier and earlier substances. In my dissertation, I defended a theistic rooted theory, but in the conclusion mentioned a weaker version on which there is no commitment to a root. At the time, I thought that not many would be attracted to an unrooted version, but when I gave talks on the material at various department, I was surprised that some atheists found the unrooted theory attractive. And such theories have indeed been more recently defended by Oppy and Malpass.

I still think a rooted version is better. I’ve been thinking about this today, and found an interesting advantage: rooted theories can allow for a tighter connection between ideal conceivability and metaphysical possibility (or, equivalently, a prioricity and metaphysical necessity). Specifically, consider the following appealing pair of connection theses:

1. If a proposition is metaphysically possible (i.e., true in a metaphysically possible world), then it is ideally conceivable.

2. If a proposition is ideally conceivable, it is true in a world structurally isomorphic to a metaphysically possible one.

The first thesis is one that, I think, fits with both the rooted and unrooted theories of metaphysical possibility. I will focus on the second thesis. This is really a family of theses, depending on what we mean by “structurally isomorphic”. I am not quite sure what I mean by it—that’s a matter for further research. But let me sketch how I’m thinking about this. A world where dogs are reptiles is ideally conceivable—it is only a posteriori that we can know that dogs are mammals; it is not something that armchair biology can reveal. A world where dogs are reptiles is metaphysically impossible. But take a conceivable but impossible world w₁ where “dogs are reptiles”—maybe it’s a world where the hair of the dogs is actually scales, and contrary to immediate appearances the dogs are cold-blooded, and so on. Now imagine a world w₂ that’s structurally isomorphic to this impossible world—for instance, all the particles are in the same place, corresponding causal relations hold, etc.—and yet where the dogs of w₁ aren’t really dogs, but a dog-like species of reptile. Properly spelled out, such a world will be possible, and denizens of that world would say “dogs are reptiles”.

Or for another example, a world w₃ where Napoleon is my child is conceivable (it’s only a posteriori that we know this world not to be actual) but impossible. But it is possible to have a world w₄ where I have a Napoleon-like child whom I name “Napoleon”. That world can be set up to be structurally isomorphic to w₃.

Roughly, the idea is this. If something is conceivable but impossible, it will become possible if we change out the identities of individuals and natural kinds, while keeping all the “structure”. I don’t know what “structure” is exactly, but I think I won’t need more than an intuitive idea for my argument. Structure doesn’t care about the identities of kinds and individuals.

Now suppose that unrooted branching actualism is true. On such a theory, there is a backwards-infinite sequence of contingent events. Let D be a complete structural description of that sequence. Let p_D be the proposition saying that some infinite initial segment of the world fits with D. According to unrooted branching actualism, p_D is actually a necessary truth. But p_D is clearly a posteriori, and hence its denial is ideally conceivable. Let w₅ be an impossible world where p_D is false. If (2) is true, then there will be a possible world w₆ which is a structural isomorph of w₅. But because p_D is a structural description, if p_D is false in a world, it is false in any structural isomorph of that world. Thus, p_D has to be false in w₆, which contradicts the assumption that p_D is a necessary truth.

The rooted branching actualist doesn’t get (2) for free. I think the only way the rooted branching actualist can accept (2) is if they think that the existence and structure of the root entities is a priori. A theist can say that: God’s existence could be a priori (as Richard Gale once suggested, maybe there is an ontological argument for the existence of God, but we’re just not smart enough to see it).

Modality and the Axiom of Choice

Suppose that the set theory of our world is a Solovay model, where we don’t have the Axiom of Choice (AC), and where every subset of the reals is Lebesgue measurable. Now imagine that God picks out a line in space, and defines the Vitali equivalence relation for points on that line (where two points are equivalent if and only if the distance between them is a rational number). It is then surely within God’s power to create a particle of some unexemplified type T at exactly one point in every equivalence class. There is nothing incoherent about that! But if God did that, then there would surely be a set of the points containing a particle of type T. And that set would be a nonmeasurable Vitali set.

So what?

Well, prima facie, there are three possibilities about the existence of nonmeasurable sets:

1. Necessarily, there are no nonmeasurable sets.

2. Necessarily, there are nonmeasurable sets.

3. It is contingent whether there are nonmeasurable sets.

My argument strongly suggests that if there are no nonmeasurable sets, it is nonetheless possible that there are nonmeasurable sets. Hence, (1) is ruled out.

