Infinite sequences of choices

Alex,

Interesting post. A minor then a major comment.

Minor: in your paragraph 4, you mention that your problem only arises for the case of infinitely many l-free choices. I think more accurately, the problem arises for the case of no earliest l-free choice.

Major: we could approach the problem by asking whether God could know the contents of a world containing no earliest l-free choice. (If he can't know it, he can't create it. Sketch of proof: He has to create some segment or slice of the world; and whatever he creates, he knows what he is creating, since creation is an intentional action. But he has to also create anything in the past of that segment or slice, since it won't come into existence any other way [whereas the future could evolve or grow]. But the past of the world we are considering is infinite. So God has to know an infinite history in order to create the world, and he might as well know the whole thing. OK, that is very loose.)

Now, I think God cannot know what he is creating, in creating a world with l-free choices in it, unless Molinism is true. That is, if God knowingly creates a world in which a free creature S chooses X in circumstances C, God will ipso facto know the counterfactual, "If it were the case that C, S would freely choose X."

So it seems to me, at present, like this: If Molinism is true, God could create a world with an l-free choices infinitely far into the past. If Molinism is false, he could not. I am more sure of the latter claim than the former.

Alex,

Imagine a world like this. God strongly actualizes an infinite number of agents at T each of whom would perform his best action A. Suppose that no agent can do A at T, but the sooner an agent does A after T the better. Suppose it is true that, for each agent that can do A, there is another agent that can do A faster. If you can do A faster, then in order to do your best possible action, you must do it faster. So we have an infinite set of agents each completing A closer and closer to T. There's an actualizable backward infinite sequence in a finite amount of time.
Really, you don't need these agents. Let God do it. Suppose that for each temporal instant after the Big Bang at T in the interval (T, 1 second] God freely utters the word 'Good'. At each such utterance, we might suppose, God creates a smallest particle of matter (lots of particles, we can suppose, are tied for smallest). Perhaps that is God's way of creating matter in some worlds. Again, we have an infinite sequence of liberatarian free actions in the finite interval (T, 1 second].

Mike:

In your examples, how do you get the requisite interdependence? In the series that interests me, each choice affects later choices.

Oh, I see. To get interdependence require that each best action in the sequence depend on the preceding action. It is equally good if S1 does A or B. But if S1 does A at T', then the best that S2 can do is B at T''. And if S1 does B at T', then the best that S2 can do is A at T''. And so on down. The infinite sequence will either be A, B, A, B . . . or B, A, B, A . . . depending on what S1 does at T'. I'm sure that some sequence like this can be defined more rigorously.

Alex,

I just saw an announcement of your move from Georgetown to Baylor at the Leiter Report. Congratulations! Very good to have you in Texas--and just a couple of hours north! Again, congrats!

Mike:

Thanks for the congratulations. I am excited at the move.

Your example is clever, but I am not convinced, though I do not have a full refutation (just as I didn't really have a full argument to start off with, but just a worrying intuition.)

One worry is that the example may be structurally too similar to grim reaper examples and the same method you give can be used to generate a grim reaper paradox. Take your original version without interdependence. But now suppose that one of the actions open to each of the agents is to kill Jones who is alive at T. Moreover, suppose that killing Jones is good (e.g., Jones is an enemy dictator against whom a just war is being waged) and each agent has an independent probability 1/2 of doing so. For simplicity suppose Jones is going to die anyway within a hundred years. Intuitively, if your case is coherent, so is this one.

But this new case seems problematic. For at what time would we expect Jones to die? This seems to be a precise question: What is the expected value of the time of Jones' death? A random variable whose range is bounded has an expected value. But take any time T' after T. The probability is 1 that someone will have killed Jones before T', since there were infinitely many agents who could have done this, and each had probability 1/2 of doing it. This argument shows that the expected value of the time of Jones' death is T. But this is incompatible with Jones' being certain to be alive at T. (Theorem: A bounded random variable whose value is certain to be greater than T has an expected value greater than T.)

Hmm. Maybe grim reaper considerations in general can be used in Kalaam arguments? I should ask Bill Craig if he's seen that done before.

I am also worried by the epistemic possibility of the claim that, necessarily, if one is free to choose at T' then one chooses at T'--at least, one chooses whether to act or not to act. If so then the arrangement involves an uncountable infinity of choices.

I hope this entry is not, as for comments, dead yet. Alex said he has „never had trouble with the idea of God creating a world with an infinite past.“ Because I’m interested in kalam arguments, I’d like to know your opinion about the following three lines of reasoning which, as it seems to me, are quite promising. They do not relate Alex’s to entry with 100% relevance, nevertheless, this is a welcome opportunity for discussion that I do not want to throw away.

