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).

I can jump 100 feet up in the air

Consider a possible world w₁ which is just like the actual world, except in one respect. In w₁, in exactly a minute, I jump up with all my strength. And then consider a possible world w₂ which is just like w₁, but where moments after I leave the ground, a quantum fluctuation causes 99% of the earth’s mass to quantum tunnel far away. As a result, my jump takes me 100 feet in the air. (Then I start floating down, and eventually I die of lack of oxygen as the earth’s atmosphere seeps away.)

Here is something I do in w₂: I jump 100 feet in the air.

Now, from my actually doing something it follows that I was able to do it. Thus, in w₂, I have the ability to jump 100 feet in the air.

When do I have this ability? Presumably at the moment at which I am pushing myself off from the ground. For that is when I am acting. Once I leave the ground, the rest of the jump is up to air friction and gravity. So my ability to jump 100 feet in the air is something I have in w₂ prior to the catastrophic quantum fluctuation.

But w₁ is just like w₂ prior to that fluctuation. So, in w₁ I have the ability to jump 100 feet in the air. But whatever ability to jump I have in w₁ at the moment of jumping is one that I already had before I decided to jump. And before the decision to jump, world w₁ is just like the actual world. So in the actual world, I have the ability to jump 100 feet in the air.

Of course, my success in jumping 100 feet depends on quantum events turning out a certain way. But so does my success in jumping one foot in the air, and I would surely say that I have the ability to jump one foot. The only principled difference is that in the one foot case the quantum events are very likely to turn out to be cooperative.

The conclusion is paradoxical. What are we to make of it? I think it’s this. In ordinary language, if something is really unlikely, we say it’s impossible. Thus, we say that it’s impossible for me to beat Kasparov at chess. Strictly speaking, however, it’s quite possible, just very unlikely: there is enough randomness in my very poor chess play that I could easily make the kinds of moves Deep Blue made when it beat him. Similarly, when my ability to do something has extremely low reliability, we simply say that I do not have the ability.

One might think that the question of whether one is able to do something is really important for questions of moral responsibility. But if I am right in the above, then it’s not. Imagine that I could avert some tragedy only by jumping 100 feet in the air. I am no more responsible for failing to avert that tragedy than if the only way to avert it would be by squaring a circle. Yet I can jump 100 feet in the air, while no one can square a circle.

It seems, thus, that what matters for moral responsibility is not so much the answer to the question of whether one can do something, but rather answers to questions like:

1. How reliably can one do it?

2. How reliably does one think (or justifiably think or know) one can do it?

3. What would be the cost of doing it?

A tweak to the ontomystical argument

In an old paper, I argued that we do not hallucinate impossibilia: if we perceive something, the thing we perceive is possible, even if it is not actual. Consequently, if anyone has a perception—veridical or not—of a perfect being, a perfect being is possible. And mystics have such experiences. But as we know from the literature on ontological arguments, if a perfect being is possible, then a perfect being exists (this conditional goes back at least to Mersenne). So, a perfect being exists.

I now think the argument would have been better formulated in terms of what two-dimensional semanticists like Chalmers call “conceivability”:

1. What is perceived (perhaps non-veridically) is conceivable.

2. A perfect being is perceived (perhaps non-veridically).

3. If a perfect being is conceivable, a perfect being is possible.

4. A perfect being is possible.

5. If a perfect being is possible, a perfect being exists.

6. So, a perfect being exists.

Premise (3) follows from the fact that the notion of a perfect being is not twinearthable, so conceivability and possibility are equivalent for a perfect being (Chalmers is explicit that this is the case for God, but he concludes that God is inconceivable). Premise (1) avoids what I think is the most powerful of Ryan Byerly’s four apparent counterexamples to my original argument: the objection that one might have perceptions that are incompatible with necessary truths about natural kinds (e.g., a perception that a water molecule has three hydrogen atoms).