So we have an argument for the disjunction of (2) and (3).

Now, I think a lot of people have the intuition that mathematical facts are necessary. If so, then (3) is ruled out. They will see this as an argument for (2).

I don’t see it that way myself: I am quite open to contingent mathematical truths.

More generally, the argument shows that:

4. For any set of disjoint nonempty subsets of the reals, it is possible that there is a choice function.

Again, if the existence of pure sets is not a contingent matter, we conclude AC is true for all subsets of the reals.

The Principles of Sufficient and Partial Reasons

I have argued that the causal account of metaphysical possibility implies the Principle of Sufficient Reason (see Section 2.2.6.6 here). The argument was basically this: If p is contingently true but unexplained, then let q be the proposition that p is unexplained but true. Consider now a world w where p is false. In w, the proposition q will be possible (by the Brouwer axiom). So by the causal account of modality, something can start a chain of causes leading to q being true. Which, I claimed, is absurd, since that chain would lead both to p being true and to p being unexplained. But the chain would explain p, so we have absurdity.

But it isn’t absurd, or at least not immediately! For the chain need not explain p. It might only explain the aspects of p that do not obtain in w. For a concrete example, suppose that p is a conjunction of p₁ and p₂, and p₁ is false in w but p₂ is true. Then a chain that leads to p being true need not explain p₂: it might only explain p₁, and might leave p₂ as is.

I think what my argument for the PSR establishes is a weaker conclusion than the PSR: the Principle of Partial Reason (PPR), that every contingent truth has a partial explanation.

I am pretty sure that PPR plus causal finitism implies PSR, and so the modality argument for PSR can be rescued, albeit at the cost of assuming causal finitism. And, intuitively, it would be weird if PPR were true but PSR were not.

The infinite future problem for causal accounts of metaphysical possibility

Starting with my dissertation, I’ve defended an account of metaphysical possibility on which it is nothing other than causal possibility. I would try to define this as follows:

• p is possible₀ iff p is actually true

• p is possible_n + 1 iff things have the causal power to make it be that p is possible_n.

• p is possible iff p is possible_n for some n.

I eventually realized that this runs into problems with infinite future cases. Suppose a coin will be tossed infinitely many times, and, as we expect, will come up heads infinitely many times and tails infinitely many times. Let p be the proposition that all the tosses will be heads. Then p is false but possible. Moreover, it is easy to convince oneself that it’s not possible_n for any finite n. Possibility_n involves n branchings from the actual world, while p requires infinitely many branchings from the actual world.

This has worried me for years, and I still don’t have a satisfying solution.

But yesterday I realized a delightful fact. This problem does nothing to undercut the basic insight of my account of metaphysical possibility, namely that metaphysical possibility is causal possibility. All the problem does is undercut one initially plausible way to given an account of causal possibility. But if we agree that there is such a thing as causal possibility, and I think we should, then we can still say that metaphysical possibility is causal possibility, even if we do not know exactly how to define causal possibility in terms of causal powers.

(There is one danger. Maybe the true account of causal possibility depends on metaphysical possibility.)

Ontology as a contingent science

Consider major dividing lines in ontology, such as between trope theory and Platonism. Assume theism. Then all possibilities for everything other than God are grounded in God.

If God is ontologically like us, and in particular not simple, then it is reasonable to think that the correct ontological theory is necessarily determined by God’s nature. For instance, if God has tropes, then necessarily trope theory holds for creatures. If God participates in distinct Platonic forms like Divinity and Wisdom, then necessarily Platonism holds for creatures.

But the orthodox view (at least in Christianity and Judaism) is that God is absolutely simple, and predication works for God very differently from how it works for us. In light of this, why should we think that God had to create a tropist world rather than a Platonic one, or a Platonic one rather than a tropist one? Neither seems more or less suited to being created by God. It seems natural, in light of the radical difference between God and creatures, to think that God could create either kind of world.

If so, then many ontological questions seem to become contingent. And that’s surprising and counterintuitive.

