Doomsday argument
A case study on what the case would be (if it were not the case)

István Danka
Philosophy, University of Pécs
E-mail: danka.istvan@freemail.hu

I claim that so-called Doomsday Argument hangs on some kind of verificationist antirealism. In this case, the argument aims only at antirealism, in its verificationist form. Realists as well as non-verificationist antirealists could steady down. Doomsday is only the doomsday of verificationist antirealism.

First I will sketch two more or less different versions of DA, figuring out what the purpose of the argument(s) could be. I formulate a very very simple preformulation of the argument called Doomsday Claim, which is presupposed by every (logically coherent) version of DA. Then I discuss some previous counterarguments - so-called SIA, first of all. One of the main objections to SIA is that it involves that there is an infinite mass of observers; however, DA itself presupposes a finite mass. Finitism is an important part of antirealist views.
Another important consequence of the debate on SIA is to see that SIA and DA differs in judging the importance of the epistemic position of an observer. Confusing what is spoken about facts from a certain epistemic position on one hand, and what happens in fact on the other (i.e., by and large, confusing being justified and being true) is also a significant feature of antirealism.
Thirdly, Doomsday argument sponges on the concept of probability. Some form of probability theories hang on a verificationist theory of truth. I claim, we should choose either to make a clear distinction between speaking about facts on one hand, and speaking about probability of facts on the other, or to accept a verificationist theory of truth. Finally, this would yield the most counterintuitive claim of antirealism, namely denial of the principle of bivalence.

If someone has a realist attitude toward facts and therefore she accepts that principle of bivalence is valid for speaking about facts, she is able to make a methodological distinction between a direct realist way of speaking about how things are on one hand, and an antirealist way of speaking about epistemological problems of how probable a certain belief is. Probability of whether doomsday is coming soon is a question of the latter, whereas whether doomsday comes on in fact, is a question of the former. An answer to the question whether or not doomsday comes on is quite different from an answer to how probable it is. Therefore if someone is not a verificationist or an antirealist, she doesn’t need to argue against the probability of the Doomsday Soon scenario, for arguing that it is unreasonable to believe.

The Doomsday Argument

In a simplified form (DA1), DA says that Doomsday’s coming is much more probable than it is ordinarily thought. DA says that there is an observer O, whose chance to be in the first 10% of human beings is, according to a very simple probability operation, exactly 10%, whereas to be in the first 50% is exactly 50%, etc. It is not too hard to see that O has a very little chance to be among the first humanoids but much more to be in the first tenth of all human beings, and even more to be in the first half. Then, the argument says, it seems to be very probable that she is among the last ones. If it is so, human species is becoming extinct, Doomsday is coming soon.

There are, of course, some variants but that is, as mentioned above, the main common structure of the argument. This version is attributed to Richard Gott. However, Nick Bostrom claims, Gott’s argument is an oversimplification. For my present purposes, there can be found every important feature of DA in this simplified version. However, I would like to avoid the objection that my argument is not good enough to some more sophisticated versions. Hence I also sketch the so-called Carter-Leslie version (preferred also by Bostrom).

In Carter-Leslie version (DA2), there is a prior probability of how near the Doomsday is. Nevertheless, if you update your beliefs by Bayes’ Theorem, you will find that a new probability appears which is much higher than the prior was. According to Bostrom, “You also have to take account of the empirical prior probability, and the way to do that is by using Bayes' theorem. Failing to do that, you do indeed get the absurd conclusion that there is nothing we can do improve the odds”.

To this objection of DA2 against DA1, I have two replies. One of them is that even in the Gott scenario, there is a prior probability. Since we have started with that ‘there is an observer O’ then there surely is some probability that observers are becoming extinct (if the certain observer is mortal, at least). Consider if there is at least one observer, she can die but if there is none then it cannot be the case. The probability of Doomsday, if there is no observer, is exactly zero. But since the first observer has been coming to existence, there is some chance to become extinct.
Second, it is an important question, what the argument is about. If DA says that Doomsday is more and more probable, the first-mentioned version is enough to demonstrate it. Of course, DA could say something else; for example in Bostrom’s case, it says that Doomsday is probably much nearer than it is thought. In this case there really is a significant difference but it is not too important from my perspective because I don’t think that it would be the most interesting part of DA whether it comes now, soon, or later; if DA is right in any slight form, then we must be ready for Doomsday which can come at any time and if it hadn’t come, it would be more and more sure that it should come soon. I’d like to argue against the point which claims that applying the formal structure of DA, independently of any background theories, lead us to be almost sure that Doomsday must ever come.