A serious difficulty for some powers accounts of modality

On my causal powers account of modality, p is possible provided that either p is true, or something has an nth order power to make p true for some n. Here, a first order power to make p true is just a power to make p true. A second order power to make p true is a power to make there be a first order power to make p true. A third order power to make p true is a power to make there be a second order power to make p true. And so on.

Here's a serious problem. It seems quite possible that there is a sequence of false propositions p₁,p₂,p₃,... with the following properties:

1. It is possible that all the propositions are true.
2. For each n, something has an nth order power to make p_n true, but nothing has a lower order power to do so.

Now, let P be the proposition that all the p_n are true. Then P is false. But for every n it is true that nothing has an nth order power to make P true, as nothing has an nth order power to make p_n+1 true.

For concreteness, we might suppose that there are infinitely many planets, and on the nth planet there is a fertile asexually reproducing person x_n who has no children. Let p_n be the proposition that x_n has nth level descendants (where first level descendants are children, second level descendants are grandchildren, and so on).

What should I do? I can think of one option I don't like and two I can live with.

The option I don't like is to adopt strong assumptions about the nature of time that rule out the above story, such as an open future view plus discreteness assumptions.

The two I can live with both involve my scrapping the nth order power stuff. The first option is to make my thesis more modest: causal powers are the ground of metaphysical possibility, but I eschew giving an account of metaphysical modality in terms of causal powers. Then I can say that the possibility of P is grounded in powers, without giving a specific account. This is unattractive because it's unambitious.

The second option I can live with is to say that a thing can have a causal power to produce an effect even when it cannot directly produce the effect. Thus, I not only have a causal power to have children, but a causal power to have grandchildren. Then in the infinitely many planets scenario, the childless people jointly have a power to make P true. This seems to be the best option, but I really liked the iterated account.

Is a necessary being inconceivable?

Consider this argument:

1. Obviously necessarily, if N is a necessary being that exists, it is impossible that N doesn't exist.
2. It is conceivable that N doesn't exist.
3. So it is inconceivable that N exists.

For this argument to work, we need to be able to make the inference from:

4. Obviously necessarily, if p, then necessarily q.
5. Conceivably not q.
6. So, not conceivably p.

Suppose that p just is the statement that necessarily q. Then (4) is uncontroversial. If the above argument form is good, then so is this one:

7. Conceivably not q.
8. So, not conceivably necessarily q.

But why can't we conceive both of not q and of necessarily q? Why should the ability to conceive of one thing, viz., the necessity of q, preclude the ability to conceive of another, viz., not q?

The principle that conceivability is defeasible evidence of possibility may seem relevant, but I don't think it establishes the point. That I can conceive of necessarily q is evidence of the necessity of q. That I can conceive of not q is evidence if the possibility of not q. So, if both, then I have evidence for two contradictory statements. Nothing particularly surprising there: quite a common phenomenon, in fact!

Suppose A and B are contradictory statements. It may be that evidence for A is evidence against B. But is evidence for A evidence against there being evidence for B? If it is, it is very weak evidence. Likewise, even given the principle that conceivability is evidence for possibility, the argument from (7) to (8) is very weak, much weaker than the inferential strength of this principle.

To summarize: The strength of the inference from (1) and (2) to (3) in the original argument is about equal to the strength of evidence that the existence of evidence for A provides against the existence of evidence against A. But the existence of evidence of A provides very little evidence against the existence of evidence against A. So the original argument is a very weak one. It would be improved if the conclusion were weakened to the claim that it is impossible that N exists, and then I would focus my attack on (2).

Possibility, Aristotelian propositions and an open future

Aristotelians think that tensed sentences like "It is sunny" expressed "tensed propositions" capable of changing in truth value between true and false as the facts alter. The proposition that it is sunny is false today but was true two days ago. Anti-Aristotelians, on the other hand, roughly say that the sentence "It is sunny" expresses the proposition that it is sunny at t₀, where t₀ is the time of utterance, a proposition whose truth value does not vary between true and false as the facts alter.