Well, maybe. But I think there is still a way—perhaps not fully satisfactory—of bringing some of these questions back to the realm of necessity. Our language is tied to our reality. Suppose that we live in a tropist world. It seems that the correct account of predication is then a tropist one: A creature is wise if and only if it has a wisdom trope. A Platonic world has no wisdom tropes, and hence no wise creatures. Indeed, nothing can be predicated of any creature in it. What might be going on in the Platonic world is that there are things there that are structurally analogous wise things, or to predication. We can now understand our words “wise” and “predicated” narrowly, in the way they apply to creatures in our world, or we can understand them broadly as including anything structurally analogous to these meanings. If we understand them narrowly, then it is correct to say that “Nothing in the Platonist world is wise” and “Nothing is correctly predicated of anything in the Platonist world.” But in the wide, analogical sense, there are wise things and there is predication in the Platonist world. Note, too, that even in our world it is correct to say “God is wise” and “Something is correctly predicated of God” only in the wide senses of the terms.

On this account, necessity returns to ontology—when we understand things narrowly. But the pretensions of ontology should be chastened by realizing that God could have made a radically different world.

And maybe there is an advantage to this contingentism. Our reasoning in ontology is always somewhat driven by principles of parsimony. But while one can understand why parsimony is appropriately pursued in study of the contingent—for God can be expected to create the contingent parsimoniously, both for aesthetic reasons and to fit reality to our understanding—I have always been mystified why it is appropriately pursued in the study of the necessary. But if ontology is largely a matter of divine creative choice, then parsimony is to be sought in ontological theories just as in physical ones, and with the same theological justification.

The above sounds plausible. But I have a hard time believing in ontology as a contingent science.

An observation about the backwards-infinity branching view of possibility

In my dissertation, I defended a causal power account of modality on which something is possible just in case either it’s actual or something can bring about a causal chain leading to its being actual. I noted at the time that unless there is a necessary first cause, this leads to an odd infinite branching view on which any possible world matches our world exactly once you get far enough back, but nonetheless every individual event is contingent, because if you go back far enough, you get a causal power to generate something else in its place. Rejecting this branching view yields a cosmological argument for a necessary being. To my surprise when I went around giving talks on the account, I found that some atheists were willing to embrace the branching view. And since then Graham Oppy has defended it, and Schmid and Malpass have cleverly used it to attack certain cosmological arguments.

I want to note a curious, and somewhat unappealing, probabilistic feature of the backwards-infinite branching view. While it is essential to the view that it be through-and-through contingentist, assuming classical probabilities can be applied to the setup, then the further back you go on a view like that, the closer it gets to fatalism.

For let S_t be a proposition describing the total state of our world at time t. Let Q_t be the conjunction of S_u for all u ≤ t: this is the total present and past at t. Here is what I mean by saying that the further back you go, the closer you get to fatalism on the backwards-infinite branching view:

1. lim_{t→− ∞} P(Q_t) = 1.

I.e., the further back we go, the less randomness there is. In our time, there are many sources of randomness, and as a result the current state of the world is extremely unlikely—it is unlikely that I would be typing this in precisely this way at precisely this time, it is unlikely that the die throws in casinos right now come out as they do, and so on. But as we go back in time, the randomness fades away, and things are more and more likely.

This is not a completely absurd consequence (see Appendix). But it is also a surprising prediction about the past, one that we would not expect in a world with physics similar to ours.

Proof of (1): Let t_n be any decreasing sequence of times going to  − ∞. Let Q be the infinite disjunction Q_t1 ∨ Q_t2 ∨ .... The backwards-infinite branching view tells us that Q is a necessary truth (because any possible world has Q_t is true for t sufficiently low). Thus, P(Q) = 1. But now observe that Q_t1 implies Q_t2 implies Q_t3 and so on. It follows from countable additivity that lim_n→∞ P(Q_tn) = P(Q) = 1.

Appendix: Above, I said that the probabilistic thesis is not absurd. Here is a specific model. Imagine a particle that on day  − n for n > 0 has probability 2⁻ⁿ of moving one meter to the left and probability 2ⁿ of moving one meter to the left, and otherwise it remains still. Suppose all these steps are independent. Then with probability one, there is a time before which the particle did not move (by the Borel-Cantelli lemma). We can coherently suppose that necessarily the particle was at position 0 if you go far enough back, and then the system models backwards-infinite branching. However, note an unappealing aspect of this model: the movement probabilities are time-dependent. The model does not seem to fit our laws of nature which are time-translation symmetric (which is why we have energy conservation by Noether’s theorem).