The First Argument

The basic pattern of this argument goes as follows: A. there is a metrically finite past; so, B. this past, taken as a whole, began to exist; so, C. the whole has a cause which is not a part of the whole; so, D. the cause is not temporal; so, E. the cause is God-like, etc.

Now, the argument for (A).

1. If there is a series of past temporal intervals of the same length which is metrically actually infinite (i.e., the series is of actually infinite duration), then it is possible that there is a token A (e.g., of machine-type) which assigns (in one-to-one correspondence and successively in time) to all of the past temporal intervals negative integers.
Like this:
Past intervals
... -4 -3 -2 -1
Premise.

2. There is a series of past temporal intervals of the same length which is metrically actually infinite. A premise for reductio.
3. It is possible that there is a token A which assigns to all of the past intervals of the series negative integers. From 1 and 2, by modus ponens.
4. It is possible that every past interval of the series has assigned by a token A just one specific (precise, particular) negative integer. From 3.
5. It is not possible that every past interval of the series has assigned by a token A just one specific negative integer.
Why? Because it holds for any past interval of the series and for any negative integer that an interval doesn‘t have assigned just one specific negative integer - because (in such an actually infinite series) every negative integer has already been assigned to some previous interval.
6. It is not the case that 2. From 4 and 5.

(Cont.)
However, Wes Morriston objects: "The Principle of Correspondence entails at most that all the numbers could have been counted by now, not that they would have been.” (http://spot.colorado.edu/~morristo/infpast.html) Thus, 5 does not follow.

Last year, W. L. Craig answered to me briefly (by e-mail): “I've defended the position that given infinite time, the counter would (not merely could) have completed its super-task. I note that this sort of inference is used frequently in physical science.” How this is supposed to solve the Morriston's objection? Craig writes about the practice in physical science. But what is the rationale for this practice?

But let me try to use Craig’s idea:

4.1. It holds for any past interval of the series and for any negative integer assigned to this interval, that it is possible that the negative integer is assigned to some previous interval.
Why? Because of the Principle of Correspondence.

4.2. If something can happen somewhere in a metrically actually infinite temporal series and there is a metrically actually infinite temporal series, then at any temporal state (say, t(x)) of this series that something happened already (i.e., it happened at some t(y) for y lower than x).
Premise.

4.3. It holds for any past interval of the series and for any negative integer that the negative integer has already been assigned to some previous interval.
From 4.1 and 4.2, somehow.

5. It is not possible that every past interval of the series has assigned by a token A just one specific negative integer.
From 4.3, somehow.

It seems that it is 4.2 that is mentioned in Craig’s anwer to me. It would be interesting to see how Craig would argue for 4.2 in detail. But he has given a sketch already: 4.2 is “used frequently in physical science.”

The argument aims to show only that it is not true that there is a series of past temporal (non-overlapping) intervals of a same length (e.g., 1 hour) such that the series is metrically actually infinite (i.e., the number of the intervals in the series is aleph-zero). Even if the argument is successful, it does not follow that there is no metrically amorphous time - i.e., time series of temporal states which is ordered according to the relation earlier than, but such that the distance between non-overlapping intervals is not defined. Sure, metrically amorphous time seems bizzare (and I don't know what is the usefulness of such a concept), but it is quite standard objection to kalam arguers. The objection can run like as follows. If there there is a metrically amorphous past earlier than metrically finite past, you cannot argue like this: A. there is a metrically finite past; so, B. this past, taken as a whole, began to exist; so, C. the whole has a cause which is not a part of the whole; so, D. the cause is not temporal; so, E. the cause is God-like, etc. 4 does not follow because a cause of the whole metrically finite past could be (some part of) a metrically amorphous past. Still, this seems to me to be the most promising metaphysical kalam argument by W. L. Craig.

The Second Argument

It's more "empirical" and less "metaphysical".

1. Whatever begins to exist has a cause of its existence. 2. The universe began to exist. 3. Therefore, the universe has a cause of its existence. Grant 1. 2 is confirmed by contemporary physics. E.g., „Borde, Vilenkin, and Guth were able in 2001 to generalize their earlier results on inflationary models in such a way to extend their conclusion to other models. Indeed, the new theorem implies that any universe which has on average been globally expanding at a positive rate is geodesically incomplete in the past and therefore has a past boundary“ (W. L. Craig, „Naturalism and the Origin of the Universe“, forthcoming). The universe has some temporal beginning. Spacetime is restricted to the universe, no spacetime outside the universe (maybe because of Ockham's razor?). So, the cause is outside space and time, So, it is changeless. So, it is immaterial. Since we should not multiply entities byond necessity, the cause is one single being, not a collection of many beings. The cause is very powerful - it's the cause of the universe. There are only two (known) kinds of immaterial beings: abstract objects and minds; so, it is a mind. Physical constants permitting intelligent life are tuned very finely and precisely, so, the cause is very intelligent. Thus, we have an ultramundane, immaterial, very powerful and intelligent person, who created the universe. A God-like being.