Most presentists are Aristotelians about propositions, and most open futurists these days seem to be presentists. I will argue, however, that an open futurist should not be an Aristotelian about propositions. I think this means that an open futurist should not be a presentist.

Consider the sentence

1. I will freely put on a pink shirt in one day.

Let p be the proposition expressed by this. Clearly:

2. p is possible.

(Also, the negation of p is possible.)

According to the anti-Aristotelian open futurist, p is the proposition that I will put on a pink shirt on day d₀+1, where d₀ is November 14, 2014. The anti-Aristotelian open futurist holds that on November 14, p is not true, but that on November 14 it may become true. So the anti-Aristotelian open futurist has a nice way of accounting for (2). While it's impossible that today p is true, it is possible that p be true tomorrow, and that's enough to make p possible.

But the Aristotelian open futurist is in trouble. For on her view, on November 15, p doesn't tell us about how things are on November 15, but about how things are on November 16: it's a tensed proposition that on any day says how things will be on the next. But on no day is it true that on the next day I will freely put on a pink shirt, if open futurism is true. And open futurism isn't just a contingent thesis. So given open futurism:

3. It is impossible that p ever be true.

(And what cannot ever be true cannot become true either, since if something were to become true, it would then be true.) But surely:

4. If it is impossible that p ever be true, then p is not possible.

And that contradicts (2).

Two applications of 'ought implies can'?

First application:
1. God ought to exist.
2. So, God can exist. (Ought implies can)
3. So, God exists. (By Ontological Argument)

Second:
4. It's an evil for a person to die.
5. If naturalism is true, people can't live forever.
6. Evils ought not be there.
7. So, people ought not die.
8. So, people can live forever. (Ought implies can)
9. So, naturalism is not true.

In the second argument, for 5 to be true, the "can" must be stronger than metaphysical possibility. Thus that argument requires a stronger ought implies can principle.

All that said, I am sceptical of 1 and 7. An impersonal "ought" not addressed to any person is weird.

The core of the second argument, however, can perhaps be rescued:
10. It's an evil for a person to die.
11. What is normal for a kind of being is not an evil.
12. If naturalism is true, it's normal for people to die.
13. So, if naturalism is true, it's not an evil for a person to die.
14. So, naturalism is not true.

Possibility, probability and propensity

I have defended at length the idea that metaphysical possibility is grounded in the causal powers of things. It just occurred to me that this view is very naturally connected to the view that objective probability is grounded in causal propensities. We can think of probability as a measure of the degree of possibility, and of possibility as an attenuated kind of probability. If we see things in this very natural way—and hopefully its naturalness isn't just due to alliteration—then we have a unified and mutually supporting story about probability and possibility. Both are grounded in causal powers, but differently. Possibility is grounded in the bare existence of causal powers. Probability is grounded in the propensities of causal powers. If we have reason to accept one view, that tends to give us reason to accept the other.

Clearly anything that's made possible by the causal powers account of possibility—let's call this "causally possible"—is possible. So the only question about the causal powers account of possibility is whether it captures all possibilities. Suppose some things are possible but not causally possible. Then we can ask about their probabilities. If probabilities are propensities, then we should say that such things have zero probability, since nothing has a propensity to produce them. And not just the kind of "numerical zero" that classical probability assigns to a sequence of infinitely many heads, but the deep kind of zero that is had by the probability that one equals two. It's plausible, though, that things with this deep zero probability just can't happen. So the propensity account of probability neatly suggests the causal powers account of possibility.

And the converse is also plausible. If all possibilities are causal possibilities, it is very natural to measure the degree of their possibility by the causal propensities.

Of course the above is very vague. It may be that particular details of how one works out a causal powers account of possibility don't sit well with the particular details of a propensity account of objective probability.

Salmon's argument against S4

Start with:

1. If x originates from chunk α of matter and β is a non-overlapping chunk of matter, then x couldn't have originated from β.
2. If x originates from chunk α of matter and α' is a chunk of matter that almost completely overlaps α, then x could have originated from α'.