Might well

It’s occurred to me that the “might well happen that” operator makes for an interesting modality. It divides into an epistemic and a metaphysical version. In both cases, if it might well happen that p, then p is possible (in the respective sense). In both cases, there is a tempting paraphrase of the operator into a probability: on the epistemic side, one might say that it might well happen that p if and only if p has a sufficiently high epistemic probability, and on the metaphysical side, one might say that it might well happen that p if and only if p has a sufficiently high chance given the contextually relevant background. In both cases, it is not clear that the probabilistic paraphrase is correct—there may be (might well be!) cases of “might well happen that” where numerical probabilities have no place. And in both cases, “might well happen that” seems context-sensitive and vague. It might well be that thinking about this operator could lead to progress on something interesting.

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.

Necessity and the open future

Suppose the future is open. Then it is not true that tomorrow Jones will freely mow the lawn. Moreover, it is necessarily not true that Jones will freely mow the lawn, since on open future views it is impossible for an open claim about future free actions to be true. But what is necessarily not true is impossible. Hence it is impossible that Jones will freely mow the lawn. But that seems precisely the kind of thing the open futurist wishes to avoid saying.

Yet another formulation of my argument against a theistic multiverse

Here’s yet another way to formulate my omniscience argument against a theistic multiverse, a theory on which God creates infinitely concretely real worlds, and yet where we have a Lewisian analysis of modality in terms of truth at worlds.

1. Premise schema: For any first order sentence ϕ: Necessarily, ϕ if and only if God believes that ϕ.

2. Premise schema: For any sentence ϕ: Possibly ϕ if and only if ∃w(at w: ϕ).

3. Premise: Possibly there are unicorns.

4. Premise: Possible there are no unicorns.

5. Necessarily, there are unicorns if and only if God believes that there are unicorns. (Instance of 1)

6. Possibly, God believes that there are unicorns. (3 and 5)

7. Possibly God believes that there are unicorns if and only if ∃w(at w: God believes that there are unicorns). (Instance of 2)

8. ∃w(at w: God believes that there are unicorns). (6 and 7)

9. ∃w(at w: God believes that there are no unicorns). (from 1, 2, 4 in the same way 8 was derived from 1, 2, 3)

So, either there is a world at which it is the case that God both believes there are unicorns and believes that there are no unicorns, or what God believes varies between worlds. The former makes God contradict himself. The content of God’s beliefs varying across worlds is unproblematic if the worlds are abstract. But if they are concrete, then it implies a real disunity in the mind of God.

Premise schema (1) is restricted to first order sentences to avoid liar paradoxes.

What I think is wrong with the proof of the Barcan formula

The Barcan formula says:

0. ∀x□ϕ → □∀xϕ.

The Barcan formula is dubious. Suppose, for instance, that the only things in existence are a, b and c, and let ϕ(x) say that x = a ∨ x = b ∨ x = c. Then the left-hand-side of (0) is true, since necessarily a = a, b = b and c = c. However the right-hand-side is not true, since it’s false that necessarily everything is one of a, b and c: even if there are only three things in existence, there could be more.

The Barcan formula can be proved in the Simplest Quantified Modal Logic (SQML) with S5.

Recently, a correspondent asked what I do about the fact that I accept S5 and yet presumably reject the Barcan formula. This gnawed at me for a bit, and I thought about the proof of the Barcan formula as presented by Menzel. I think I now have a pretty firm idea of where I get off the boat in the proof, and it has nothing to do with S5.

The first two steps of the proof are:

1. ∀x□ϕ → □ϕ (quantifier axiom)

2. □(∀x□ϕ→□ϕ) (from (1) by Necessitation).

Claim (1) is hard to dispute. But claim (2) isn’t right. Let ϕ be the formula D(x), where D(x) says that x is divine. Then (2) says:

2. □(∀x□D(x)→□D(x)).

By Generalization, which I think is hard to dispute, we get:

3. ∀x□(∀x□D(x)→□D(x)).

But (3) is false. For let a be me. Then (3) says the following about me:

3. □(∀x□D(x)→□D(a)),

i.e., that in the possible worlds where everything is necessarily divine, I am necessarily divine. But that’s just false. For I don’t exist in possible worlds where everything is necessarily divine. Only God exists in those worlds.

So I think the problem lies with Necessitation, which is the rule that says that theorems are necessary and yields (2) from (1). Here is my story as to what the problem with Necessitation is. Some logics have presuppositions. We can, for instance, imagine a theological logic that presupposes the existence of God. If a logic has presuppositions, then unless we have established that the presuppositions are themselves necessary truths, we are not entitled to assume that the theorems of that logic are themselves necessary. Instead, all that we are entitled to assume that the theorems of that logic necessarily follow from the presuppositions.