The Third Argument

Now a kalam argument against the infinitude of the past from the principle of sufficient reason. It is inspired by D. S. Oderberg's paper "Traversal of the Infinite, the "Big Bang," and the KCA", Philosophia Christi 2002, pp. 310ff.

Suppose we have a person called Tristam Shandy. Shandy writes an autobiography. It's very detailed: it takes him a year to record one day of his life. Suppose further that the past is infinite: there is actually infinite number of past days. And suppose Shandy writes from eternity past.

Now, Oderberg asks (p. 311), could Shandy complete his life story? (1) we see Shandy, before us, putting the final full stop to the final page of the last page of his autobiography; (2) Shandy has been writing for an infinitely long time. Are (1) and (2) compatible? Oderberg says they are not - at least if the principle of sufficient reason (which reads: every event has an adequate explanation) holds.

Pp. 315f.: "The thought is that there is no adequate explanation for Shandy's having completed his task now rather than yesterday, or the day before, or ... The fact that he has "had long enough" does not suffice ... But what else could explain the precise time of his finishing? Not the the Principle of correspondence ... because whatever time in the past we select as a possible finishing time, that and all prior days can be correlated; so far as the principle is concerned, any finishing day is as good as another. But it is hard to see what else can be advanced as an adequate explanation of Shandy's finishing at one time rather than another. Yet ... (1) ... just is the hypothesis that he does finish at a particular time ... [T]he hypothesis, in conjunction with part (2), violates the Principle of Sufficient Reason ...“

Here's a formal presentation. First, some abbreviations.
"E(p)" stands for "There is an adequate explanation of the fact that p". p is a proposition about events to be explained.
Let's have two concrete proposition f and g. "f" stands for "Tristam Shandy finishes at a particular time, say, t(0)". "g" stands for "Tristam Shandy finishes at a previous particular, say, t(-1)".
"not-p" stands for "It is not true that p."

Now, the argument.

(1) We see Shandy, before us, putting the final full stop to the final page of the last page of his autobiography.
Premise.

(2) Shandy has been writing for an infinitely long time.
Premise.

(3) not-E(f and not-g) and not-E(g and not-f)
From (2), the Principle of Correspondence, and the fact that only the Principle of Correspondence can be useful in the given context. (1) and f mean the same.

(4) From
not-E(f and not-g) and not-E(g and not-f)
we can infer
not-E((f and not-g) or (g and not-f)).
Why? Because:"If neither of two events have an adequate explanation, there is no adequate explanation of their disjunction." (p. 316)

(5) not-E((f and not-g) or (g and not-f))
From (3) and (4), by modus ponens.

(6) (f or g) and not-E(f or g)
From (1) and (5)

(7) The principle of sufficient reason (Oderberg-version): every event has an adequate explanation.
Premise.

(6) contradicts (7). Thus, ((1) and (2) and (7)) is not consistent.

The Third Argument Modified

Well, Oderberg suggests that a similar argument can be made for the general conclusion that it is not true that the series of past temporal states is actually infinite. He wrote (pp. 316f.): "the hypothesis of such a series violates PSR". There is "no adequate explanation of how the series can terminate at any specific point" (just as there is no adequate explanation of the fact that Shandy finishes his autobiography at a particular time); thus, "there is no adequate explanation of how it can terminate tout court".

I've wondered how would the argument look like - because I haven't been able to read it off Oderberg's paper.

I've tryed to make use of the kalam argument (against the infinite past) from the task of counting all negative integers (see above, section I). This argument seems relevantly similar to the case of Tristam Shandy. I started with this idea: If the series of past temporal states (ordered according to the relation earlier than) is actually infinite (i.e., having aleph-zero cardinality), then it is metaphysically possible that there is a token entity P (e.g., of machine-type) which assigns (in one-to-one correspondence) to all of the past temporal states negative integers. (By "temporal states" I mean really different maximal complexes of simultaneous occurences - i.e., really different complexes of whatever occurs at a time.) Maybe I can show, I thought, that there is no explanation why P finishes its counting (more accurately, its putting into one-to-one correspondence with all temporal states) of all negative integers at a particular time. But I found out that I am not able to deduce the conclusion (that there is no explanation why P finishes at one specific time or at the previous time) from metaphysical possibility of P and that I need P to be in actual world (details apart). Now, If P is in actual world and if it suffices for P to exist that the past is infinite, it would be strange to say that P is a concrete (as opposed to abstract) entity or that P is a machine. P should be something whose existence is more ready to be entailed. What about some abstract process? Maybe. I also found out that someone might object that finishing at one specific time is essential for P - this would block the argument because in such a case it would be strange to say that there is no explanation why P finishes its task at a particular time. From these considerations arose the premise 1 below.