Iterating (2), and assuming a finite sequence of almost completely overlapping chunks between α and β, we conclude that an object x that originates from α possibly possibly ... possibly (with a finite number of possiblys) originates from β. By S4, we conclude that x could have originated from β, contrary to (1). Nathan Salmon uses this as an argument against S4.

But this is a mistaken line of thought. For (2) is not significantly more plausible than:

3. If x could have originated from chunk α of matter and α' is a chunk of matter that almost completely overlaps α, then x could have originated from α'.

Both (2) and (3) embody the same prima facie plausible small-variation intuition. If one thought (3) was false, one would have little reason to think (2) is true.

But given (3), Salmon's argument can be run without S4—all we need is T (what is actually true is possible). Iterating uses of (3) and modus ponens, we conclude that (1) is false. In other words, we cannot hold both (1) and (3). And since (2) has little plausibility apart from (3), we shouldn't hold both (1) and (2). Thus, Salmon's argument is not an argument against S4, but an argument against the conjunction of (1) and (2). And I say we should reject (2).

The possibility of unicorns

Kripke argued that it is not possible for there to be unicorns. For "unicorn" is a natural kind term. But there are many possible natural kinds of single-horned equines that match our unicorn stories, and there is no possible natural kind that has significantly more right to count as the kind unicorn. So none of them count, and no possible world contains unicorns.

But there is another approach, through vagueness and supervaluationism. Let's say that the term "unicorn" is vague. It can be precisified as a full description of any one of the possible natural kinds of single-horned equines that match our unicorn stories.

Now consider the sentence that Kripke is unwilling to assert but which seems intuitively correct:

1. Possibly there is a unicorn.

We get to say (1) if we can correctly affirm:

2. Definitely, possibly there is a unicorn.

And we certainly do get to say this. For (2) holds if and only if:

3. For every precification U of "unicorn", possibly there is a U.

And assuming that all the precifications of "unicorn" are natural kinds that are possibly instantiated (we don't allow as a precification of "unicorn" something whose eyes are square circles, etc.), (3) is true. And so even if there is no world at which definitely there is a unicorn (though there might be—maybe that world is very rich and contains animals that fall under all possible precifications of "unicorn"; Blake McAllister suggests a multiverse), it is definite that there is a world at which there is a unicorn.

Solipsism, presentism, actualism

Consider three debates: solipsism vs. other minds; presentism vs. eternalism; actualism vs. extreme modal realism. Let say that, like most people, we want to go for other minds and actualism. The sane view is that of course other people exist but unicorns don't. Can we get any guidance from this decision as to the presentism vs. eternalism debate? Is "now" more like "I", in which case we get the hint that we should be eternalists, or is "now" more like "actual", in which case the hint is that we should be presentists?

Here is one important way in which "now" is more like "I". I communicate with people who are other than I. I do not communicate with people who are other than actual. But I do communicate with people who are other than now: I read Plato and maybe even aspire to writing for people yet to be conceived. And even when we communicate with people who are now alive, typically—unless we're speaking at each other at the same time—we do so diachronically. I speak now and they will respond later. I respond now and they spoke earlier. So our conversation reaches across times, just as it reaches across people. But it does not reach across worlds.

Suppose we think of sentences like "I am sitting" as expressing self-locating propositions (I actually probably don't want to think of them like that). Here, then, is a closely related point, inspired (as really is the above stuff) by this article. I tell you on the phone: "I am sitting." In so doing, I express a de se self-locating proposition, a proposition that locates me. But while I express a self-locating proposition, that isn't what I communicate or even try to communicate. For if you accept the self-locating proposition that I am expressing, you will thereby take yourself to be sitting, and that's not what I am trying to communicate. So there is a difference between what I am expressing and what I am communicating: I am expressing a self-locating proposition that I am sitting, but I am communicating the non-self-locating proposition that Alex is sitting. Moreover, the two are closely related. Quite plausibly, in accepting the self-locating proposition that I am sitting, I am also accepting a non-self-locating version of it. It could even be that I express both.