Now, infamously, classical logic has an existential presupposition: all the names and terms are names and terms for existing things. Because it has an existential presupposition, unless we have established the necessity of the existential presupposition, all we can say about theorems is that they necessarily follow from the existential presupposition, not that they are actually necessary.

Assuming we have a name for me in the language, it is indeed a theorem of classical logic that if everything is necessarily divine, then I am necessarily divine. But we cannot conclude that it is necessary that if everything is necessarily divine, then I am necessarily divine. For that would imply that in the world where only God exists, I would exist as well and be God. Rather, all we can conclude is that:

4. It is necessary that: if I and all the other things whose existence is presupposed exist, then if everything is necessarily divine, I am necessarily divine.

And that is trivially true, because in the worlds where everything is necessarily divine, I don’t exist.

A cosmological argument from a PSR for ordinary truths

Often in cosmological arguments the Principle of Sufficient Reason (PSR) is cleverly applied to vast propositions like the conjunction of all contingent truths or to highly philosophical claims like that there is something rather than nothing or that there is a positive contingent fact. But at the same time, the rhetoric that is used to argue for the PSR is often based on much more ordinary propositions, such as Rescher’s example of an airplane crash which I re-use at the start of my PSR book. And this can feel like a bait-and-switch.

To avoid this criticism, let’s suppose a PSR limited to “ordinary” propositions, i.e., the kind that occur in scientific practice or daily life.

1. Necessarily we have the Ordinary PSR that every contingent ordinary truth has an explanation. (Premise)

2. That there is an electron is an ordinary proposition. (Premise)

3. It is possible that there is exactly one contingent being, an electron. (Premise)

4. Necessarily, if no electron is a necessary being, then any explanation of why there is an electron involves the causal activity of a non-electron. (Premise)

5. Let w be a possible world where there is exactly one contingent being, an electron. (By 3)

6. At w, there is an explanation of why there is an electron. (By 1, 2 and 4)

7. At w, there is a non-electron that engages in causal activity. (By 4, 5 and 6)

8. At w, every non-electron is a necessary being. (By 5)

9. At w, there is a necessary being that engages in causal activity. (By 7 and 8)

10. So, there is a necessary being that possibly engages in causal activity. (By 9 and S5)

So, we have a cosmological argument from the necessity of the Ordinary PSR.

Objection: All that the ordinary cases of the PSR show is that actually the Ordinary PSR is true, not that it is necessarily true.

Response: If the Ordinary PSR is merely contingently true, then it looks like we are immensely lucky that there are no exceptions whatsoever to the Ordinary PSR. In other words, if the Ordinary PSR is merely contingently true, we really shouldn’t believe it to be true—we shouldn’t think ourselves this lucky. So if we are justified in believing the Ordinary PSR to be at least contingently true, we are justified in believing it to be necessarily true.

An ontological argument from the possible nondefectiveness of modality

1. Necessarily, if it is necessary that there is no God, then modal reality is bad. (Making the existence of God impossible is terrible!)

2. Necessarily, if something is bad, it is possible for it not to be bad. (The bad is a flaw in something that ought to be better than it is, and what ought to be can be.)

3. So, if modal reality is necessarily bad, then it is possible for modal reality not to be bad. (by 2)

4. So, if modal reality is necessarily bad, then modal reality is not necessarily bad. (by 3)

5. So, modal reality is not necessarily bad. (by 4)

6. So, possibly, modal reality is not bad. (by 5)

7. So, possibly, it is not necessary that there is no God. (by 1 and 6)

8. So, possibly, it is possible that God exists. (by 7)

9. So, it is possible that God exists. (by 8 and S4)

10. Necessarily, if God exists, it is necessary that God exists. (God is a necessary being and essentially divine.)

11. So, it is possible that it is necessary that God exists. (by 9 and 10)

12. So, God exists. (by 11 and Brouwer)

The possibility test for intentions

This test for whether one is intending some effect E of an action is often employed (e.g., by Germain Grisez) in the Double Effect literature:

1. If it is logically possible for an action with an intention J to be fully successful even though E does not happen, then E is not included in J.

Claim (1) follows in standard modal logic (with no need for anything fancy like S5) from:

2. If an intention J includes E, then the inclusion of E is an essential property of J.

3. Necessarily, if an action is done with an intention that includes E and E does not occur, then the action is not fully successful.