If I am right, it is not the main question of DA whether it is formally correct or not. The argument, as any formal lines of thought, can be interpreted in several ways (even if it cannot mean that ‘anyhow’). How to interpret the argument hangs on what it is claimed to prove. It can be then highly relevant, what is our reason to deal with the argument.
DA itself, without an interpretation, has probably only some particular interest, even among logicians. Why it seems to be so interesting and important is that it is applied to the question of whether and when Doomsday comes on. Accoding to Bostrom, “one of the reasons why philosopher John Leslie wrote a book about the Doomsday argument was that he hoped it could influence people to put more effort into preventing disasters such as nuclear or germ warfare, a runaway greenhouse effect etc”.

However, if the argument is right, we cannot do anything against Doomsday. It is, independent of what we are doing, more and more probable that Doomsday is coming soon. If there is any chance of Doomsday, it becomes necessarily more and more probable till it will have be fulfilled. It is quite easy to see why. Suppose there is a finite set of human beings who will have ever born, and our observers O1-On are random samples (where n is the number of human beings who will have ever lived but O1-On is not well-ordered; indexes show how your almost fully random sampling method actually goes on). You would like to decide who will be the last human being who will have ever lived. Suppose you have a method to do this. Since you hope the last humanoid will not be you because you don’t want to die lonely, you decide to see whether you are the last one at the end of your method - so the only non-randomized index is n, and you are labelled On. How do your chances change during your investigations? At the beginning, the probability of you are the last humanoid is p(On)=1/n. After you see whether O1 is this unlucky guy, and you find it false, your chances will have become worse. There is now an evidence that O1 is surely not the last human being. Since the number of possible last human beings is now less than it previously was, the more observers you find not to be the last, the more probable is that it is you. In any case, the simple fact that you still alive is a strong reason to believe that you can easily be among the applicants for this undesirable position.
Consider now that whatever the case is, there surely is a last human being if there is a finite set of observers. In this case, Leslie’s purpose is unsatisfiable. DA does not only show that there is a very high probability that Doomsday is coming soon, but it also show that Doomsday will necessarily have come, and we cannot do anything against. In this case, the most reasonable thing which can be done is either to pray for God or to disbelieve that the argument is right, independent of how well-established it is. I am not satisfied with such solutions which claim that Doomsday is surely coming, but, fortunately, it is still far. If humankind was determined to go extinct, a loophole that it would be but done only after some thousands of centuries, would be not too assuring, for humankind in general, at least. So I assert that there is a common ground in DA1 and DA2, which is theoretically much more important than the difference in details. If any of them is true, then we are determined to become extinct. All we can do is only to postpone but not to avoid it. This is, I think, cannot be reasonable at all. I have no reason to think that I am unable to do anything against; if I am really unable, my life itself seems to be unreasonable.



The common ground can be formulated even more simply than it is in DA1. I will then discuss this simple version which, if the above-mentioned are right, is presupposed by both DA1 and DA2. It asserts that

Doomsday Suggestion: The fact that there is at least one observer makes it reasonable to believe that Doomsday is coming (either soon or late).

DS is accepted by everyone who holds any versions of DA because the only presupposition of both DA1 and DA2 is the existence of an observer, and the conclusion of them is that Doomsday is probably coming. Every version which I know is only some more sophisticated elaborations of this intuition. However, DS often has an additional criteria which is not argued. It makes versions of DA too strong:

Doomsday Claim: The fact that there is at least one observer involves that Doomsday is coming (either soon or late).

The strongness of this claim lies in the connection between the condition and the consequence. If such a strong connection was not suggested, the argument would not be so harsh. There are some very good reasons to accept DS, but there are very slight evidences to accept DC. It seems to me that rejecting DC is denying the common interpretations of DA; but I do not think that it is equal to rejecting the argument itself, or weaker interpretations like DS. (However, I also make it questionable whether DS is really so reasonable but not in details.) Below I try to figure out on what presuppositions DC depends. I will distinguish four different presuppositions of DC, all of them are mostly attributed to some forms of antirealism.