But a similar thing happens with time. I hereby write: Alex is now sitting. In so doing, I express a tensed proposition (apologies for the use of "tensed" for non-linguistic entities), a temporally-locating proposition. But I do not communicate the same proposition to you. For while you might infer that I may still be sitting when you read the message, that's a risky inference of yours, not just what I communicated to you. If you just want to believe what I have informed you of, you will believe something that you may express with words like "Alex was sitting then." So there is a difference between what I expressed and what I communicated.

You might think that the difference is not a difference in kind. After all "Alex was sitting then" itself expresses a temporary proposition because of the past tense "was". But that, I think, is just an artifact of the fact that when you went from my "Alex is now sitting" to your "Alex was sitting then", you didn't just accept something that I communicated. Rather, you took what I communicated and combined it with the fact that my communication temporally precedes your reception of it, a fact you know empirically (but it would not affect my argument if you knew it a priori—it's still a fact over and beyond my communication). Sticking to what I communicate to you, you cannot think more than some proposition like that Alex is sitting at that time (where "that" refers to the time of my utterance).

There are now two options. We could go the presentist route and say that both what I expressed and what I communicated are tensed propositions. On this reading, what I expressed was that Alex is now sitting, but what I communicated was that Alex was, is or will be sitting then. But this doesn't seem to me to be a very attractive theory. For when things go right in communication, I shouldn't be communicating a proposition I didn't express, while to claim that I expressed two tensed propositions, though only one was communicated, seems odd. It makes it sound as if you only half believed what I said.

The superior reading, I think, is that I expressed both a tensed and an untensed proposition, and what I communicated was the untensed one.

These things combine. When I say: "I am sitting now", I express three things: a self-locating tensed proposition that I am sitting now, a non-self-locating tensed proposition that Alex (or that guy/gal) is sitting, and a non-self-locating untensed proposition that Alex is (tenselessly) sitting then. But only the last of these do I communicate when I communicate across a relevantly large time delay. But that's all predicated on the view on which there are de se propositions.

From ease to counterfactuals?

Consider the concept of how easy it is for a proposition to be made true, given how things are. It is by far easiest for propositions that are already true: nothing more needs to happen. It is hardest for self-contradictory propositions, like that Socrates is not Socrates: there is no way at all for it to happen. Contingently false propositions that require changes that go far back in time are going to be harder to be made true than ones that don't. And we can talk of the ease of p being made false as just the ease of not-p being made true. So, we can offer this account of counterfactuals:

• p→q holds if and only if it is easier for p to be made true than for the material conditional p⊃q to be made false.

This yields the Lewis-Stalnaker account of counterfactuals provided that we stipulate that a is easier to be made true than b if and only if there is a world where a holds which is closer than every world where b holds.

But we need not make this stipulation. We might instead take the easier to be made true relation as more fundamental. (And while we might define a closeness relation in terms of it—say, by saying that w₁ is closer than w₂ iff <w₁ is actual> is easier to be made true than <w₂ is actual>—depending on which axioms easier to be made true satisfies, that might not yield an account equivalent to the Lewis-Stalnaker one.)

On some assumptions, this is a variant of the central idea in yesterday's post.

The unavoidability of sin

Aquinas says that without grace we can avoid each individual mortal sin but not all mortal sin, at least not for a significant length of time.

On its face, this seems contradictory. After all, I can avoid the first mortal sin. If I avoid it, then I should be able to avoid the next one. And so on. And hence I should be able to avoid all mortal sin.

But this argument mistakenly agglomerates what one can do. For instance, suppose that there is a mine field with a thousand mines. I know how to defuse a mine, but I have an independent probability of 10% of slipping and detonating the mine. It's correct to say that each mine can be defused by me—being able to do this with 90% reliability is sufficient for this—but it is incorrect to say that I can defuse the whole minefield. It is appropriate to look at the minefield that one is to defuse and think: "This is an impossible task without help."