For suppose that E is included in J. Then in every possible world where an action is done with J, the action is done with an intention that includes E by (2)) and so in every possible world where an action is done with J, the action is not fully successful if E does not occur, by (3). Hence, there is no possible world where an action is done with J and is fully successful even though E does not happen. Thus, we have (1).

At the same time, (1) sounds awfully strong. Even if the possible world where the action is successful despite the lack of E requires a miracle, E is not included in J. For instance, suppose God is able to keep the soul of a human being bound to a single atom. That means that someone whose intention was to blow the man blocking the mouth of the cave literally to single atoms was not intending death, since there is a possible world where the person’s soul remains bound to a single atom, and in that world the action is clearly successful.

To deny (1), one needs to deny (2) or (3). I think the best route to denying (2) is a strong dose of semantic externalism: the content of an intention is dependent in part on things outside the individual. Perhaps on Earth the very same intention may be an intention to drink water, while on Twin-Earth the very same intention may be an intention to drink XYZ. I am sceptical of this: it seems to me that the best way to understand the water-XYZ issue is that intentions are partly grounded in facts outside the individual, and so it is a different intention on Twin-Earth than on Earth, even if it is partly grounded in the same facts in the individual.

But even if one is impressed by the water-XYZ issue, it seems one should be willing to accept the following variant on 2:

4. If an intention J includes E and occurs at t, then in any possible world that exactly matches the actual world up to an including t the intention J includes E at t.

The argument for (1) can now be modified to yield an argument for:

5. If an action with an intention J occurs at t, and if there is a possible world that matches the actual world up to and including t and where the action with J is fully successful but where E does not happen, then E is not included in J.

And if one’s motivation for denying (1) is to avoid the conclusion that intending to blow the man in the mouth of the cave to single atoms does not include intending death, then (5) is just as bad. For God could miraculously keep the soul bound to a single atom without anything being any different up to and including the time of the action.

If we don’t want (1), we won’t want (5), either.

So a better bet is to deny (3). A start towards a denial of (3) would be to talk of something like “stretch goals”. It seems that an action may have a stretch goal and yet be successful even if that stretch goal is unachieved. However, the stretch goal is surely intended.

I am not sure. If the stretch goal is intended, then it seems that the right thing to say is that the action is successful but not fully successful if the stretch goal is not met.

In any case, we might grant the claim about stretch goals, and introduce the concept of an intention being perfectly satisfied, which includes the satisfaction of all stretch goals, and then replace “fully successful” with “perfectly successful” in (1) and (5). And I think this will still generate the result about blowing the fat man to atoms, because the death of the fat man—the separation of soul from body—is not a stretch goal either. (If anything, one might imagine that his survival is a stretch goal.)

All this makes me want to say that (3) really is true, and we cannot avoid the conclusion that it is possible to intend to blow the man in the mouth of the cave to single atoms without intending to kill him. But I am now inclined to think that an intention to kill is not a necessary condition for murder, and so the action could still be a murder.

Arbitrariness and contingency

I’ve come to be impressed by the idea that where there is apparent arbitrariness, there is probably contingency in the vicinity.

The earth and the moon on average are 384400 km apart. This looks arbitrary. And here the fact itself is contingent.

Humans have two arms and two legs. This looks arbitrary. But it is actually a necessary truth. However there is contingency in the vicinity: it is a contingent fact that humans, rather than eight-armed intelligent animals, exist on earth.

Ethical obligations have apparent arbitrariness, too. For instance, we should prefer mercy to retribution. Here, there are two possibilities. First, perhaps it is contingent that we should prefer mercy to just retribution. The best story I know which makes that work out is Divine Command Theory: God commands us to prefer mercy to just retribution but could have commanded the opposite. Second, perhaps it is necessary that we should prefer mercy to retribution, because our nature requires it, but it is contingent that we rather than beings whose nature carries the opposite obligation exist.

Now here is where I start to get uncomfortable: mathematics. When I think about the vast number of possible combinations of axioms of set theory, far beyond where any intuitions apply, axioms that cannot be proved from the standard ZFC axioms (unless these are inconsistent), it’s all starting to look very arbitrary. This pushes me to one of three uncomfortable positions:

• anti-realism about set theory

• Hamkins’ set-theoretic multiverse

• contingent mathematical truth.