Abbreviations:

"e": There is some collection K of past temporal states which is: actually infinite (i.e., has aleph-zero cardinality); in actual possible world; ordered according to the relation "temporally later than", in this way:
... C B A
(State A is later than state B, B is later than C, etc. State A is the latest temporal state.)

"f": There is a token P (of abstract-process-type) which: is in actual possible world; gives successively all members of K into one-to-one correspondence with all negative, whole, ordinal numbers in such a way that: there is the correspondence between ordinal numbers of members of K on the one side and temporal topological properties of members of K on the other side; state A corresponds to -1, in this way:
... C B A
... -3 -2 -1
(The numerals should be exactly below the letters.)

(Topological correspondence means: if state x has higher ordinal number than state y, than x is temporally after y. E.g., if x has ord. nr. -4 and y has or. nr. -8, then x is temporally after y. Negative number -4 is higher number than -8. But maybe the talk about topological correspondence is unnecessary.)

"g": P gives successively all members of K into one-to-one correspondence with all negative, whole, ordinal numbers in such a way that: there is the correspondence between ordinal numbers of members of K on the one side and temporal topological properties of members of K on the other side; state B corresponds to -1, in this way:
... D C B
... -3 -2 -1
(The numerals should be exactly below the letters.)

1. If e, then (f and (it is metaphysically possible that g))
Premise.

2. e
Premise, for reductio.

3. f and (it is metaphysically possible that g)
From 1 and 2, by modus ponens.

4. f
From 3, by simplification.

5. not-E(f and not-g) and not-E(g and not-f)
Premise.
Why this premise? Because, to paraphrase Oderberg, "any finishing" state is "as good as another" for completing the task. And "it is hard to see what... can be advanced as an adequate explanation of" P's finishing at one state "rather than another". Thus, "there is no adequate explanation for the fact that" P completes at the temporal state A and not at the temporal B "and no adequate explanation for the fact that" P completes at B and not at A. (p. 316)

6. (not-E(p) and not-E(q)) entails not-E(p or q)
Premise.
If "neither of two events have an adequate explanation, there is no adequate explanation of their disjunction. (Example: there is no adequate explanation of the fact that I am ill, and no adequate explanation of the fact that I failed to submit my paper, so there is no adequate explanation of the fact that either I am ill or failed to submit my paper.)" (p. 316)

7. not-E((f and not-g) or (g and not-f))
From 5 and 6.

8. (f or g) and not-E(f or g)
From 4 and 7. Cf. p. 316 (cited above).

9. Every event has an adequate explanation.
Premise.

9 is in contradiction with 8. Where is the problem? A hypothetical supporter of the argument says the problem is premise 2. And maybe the argument can be modified to deal with the past series with greater than aleph-zero cardinality, too. However, the premise 1 is quite bizzare, isn't it?

Thank you so much for your comments!

Vlastimil,

What an embarrassment of riches!

I want to say something about the principle of sufficient reason (PSR) based argument. That argument presupposes that time is not merely relational. If time were merely relational, then a world where all events were shifted over by a day would be the same world.

Note that this argument would also disprove the hypothesis that the universe had a beginning in time but there was an infinite empty past before that. For we could then ask--as Augustine does in the Confessions--why the world was not created earlier, and it seems no reason can be given.

Moreover, arguments of the same sort can be employed to rule even a universe that does have a beginning in time. Given a world w that has a beginning in time, let w(a) be a world like w but which is temporally stretched out by a factor a, where a is a positive real number. What do I mean? Well, when an event happens at time t in w, it happens at time at

A relationalist about time can say that w(a)=w for all positive w. There is no difference between a world and a stretching of it.

Vlastimil,

This is a fascinating series of arguments!

I'm not quite clear on the meaning of 4.2, but I can see at least one thing it might mean that, with the aid of appropriate assumptions, can be shown to have probability 1. Here's the claim:

4.2. If something can happen somewhere in a metrically actually infinite temporal series and there is a metrically actually infinite temporal series, then at any temporal state (say, t(x)) of this series that something happened already (i.e., it happened at some t(y) for y lower than x).