The person without grace is unable to presently control her future actions in favor of the good. Each individual action is in her power, but she cannot control them all at once. Hence she can rightly look with trepidation at her future moral life and say: "This is an impossible task without help."

But a virtuous person can control future actions in favor of the good en masse, by growing in virtue and resolving to do good.

Hiddleston's review of Actuality, Possibility, and Worlds

NDPR has a review of my Actuality, Possibility, and Worlds by Eric Hiddleston.  I think it's quite a helpful review--the concerns about my account are powerful and interesting.

This post is a very rough bunch of responses to Hiddleston, and will not be comprehensible without reading his review.

I am inclined to endorse some version of the "externalist" way out that Hiddleston gives.  I think this will damage at least one of my arguments against Platonism, the one that says that opponents of Platonism are "horribly confused" if Platonism is true. But Hiddleston is right that that's a bad argument.

I think Hiddleston doesn't give enough credit to my dogs argument against Platonism.  There, I am imagining that the Platonist heaven is augmented with the necessity of there being no dogs, but all earthly stuff is unchanged.  I claim in the book that nonetheless dogs would remain possible.  My line of thought behind that was that dogs would remain possible, because they would remain actual, and the actual is possible, no matter what the Platonist heaven says.  I think Hiddleston's Little-P = Big-P position doesn't help here.

(It occurs to me, by the way, that the Platonist could have a theory that escapes my dogs argument as it stands in the book. Here's the theory. The primitive property is mere possibility. Possibility is then defined in terms of mere possibility: a proposition is possible provided that it is either true or merely possible. But I think this version still has a problem. The original Platonic version has the puzzle of why it is that actuality implies possibility. This version doesn't have that problem. Instead it has the problem of why it is that that mere possibility implies non-actuality.)

Hiddleston also worries a lot about Euthyphro-type questions, like:

• (E1) Why should God be incapable of bringing about really impossible propositions, such as contradictory ones?
• (E2) Why should God be capable of bringing about really possible propositions?

I think there is a neat counter to the "contradictory ones" part of E1 that favors my view over other views. The following seems true to me:

1. If, per impossibile, God or any other agent were capable of bringing about a contradictory proposition, that proposition would be possible.

This suggests to me that what agents can bring about is actually more fundamental than what is contradictory. Notice that the plausibility of (1) highlights a difference between my view and divine command theory. In the case of divine command theory, the following seems false:

2. If, per impossibile, God were to command a horrendous deed, that horrendous deed would be obligatory.

(But see this paper of mine for a more detailed discussion of whether (2) is a good objection to divine command theory.)

I do think Hiddleston's question about what explains why God can do contradictory things is a good and difficult question, but I don't think they're quite species of the Euthyphro problem. I think I can say that there just does not exist any being with the power to bring about contradictory propositions. This is in need of no more grounding than the fact that there are no unicorns--it's just a negative existential. Is it in need of an explanation? My official line on the PSR restricts it to contingent truths. But maybe there still is an explanation of it in terms of some deep facts about the divine nature (maybe its beauty, say). Maybe Hiddleston's best bet here would be to push me in a way that Josh Rasmussen has done: I can't explain why God can't create square circles, just as the Platonist can't explain why everything that's actual is also possible, and so I don't have an advantage over the Platonist here. I don't know exactly what to say here, but I think one difference is with regard to per impossibile counterfactuals.

3. If dogs existed but the Platonic heaven didn't say that they were possible, dogs would still be possible.
4. If God were capable of creating a square circle, square circles would be possible.

I think (4) favors my view and (3) disfavors Platonism. But I am not happy to hang too much on per impossibile counterfactuals.

As for (E2), I don't feel the force of that. First of all, the primary view doesn't mention God: it's quantified over all agents. So the modified question is:

• (E2b) Why should every really possible proposition be such that there is an agent who can bring it about (or, more precisely, bring about a chain of causes leading to it)?

