## Thursday, September 21, 2023

### Dry eternity

Koons and I have used causal paradoxes of infinity, such as Grim Reapers, to argue against infinite causal chains, and hence against an infinite causally-interconnected past. A couple of times people have asked me what I think of Alex Malpass’s Dry Eternity paradox, which is supposed to show that similar problems arise if you have God and an infinite future. The idea is that God is going to stop drinking (holy water, apparently!) at some point, and so he determines henceforth to act by the following rule:

1. “Every day, God will check his comprehensive knowledge of all future events to see if he will ever drink again. If he finds that he does not ever drink again, he will celebrate with his final drink. On the other hand, if he finds that his final drink is at some day in the future, he does not reward himself in any way (specifically, he does not have a drink all day).”

This leads to a contradiction. (Either there is or is not a day n such that God does not drink on any day after n. If there is such a day, then on day n + 1 God sees that he does not drink on any day after n + 1 and so by the rule God drinks on day n + 1. Contradiction! If there is no such day, then on every day n God sees that he will drink on a day later than n, and so he doesn’t drink on n, and hence he doesn’t ever drink, so that today is a day such that God does not drink on any day after it. Contradiction, again!)

Is this a problem for an infinite future? I don’t think so. For sonsider this rule.

1. On Monday, God will drink if and only if he foresees that he won’t drink on Tuesday. On Tuesday, God will drink if and only if he remembers that he drank on Monday.

Obviously, this is a rule God cannot adopt for Monday and Tuesday, since then God drinks on Monday if and only if God doesn’t drink on Monday. But this paradox doesn’t involve an infinite future, just two days.

What’s going on? Well it looks like in (2) there are two divine-knowledge-based rules—one for Monday and one for Tuesday—each of which can be adopted individually, but which cannot both be adopted, much like in (1) there are infinitely any divine-knowledge-based rules—one for each future day—any finite number of which can be adopted, but where one cannot adopt infinitely many of them.

What we learn from (2) is that there are logical limits to the ways that God can make use of divine foreknowledge. From (2), we seem to learn that one of these logical limits is that circularity needs to be avoided: a decision on Monday that depends on a decision on Tuesday and vice versa. From (1), we seem to learn that another one of these logical limits is that ungrounded decisional regresses need to be avoided: a decision that depends on a decision that depends on a decision and so on ad infinitum. This last is a divine analogue to causal finitism (the doctrine that nothing can have infinitely many things in its causal history), while what we got from (2) was a divine analogue to the rejection of causal circularity. It would be nice if there were some set of principles that would encompass both the divine and the non-divine cases. But in any case, Malpass’s clever paradox does no harm to causal finitism, and only suggests that causal finitism is a special case of a more general theory that I have yet to discover the formulation of.

Walter Van den Acker said...

Alex

"Foreknowledge of libertarian free choices is impossible"

Majesty of Reason said...

Hey Alex!

Great post! Here are two comments, which I'll break up into two parts due to character limits. This is the first comment. (Terminology: ’CFist’ means causal finitist.)

(1) You write: “What’s going on? Well it looks like in (2) there are two divine-knowledge-based rules—one for Monday and one for Tuesday—each of which can be adopted individually, but which cannot both be adopted, much like in (1) there are infinitely any divine-knowledge-based rules—one for each future day—any finite number of which can be adopted, but where one cannot adopt infinitely many of them.

What we learn from (2) is that there are logical limits to the ways that God can make use of divine foreknowledge.”

But it seems to me like the non-CFist could then say the same thing about the GR Paradox. In the GR paradox, there are two conditions: (i) there’s an infinite set S, linearly ordered by R (e.g., causation), that is beginningless (so that for every member m, there is another member n such that nRm); and (ii) for each member of set S, that member is F iff no prior member is F. (Note that n is prior to m iff nRm or else there is a linear sequence of members connected by R starting with n and ending with m). Each of these may be possible individually, but they cannot both be true of S. What we learn from the GR Paradox is simply that there are logical limits to the ways that infinite linearly ordered collections of items can behave. This doesn’t mean that there cannot be infinite collections of items linearly ordered by R (where R is, e.g., causation); it simply means that in any world in which there are such collections of items, they don’t *also* satisfy (ii), just as in any world in which God satisfies one intention, he doesn’t *also* satisfy the other intention (even though each may be individually possible for God to satisfy).

