An Objection to the Eternal Society Paradox

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Intro

The Eternal Society Paradox is one in which an eternal society with the abilities of ordinary human beings in each year of its existence uses its modest abilities to create a logical contradiction, thereby casting doubt on the metaphysical possibility of an infinite past. If the past is finite, this in turn can be used as part of an argument for the universe having a transcendent personal cause. Jimmy Akin wrote an article that among other things critiqued the Eternal Society Paradox argument against an infinite past. The original link is here but in case something disastrous happens, the archived link is here.

Background

Roughly (in the paper the Eternal Society Paradox was published), an Eternal Society is a society that has existed for a beginningless, infinite duration of time and has the abilities of ordinary human beings in each year of its existence; e.g. in each year people in the society can flip coins, write books, sing songs, and pass on information possessed in the current year to the next year. Because of the society’s extremely modest abilities, it seems like an Eternal Society would be possible if an infinite past were possible (note that by “possible” in this article I’ll be referring to metaphysical possibility, as opposed to e.g. physical possibility).

Now imagine the Eternal Society has the following Annual Coin Flipping Tradition (ACFT):

For each year y in the beginningless, infinite past: the society flips a coin, and they do the chant if and only if these two conditions are met: (a) the coin comes up heads; (b) they never did the chant in a year prior to y.

Less technically but somewhat more roughly: ACFT is where every year they flip a coin and they do a chant in that year if and only if (a) the coin comes up heads; and (b) they never did the chant in a prior year. Even more crudely put: each year they flip a coin. If the coin comes up heads and they never did a particular chant before, then they do the chant; otherwise they do not do the chant for that year.

The coin flips are probabilistically independent events, so any particular infinite permutation of coin flips is equally unlikely but also equally possible. Consider scenario S1 in which the coin came up heads for the first time last year for the Eternal Society practicing the aforementioned Annual Coin Flipping Tradition. The Eternal Society gets together to do the chant for the first time. This seems like it would be possible if an infinite past were possible (an eternal society with the ability of ordinary humans, by which I mean the society has the ability of ordinary humans in each year of its existence, could surely do something like this), but this scenario is provably not possible.

Again, the coin flips are probabilistically independent events, so if scenario S1 were possible, then another scenario, that we can call scenario S2, would be possible: the coin came up heads each year of the infinite past for the Eternal Society engaging in the Annual Coin Flipping Tradition. If the coin came up heads each year, did the Eternal Society ever do the chant? They would have had to have done the chant some year, because they would have done the chant last year if they hadn’t done it yet (since the coin came up heads last year). And yet any year you point to, there is a prior year in which they would have done the chant if they had not done the chant before. So they had to have done the chant (since the coin came up heads last year), yet they could not have done the chant (there is no year they could have done it), and so this scenario creates a logical contradiction.

Although scenario S1 is not directly self-contradictory, scenario S1 is impossible because it implies the possibility of a logical contradiction. The Eternal Society argument against an infinite past goes like this:

1. If an infinite past were possible, an Eternal Society would be possible.
2. If an Eternal Society were possible, then scenario S1 would be possible.
3. If S1 would be possible, then S2 would be possible.
4. S2 is not possible.

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5. Therefore, an infinite past is not possible.

The Eternal Society Paradox Argument Against an Infinite Past is a deductively valid argument—the conclusion (line 5) follows logically and inescapably from the premises (lines 1-4). A sound argument is a valid argument with all true premises, so the only way the argument can fail to be sound is with a false premise.

One could deny premise (1) particularly since that seems to be the most vulnerable premise, but as the Eternal Society Paradox paper says, “Surely there is something metaphysically suspicious about an infinite past if an eternal society with the abilities of ordinary humans can actualize a logical contradiction.” The idea that an infinite past is possible but an Eternal Society is not possible strikes me as overly ad hoc due to the Eternal Society’s extremely modest abilities (the abilities of ordinary humans in each year of its existence).

The Rebuttal

When using the phrase “Eternal Society Paradox” the author seems to have in mind specifically scenario S2. From the article:

…the solution [to the paradox] is straightforward: The Eternal Society Paradox is presupposing a logical contradiction.

How is this a solution? The fact that the Eternal Society Paradox (in scenario S2) entails a logical contradiction is part of the point; it’s not a solution to simply to concede part of the claim.

It presupposes a first and a last element to a supposedly infinite series, so the Eternal Society Paradox commits the First-and-Last Fallacy.