I don't feel much intuitive force to this question. Maybe I've just been thinking along the lines of my view for too long. I am tempted to say that the question is exactly like: "Why should every sample of water contain hydrogen atoms?" That's what Hiddleston labels the externalist way out, so I guess I am with him on that, and I wish I was explicit about that in the book.

An ontological argument

Start with:

1. (Premise) Possibly, an unlimited being exists.
2. (Premise) Necessarily, for every proposition q that is possibly true, there is a state of affairs p(q) such that p(q) grounds the possibility of q.
3. (Premise) Necessarily, if s grounds the possibility of x not existing or the possibility of x being limited, then s limits x.
4. (Premise) Necessarily, nothing limits an unlimited being.
5. (S5) If something is possibly necessary, then it is necessary.

It follows that there is an unlimited being. For suppose w is a world that contains an unlimited being u. Suppose it is true at w that u possibly does not exist. Then by 2 and 3, something at w limits u, which violates 4. So it is true at w that u necessarily exists. Suppose it is true at w that u is possibly limited. Then by 2 and 3, something at w limits u, which again violates 4. So, at w it is true that u exists necessarily and is necessarily unlimited. So, by S5, u exists necessarily and is necessarily unlimited.

The most controversial premises, I think, will be 1 and 2.

Two presentist ways of seeing worlds

If presentism is true, then right now, call it t₂, the proposition B that Bucephalus exists is false, but it once was true, namely at t₁. Now, at every time a token of the following sentence expresses a truth:

0. For all p, a proposition p is true if and only if it is true at the actual world.

Now, let's imagine ourselves at t₁. Then Bucephalus exists. Thus, B is true. Moreover, (0) expresses a truth, and so B is true at the actual world. So at t₁ the sentence

1. B is true at the actual world

expresses a truth. But now let's return to our time. B is false. But (0) expresses a truth, and so the sentence

2. B is not true at the actual world

does expresses a truth. Thus, (1) expresses a truth when said at t₁ but expresses a falsehood when said at t₂. This shows that either:

3. "The actual world" refers to different worlds at different times

or

4. The proposition that p is true at w can change in truth value, even if "p" and "w" refer rigidly to a proposition and a world, respectively.

Thus, the presentist has two ways of understanding possible worlds. Either possible worlds are tensed, so that at every time we inhabit a different possible world (that's option (3)) or else the "true at" relation is tensed, so that we inhabit the same world at different times, or when we say at t that p is true at w, we say something true if and only if p is true at t at w.

I think there is a problem for (4). Let p be the proposition that horses do or do not exist. Let t be the actual present time. Then p is true at every world, since it's a necessary truth. Now consider a world w where the time sequence does not include t. There are several options for this. Maybe in w, time comes to an end in 2011. Maybe time is discrete in w while in our world it is continuous, and so w either includes no times from our world or else w "skips over" t. Or maybe for some other reason the time sequence in w is radically different from our world's time sequence. Then p is true at w. But on (4), when we say that p is true at w, that is true if and only if p is true at t at w. But nothing is true at t at w, since t isn't a time at w.

Here's a slightly different way to see the point. When p is true at w, it is true either because there are or because there are not horses at w (this is an uncontroversial case of disjunctive grounding). Suppose it's true because there are not horses at w. But at which time are the horses not there at w? After all, w could have horses at some but not other times. Presumably, the relevant time is the present time. On proposal (3), every world comes along with its own present time, and this is fine. But on proposal (4), a world's relevant present time is our present time, and w doesn't have our world's present time.

One could try to solve this with counterpart theory for times. But one can suppose w won't have a counterpart to our time.

Here's a bolder move to defend (4) against our argument: The accessibility relation between worlds differs between times. The proposition p isn't true at all worlds, but only at all accessible worlds (this may or may not involve a denial of S5—S5 does not say that all worlds are accessible, but only that accessibility is an equivalence relation). And a world is only accessible if it includes the present time (or a counterpart to it?). This has the implausible consequence that what is metaphysically possible changes with time. For instance, if in w the time sequence comes to an end with 2011, then the proposition that w is actual was possible in 2011, but is no longer possible. But it's implausible that what is metaphysically possible changes with time.