Majesty of Reason said...

Here's the second comment! :)

(2) Ultimately, I think what’s needed is some reason to think that the bare individual possibility of (i) *implies* the joint possibility (i)-and-(ii). Since (i)-and-(ii) is impossible, we could then infer the impossibility of (i) for some ordering relation R (e.g., causation). But to avoid future-oriented theistic paradoxes, our reason for thinking that the bare possibility of (i) implies the joint possibility of (i)-and-(ii) should not equally motivate the claim that the bare possibility of

(a) A theistic endless future

implies the joint possibility of (a)-and-(b), where (b) is...

(b) for each member of some infinite collection of items spread out over the endless future, that member is F iff no later member is F.

For (a)-and-(b) is impossible, and yet the theist should maintain that (a) is individually possible. So the justification for the claim that ♢(i) —> ♢((i)-and-(ii)) should not equally carry over to the claim that ♢(a) —> ♢((a)-and-(b)).

The trouble is that the reasons I’ve come across for thinking ♢(i) —> ♢((i)-and-(ii)) seem to equally motivate thinking ♢(a) —> ♢((a)-and-(b)).

For instance, it’s individually possible for God to reveal whether a future Reaper scythe-swings to a particular Reaper with the intrinsic power and disposition to scythe-swing iff God reveals to it that no Reaper in its future scythe-swings. If the future could be endless and theism is true — that is, if ♢(a) — then there’s a possible world with enough spatiotemporal ‘room’ to accommodate infinitely many duplicated ‘patches’ of this individual possibility. Per recombination principles, there’ll be a possible world in which we get both (a)-and-(b) satisfied in a future-oriented paradox. So recombination reasoning seems to equally support ♢(a) —> ♢((a)-and-(b)).

Likewise, we can run ‘mysterious force’ reasoning parallel to your (2018) by equipping God and infinitely many future Reapers with indeterministic and independent coins, such that a paradox results only if their coins all land a very specific way. The reasoning runs in much the same way as your reasoning.

So the real challenge, I think, is twofold. First, the theistic CFist needs to respond to the future-oriented paradox in ways that don’t equally legitimate a corresponding move for the non-CFist in response to the GR paradox. This was my (1). Second, the theistic CFist needs to find a way to motivate ♢(i) —> ♢((i)-and-(ii)) that doesn’t equally motivate ♢(a) —> ♢((a)-and-(b)). This was my (2). In my estimation, these challenges haven’t been met, but I'm open-minded about whether they can be!

Heavenly Philosophy said...

Yeah, the CFist is denying something beyond just the contradiction in this case. They're denying that infinite divine decisional regresses are possible. That rules out the paradox. It's just denying a version of (i). The CFist is doing the exact same thing they did with the Grim Reaper Paradox. They're using it to infer that an infinite regress of a certain type is impossible. It's the same inference.

Also, one might say that divine rationality prevents God from creating any paradoxes.

Alexander R Pruss said...

Thanks for these great responses.

But I definitely wasn't saying that SPECIFICALLY the rules I give are forbidden due to the contradiction. Both cases to a general constraint: a prohibition on quasi-causal circles and a prohibition on quasi-causal infinitism (I did miswrite when I said that what's forbidden are ungrounded decisional regresses--that's not all that's forbidden), where quasi-causation is whatever it is that happens when a counterfactual connection is mediated by a divine action.

Checking back with my infinity book, it looks like I have already said all this stuff in Section 9.3.4.

All that said, I also note an interesting suggestion that there might be a way around the "mysterious force" objection in the special case of God. "While it would be ad hoc if we simply ruled out all the paradoxes one by one without invoking a single covering principle like causal finitism, in the case of God’s involvement perhaps this is not ad hoc. For maybe God’s perfection would not permit him to place a person in a situation where there is a rationality paradox.
God is himself a rational being, rational beings are made in God’s image, and to act
irrationally is to act in some sense against God. Thus, perhaps, God couldn’t put a
person in a position where rationality required two incompatible courses of action."

Heavenly Philosophy said...

It looks like we both said almost the same thing at the same time lol.

But yeah, like what Dr. Pruss and I said, the causal finitist isn't suddenly changing their attitude when this new paradox comes up. They have the same response as to the Grim Reaper Paradox. They're being perfectly consistent.