Here's the interpretation on which this comes out true. Tile the negative portion of the timeline with equal-sized intervals [-1, 0), [-2, -1), etc. Take "can happen" to mean that there is some minimum real nonzero probability r that the event of interest happens in each such interval of time given that it has not happened in any previous interval. Then the probability that it does not happen in any given interval, given that it hasn't happened up to that time, is bounded by (1 - r). (Bounded, because for any given interval the probability might be greater than r -- that is just a minimum.) And the probability that it doesn't happen in a sequence of n consecutive intervals, given that it hasn't happened before that sequence, is bounded by (1 - r)^n. But ex hypothesi we have an infinite sequence of such intervals in the past. Hence, the probability that the event hasn't happened in this sequence is zero.

Tim,

Thank you. As for 4.2, I should note that Craig has some examples of using (something like) this premise at least in physical cosmology. He talks about probabilities too, though not so exactly as you. Your interpretation is very helpful.

Here's a quotation from Craig, http://www.leaderu.com/offices/billcraig/docs/ultimatequestion.html, section Vacuum Fluctuation Models:

"Vacuum Fluctuation Models did not outlive the decade of the 1980s. Not only were there theoretical problems with the production mechanisms of matter, but these models faced a deep internal incoherence. According to such models, it is impossible to specify precisely when and where a fluctuation will occur in the primordial vacuum which will then grow into a universe. Within any finite interval of time there is a positive probability of such a fluctuation occurring at any point in space. Thus, given infinite past time, universes will eventually be spawned at every point in the primordial vacuum, and, as they expand, they will begin to collide and coalesce with one another. Thus, given infinite past time, we should by now be observing an infinitely old universe, not a relatively young one. About the only way to avert the problem would be to postulate an expansion of the primordial vacuum itself; but then we are right back to the absolute origin implied by the Standard Model. According to Isham this problem proved to be "fairly lethal" to Vacuum Fluctuation Models; hence, these models were "jettisoned twenty years ago" and "nothing much" has been done with them since."

And here's another case of using 4.2 in physical cosmology: if the past of the universe is metrically infinite, then the universe would be much colder, much darker and less dense (than it actually is) and lifeless. Ergo.

A similar idea can be found in W. L. Craig, „Naturalism and the Origin of the Universe“, forthcoming.

It is another promising ground for accepting the premise 2 in the Second Argument.

Vlastimil, I want to address your first argument presented for "A. there is a metrically finite past".

You have:

5. It is not possible that every past interval of the series has assigned by a token A just one specific negative integer.
Why? Because it holds for any past interval of the series and for any negative integer that an interval doesn‘t have assigned just one specific negative integer - because (in such an actually infinite series) every negative integer has already been assigned to some previous interval.

6. It is not the case that 2. From 4 and 5.

First for point 5. I confess I can't quite make out the justification for making it. Is the claim that the token can't assign a unique negative number for each past hour because it already used up all the negative numbers when you ran it an hour ago for all of the then past hours?

As to 6, formally, if 1 thru 6 lead to a contradiction, you should justify which premise to reject. I'm thinking of rejecting 1 or 5 (or what justifies 5) just as much as 2.

Craig,

Thanks for your very good comment.

As for the point 5 in the First Argument: I do not claim, e.g., that the past comprising aleph-zero hours and another past comprising aleph-zero + 1 hours don't have the same cardinality. I belive they do. I also belive that it holds for both of them that there is one-to-one correspondence between all negative integers and the hours. However, the claim that the correspondence can't be established/represented by the token SUCCESSIVELY (as in the described scenario) has some plausibility. Unfortunately, currently, I am not able to present this plausibility more clearly and in a way which would be relevantly different from the First Argument. If anyone is able to do it, it would be great to hear about that.

As for the point 6 in the First Argument: you’re just right. I haven’t justify rejecting the point 2 rather than, say, the point 1. Similar objection applies to the Third Modified Argument. Again, if anyone see which points are more plausible (and which points should be rejected), let us know, please.

For now, a brief addendum.

One could argue for the premise (9), i.e., the principle of sufficient reason, in the Third Argument by way of Richard Gale's and Alex Pruss' cosmological argument. See: http://agorametaphysica.blogspot.com/2006/09/atheists-cosmological-argument-for.html
Or one could argue in their manner from the premise that every contingent event has some possible explanation for the sub-conclusion that it has an explanation in the possible world in which it obtains. This would seem to allow to get along without the postulation of the existence of the process called "P" in the ACTUAL world.

More later.