If this is right, then the presentist should embrace (3). But is (3) plausible? Do we really live in different worlds at different times?

The presentist's other move is simply to abandon talking about worlds, and instead talk about, say, abstract times (in the Crisp sense).

48 arguments against naturalism

Consider this argument:

1. A desire to be morally perfect is morally required for humans.
2. If naturalism is correct, a desire to be morally perfect cannot be fulfilled for humans.
3. If a desire cannot be fulfilled for humans, it is not morally required for humans.
4. Therefore, naturalism is not correct.

This argument provides a schema for a family of arguments. One obtains different members of the family by replacing or disambiguating the underlined terms in different ways.

If one disambiguates "naturalism" as physicalism (reductive or not), one gets an argument against physicalism (reductive or not). If one disambiguates "naturalism" in the Plantinga way as the claim that there is no God or anybody like God, one gets an argument for theism or something like it. Below I will assume the first disambiguation, though I think some versions of the schema will have significant plausibility on the Plantingan disambiguation.

One can replace "morally required" by such terms as "normal", "non-abnormal" or "required for moral perfection".

One can replace "to be morally perfect" by "for a perfect friendship", "to be perfectly happy" or "to know with certainty the basic truths about the nature of reality" or "to know with certainty the basic truths about ethics" or "to have virtue that cannot be lost". While (1) as it stands is quite plausible, with some of these replacements the requiredness versions of (1) become less plausible, but the "non-abnormal" version is still plausible.

Probably the hardest decision is how to understand the "cannot". The weaker the sense of "cannot", the easier it is for (2) to hold but the harder it is for (3) to hold. Thus, if we take "cannot" to indicate logical impossibility, (2) becomes fairly implausible, but (3) is very plausible as above.

I would recommend two options. The first is that the "cannot" indicate causal impossibility. In this case, (3) is very plausible. And (2) has some plausibility for "moral perfection" and all its replacements. For instance, it is plausible that if naturalism is true, certain knowledge of the basic truths about the nature of reality or about ethics is just not causally available. If, further, moral perfection requires certainty about the basic truths of ethics (we might read these as at the normative level for this argument), then moral perfection is something we cannot have. And if we cannot have moral perfection, plausibly we cannot have perfect friendship either. Likewise, if naturalism is true, virtue can always be lost due to some quantum blip in the brain, and if moral perfection requires virtue that cannot be lost, then moral perfection is also unattainable. And perfect happiness requires certain knowledge of its not being such as can be lost. Maybe, though, one could try to argue that moral perfection is compatible with the possibility of losing virtue as long as the loss itself is not originated from within one's character. But in fact if naturalism is true, it is always causally possible to have the loss of virtue originate from within one's character, say because misleading evidence could come up that convinces one that torture is beneficial to people, which then leads to one conscientiously striving to become cruel.

The second option is that the "cannot" is a loosey-goosey "not really possible", weaker than causal impossibility by not counting as possible things that are so extraordinarily unlikely that we wouldn't expect them to happen over the history of humankind. Thus, in this sense, I "cannot" sprout wings, though it seems to be causally possible for my wavefunction to collapse into a state that contains wings. Premise (2) is now even more plausible, including for all the substituents, while premise (3) still has some plausibility, especially where we stick to the "morally required" or "required for moral perfection", and make the desire be a desire for moral perfection.

If I am counting correctly, if we keep "naturalism" of the non-Plantingan sort, but allow all the other variations in the argument, we get 48 arguments against naturalism, though not all independent. Or we can disjoin the conjunctions of the premises, and get an argument with one premise that is a disjunction of 48 conjunctions of three premises. :-)

Deep Thoughts XXXIII

Everything is possible except the impossible.

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.