Tuesday, April 19, 2016

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 pn true, but nothing has a lower order power to do so.
Now, let P be the proposition that all the pn 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 pn+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 xn who has no children. Let pn be the proposition that xn 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.


Mark Rogers said...

Is adoption out of the question? You, Alex, seem far more closely related to Abraham, than the progeny produced would be to the joint budding of childless people.

Heath White said...

I think your last option is the best, conceptually speaking.

Think of the power to make P true as a kind of modal operator. Then the point of the clause that [P is possible =df there is something with an nth-order power to make P true] is that possibility is preserved over iterations of this operator. And the problem is that if there are infinitely many iterations, there is no nth-order power for any n.

But S5, anyway, has it as a theorem that any number of iterations of the diamond (possibility) operator collapse. And this is constitutive of the notion of possibility it is modeling; anything possibly-possible is, conceptually, possible simpliciter. So anything that lost this very simple conceptual connection would be just dealing with a different concept of possibility. Your last option handles this correctly but the original account does not (which is, of course, the point of the post).

Maybe what you want is a recursive definition of "has a power to make P true":
* P is possible if there is something with the power to make P true.
* X has the power to make P true if either (a) it can do it directly or (b) it has the power to bring about a power to make P true.

Alexander R Pruss said...


Additionally, the medieval thought (popularized by Descartes) is that iff x has the power to produce A, then A is found "eminently" in x. It's plausible that "is found eminently in" is transitive.

I am not clear why my original case isn't a counterexample to your proposed recursive definition if we replace X by the plurality or fusion of the x_n. X has the power to make P true, but not directly, nor at one remove, nor at two removes, etc.

Heath White said...

I'm not sure I follow the business about the fusion of the x_n. I think recursive definitions can go to infinity; that is how numbers are defined.

I can think of two ways to argue that your original case is not a counterexample, which may come to the same thing.

1) Treat universal quantification as a series of conjunctions. So “all the pn are true” amounts to the infinite conjunction “p1 & p2 & …” . The conjunction is possible iff each conjunct is possible. But each conjunct is possible.

2) Suppose for reductio that “all the pn are true” is not possible. Then necessarily, not all pn are true. Then necessarily, some pn is not true. But every pn is possibly true. Contradiction. (Is that valid?)

Alexander R Pruss said...
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Alexander R Pruss said...

Let pn = "The tallest person has finite height and is more than n inches tall."
Then each pn is possible (arguably).
In fact the conjunction of every finite set of pn is possible.
But the conjunction of the pn is impossible.

So one cannot go from possibility of conjuncts to possibility of conjunction.

Heath White said...


You're quite right: quite generally it is fallacious to go from the possibility of conjuncts to the possibility of conjunction. Consider Mp & M~p.

OK: on reflection, so long as you are willing to countenance the possibility of an actual infinite, I think you are going to be stuck with a problem if you require possibilities to rest on finite-order powers. One reply to your original scenario is that an infinite series of planets is not possible. But of course it seems that it is.

One antidote would be to allow infinite-order powers. I'm not sure that's attractive for various reasons. I think you'd need all the orders of infinity.

Alexander R Pruss said...

And even a future infinite will be a problem, not just a simultaneous one. For instance, suppose that God were going to allow the human race to continue reproducing forever (apparently contrary to fact). Suppose I have no children but am fertile. Let p be the proposition that I will have infinitely many levels of descendants (children, grandchildren, etc.). Clearly p would be possible, and intuitively grounded in my powers in some sense, but nonetheless it's not grounded in any nth order power of me.

Heath White said...

Yes, that's very simple and clear. I think the original proposal is refuted.

Alexander R Pruss said...

Here's a reason to think that while the detailed account is refuted, the basic idea is not.

Consider this proposal:
(*) Metaphysical possibility = causal possibility.
This is the basic idea of my account. Now, my counterexamples to the original accounts are clearly cases of *causal* possibility. So they aren't counterexamples to (*). Rather, they are counterexamples to how I characterized the right hand side.

In other words, I can continue to maintain (*) while saying that the right hand side of (*) is more difficult to explain than we thought.

Heath White said...

Correct. It was the "for some [finite] n" that I thought was problematic.

Michael Gonzalez said...

My two cents: There already paradoxes aplenty when one tries to have an actual infinite (especially one in which there are causal connections among the members), so this would just be yet another one of those. Such considerations have always (rather naturally) led me to believe that actual infinites are impossible (at least, actual infinites with causal relations). Given the strength of the powers approach to modality, I would just call this yet another very good reason to think actual infinites are impossible.

Michael Gonzalez said...

As for your example about future infinites (your descendants ad infinitum), I am a little surprised that the main proponent of the Grim Reaper paradoxes doesn't just chalk this up to the impossibility of reaching an actual infinite by successive additions. You will, obviously, never actually have infinitely many descendants. You are performing additions of finite + finite... how could that ever yield an infinite??

Alexander R Pruss said...

This way out would require an open future view, and that conflicts with eternalism, Christian tradition, Scripture, and perhaps even classical logic (depending on how it's spelled out).

Michael Gonzalez said...

One more quick question (sorry to spam the page!)... I thought your account was more that modal truths are GROUNDED in causal powers, but that the truths themselves are ideas in the mind of God (the sort of "Aristotelian-Leibnizean" fusion proposed in your book on the topic... with the option of sprinkling in some Tractatus, but that's not really relevant here...). If so, then, can the ACCOUNT of possibility not be more about the ideas in God's mind, but the GROUNDING be based in causal powers of actual things (including God)?

I still think actual infinites with causal connections are impossible (it's just Hilbert's Hotel again), but at least this ought to somewhat assuage your concern that you are demoting causal powers from being the full account to being a mere grounding, no?

Michael Gonzalez said...

Have you read William Lane Craig's work on this topic? His thoughts on actual infinity might be relevant here, from a Christian perspective. For that matter, Peter van Inwagen, Richard Swinburne, Dean Zimmerman, and lots of other Christian philosophers go even further and have OPEN views of the future. For my own part, I can't think of anything in Scripture that would be endangered by this.

Alexander R Pruss said...

Craig, van Inwagen and Swinburne think there are full facts about every future time. That seems to be enough to generate the problem I have.

Open future views can get out of that difficulty. But the cost! We can't even reasonably assign high probabilities to propositions about the future, since how can one reasonably assign a high probability to a proposition that one is sure isn't true!

Alexander R Pruss said...

And, yes, my story has two parts. Worlds are divine ideas. But what makes them *possible* is that there are entities (especially God) that can make them true.

Michael Gonzalez said...

I would say that Craig thinks there are full facts in the present, about what I *would* do in circumstances that I will encounter, but I get your point. I'm more inclined toward the Zimmerman sort of totally open view. Robin Collins would be another example.

Anyway, don't you think you can assign high probabilities to fantasies that you know are false (for example, high probabilities of what will happen in the world of Harry Potter, given the right set-up and circumstances)? So, why not for the fantasy of the future? It still rightly aligns your expectations, just as probabilities are supposed to do, without extra ontological commitment.....