Simply calling something a fallacy doesn’t make it so. The “first-and-last fallacy” is described as follows:

The First-and-Last Fallacy occurs if and only if a person envisions a supposedly infinite series as having both a first and a last element.

I didn’t envision it, and neither did scenario S2. Indeed, part of the reason there’s a contradiction is that the scenario (if anything) envisions that there is no first element.

Another problem with the objection is that the Eternal Society Paradox Argument is logically valid, so if the argument is unsound, which premise is false? At first blush at least, this objection doesn’t actually attack any premise of the argument! My conclusion was initially that this objection is a red herring (for more on red herrings, see my red herring video). But see the update below:

Update

After personal communication with the author in 2023-07-31-MO I discovered I hadn’t quite correctly understood his objection. While at the time it wasn’t clear to me which premise he was attacking, it seems he was attacking the premise that says scenario S2 is impossible. Although proof of its logical impossibility can be proven via symbolic logic, we won’t need to go into something so complex here. Here was Akin’s understanding of the Annual Coin-Flipping Tradition that I’ll label ACFT*:

The society flips a coin each year of the infinite past. If the coin comes up heads, then they do the chant if and only if it’s the first time the coin comes up heads.

The objection goes as follows: scenario S2 is the conjunction of ACFT* and the coin coming up heads each year. Since there is no first year the coin came up heads, they never do the chant. There is no self-contradiction with ACFT* and the coin coming up heads each year; the society just never does the chant in any year. Therefore, S2 is not self-contradictory and premise (4) is false. To think that in S2 they had to do the chant assumes that there was a first year the coin came up heads (as well as a “last time” the coin came up heads, since there is a present year); hence a “First and Last Fallacy.”

The problem is that ACFT* is not synonymous with ACFT. To see this I’ll use a reductio ad absurdum argument, which assumes the opposite of what we want to prove (the reductio assumption) and show that it leads to an absurdity (such as a self-contradiction). Recall that ACFT says that the society does the chant when these two conditions are met: (a) the coin comes up heads; and (b) they have not done the chant in a prior year. Assume for reductio that when following ACFT the society never did the chant in any year and the coin comes up heads each year. Now consider last year: are both conditions (a) and (b) met?

• Condition (a) is met, since the coin came up heads last year.
• Condition (b) is also met since they never did the chant in a prior year. (Since they never did the chant in any year.)

So on AFCT they would have done the chant last year since both conditions are met, which contradicts the original assumption that they never did the chant in any year. Thus it is logically impossible for them to have never done the chant if the coin came up heads each year and they followed AFCT. It also follows that ACFT and ACFT* are not equivalent; when the coin comes up heads each year, the society never doing the chant is possible on ACFT* but logically impossible on ACFT (since last year, both conditions (a) and (b) would have been met and they would have done the chant last year).

Somewhat more formally, the reductio argument goes as follows, where the premise (6) is true by definition of the ACFT:

6. If ACFT is true, then for any year y, the society does the chant in year y when (a) the coin comes up heads in year y; and (b) they did not do the chant prior to y. (Follows from the definition of ACFT)

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7. In scenario S2, ACFT is true, the coin came up heads each year, and they never did the chant in any year. (Reductio assumption; we’ll show that this leads to an absurdity.)
8. Last year, the coin came up heads (follows from 7) and thus condition (a) is met.
9. Last year, they never did the chant in a prior year (follows from 7) and thus condition (b) is met.
10. Last year, conditions (a) and (b) are met. (Follows from 8 and 9.)
11. If conditions (a) and (b) are met last year, then they did the chant last year. (Follows from 6.)
12. They did the chant last year. (Follows from 10 and 11.)
13. They did not do the chant last year. (Follows from 7.)
14. They did the chant last year and they did not do the chant last year. (Follows from 12 and 13.)
15. Therefore, (7) is false. (Follows from reductio and 14.)

We know that scenario S2 includes ACFT and that the coin came up heads each year, so the part of (7) that must be false is the idea that they never did the chant in any year.

Conclusion

This illustrates an important lesson in spotting straw men; sometimes statements that might initially appear to be synonymous are actually subtly different, and sometimes that subtle difference matters when evaluating objections. When evaluating an objection, consider how that objection fares against the position as it was originally stated. In the case of the Eternal Society never having done the chant in scenario S2, when we apply ACFT’s conditions (a) and (b) to last year we see that they would have done the chant last year, whereas on ACFT* they would not have done the chant. I can understand why one might think that ACFT and ACFT* are synonymous, but that wasn’t quite true; they were strongly similar but subtly different, and that subtle difference made all the difference in the world.