Maybe the most promising way to argue against DC is using the so-called Self-Indication Assumption. Bostrom and Cirkovic formulates SIA as “The fact that one is an observer gives one some reason to believe that the world contains many observers”. Bostrom also claims that “Prima facie, the SIA is admittedly dubious, and perhaps it is no less dubious ultimo facie, but it deserves serious consideration as it might be the only consistent way of resisting the conclusion of the DC”.
Bostrom adds, “One of Leslie's favourite objections against the SIA is that it would lead to the conclusion that it is certain that the world contains infinitely many observers.” However, “...the observers in the universe were caused by a random process which was extremely unlikely to generate infinitely many of them. This would clearly be a highly counterintuitive consequence”.
I have to admit I am not sure whether I understand such an objection. If there are finitely many observers, there is surely a first and a last one; it is not something interesting that doomsday is coming on. Moreover, in that case, it is also not too surprising that doomsday is more and more probable. The only interesting question is, the above-mentioned clearly show, how probable it is. Therefore one could claim that my argument is irrelevant. However, it is not so simple. DC presupposes finite set of observers, since if observers were infinitely many, we would not only be unable to specify our birth rank, but also could not work out probability operations about doomsday.
DC is a deterministic claim in the sense that it says there is a definite number of human beings who will have ever born (or a definite (spatio)temporal extension of the existence of humanoids). In DA1, for supposing that we are within the first 10% of all humanoids, it must be presupposed that there is a determined number of them, even if it is not known. In DA2, we would need to have a prior probability of the extinction. However, if human beings are (or could be) infinitely many, there is no such evidence at all. Bostrom refers to the “counterintuitive consequence” of the supposition that there are infinitely many observers; leaving out of consideration how counterintuitive DC itself is. If space and time would be infinite, we could not exclude that there would be infinitely many observers; even if there would be a doomsday, humanoid life could be regenerated, redeveloped or, extremely, recreated.  I am not sure whether space and time is finite or infinite; however, it seems to be, if it is infinite, DC cannot be held. Bostrom wirtes, “John Leslie thinks that the DA gives us supremely strong reasons against any hypothesis implying the existence of an infinity of observers.” However, if there is an infinity of observers, it is not reasonable to believe that DC is true; it simply means that infinitism and DC are contradict to each other, not that any of them could refuse the other.



A usual way of arguing for SIA is that imagining two scenarios, in one of them there are a few human beings, in the other there are many, the simple fact that I exist makes it more probable that there are many others. (Simply because I as a random sample can be replaced for many cases and not only for a few. This increases the probability.) However, Leslie and Bostrom argues via God’s coin tosses example,  that it is either true or false whether there is an observer Om+1 (where m is the index of the last human being of the first ‘few’ ones). It seems to me that the difference is because of the question of from which epistemic position the probability is judged. If you are in God’s position, you surely know that the result of the coin toss is either heads or tails; chances are fifty-fifty. Moreover, after the toss of a coin, you will also certainly know, which is the case. It means that you can update your belief that it had been fifty-fifty since you still have known whether it is heads or tails. From God’s perspective, the probability of what the case is (after the toss, at least) is exactly one and the probability of what the case is not is exactly zero. The probability of what the case is, from a non-divine perspective, is somewhere between 0 and 1. For some particular ways of thought, it can be important, of course, what the exact number is. But for my present purposes it is quite enough to say that from God’s perspective (if there is any), ‘probability’ of a certain fact is either zero or one because every fact is well and certainly known, not only probable. There is, therefore, no one-to-one relation between probability from God’s perspective on one hand, and from a non-divine observer on the other. In other words, avoiding allusions to divine beings, there is an assymmetry between (sure) facts and (probable) observations.
This claim is, once again, not a refutation of the argument. Forms of DA is admittedly based on anthropic reasoning. All the same, in our own epistemic position, DS seems to be quite reasonable. But from God’s perspective, it is probably not the case. Even if God could not foresee what will happen, God would know a lot of (previous) facts unknown to us which would surely modify the probability of doomsday. Hence there is no unique probability of doomsday’s coming, independent of epistemic positions. (Or if there was, it surely could not be seen from our perspective.)


Probability and provability

DA, in a sense, reminds me of the so-called paradox of knowability. Without going into details, Knowability Paradox says in Cozzo’s formulation that from the statement

i) if ‘S’ is true, then it is provable (or knowable) that S

it follows that

ii) If ‘S’ is true, then it has been proved or will in fact be proved that S.

However, it is very hard to imagine that we can prove everything which will have ever been done (perhaps we don’t have enough time, there are some very old facts of minor importance which cannot be recognized (e.g. whether Caesar had done even or odd steps during his life).
The similarity to DA is that truth is defined (or supposed to be determined) by something what depends on the observer’s epistemic position. Knowability/provability of a fact, as well as its probability is partially determined by the observer’s prior hypotheses, previous experiments, cognitive capacities, spatiotemporal location, etc.
KP is mostly seen as a hard attack on verificationist antirealism. Antirealism, in its verificationist form, supposes that truth can be fully described in terms of provability (or knowability, or assertibility, or the like). It differs from realism in the question whether there is anything besides such criteria of truth. According to an antirealist, everything what can be said about truth can be said in terms of provability. It involves, according to her, that there is no theory-independent reality (or, in some weaker and more reasonable versions, we cannot speak about theory-independent reality). Therefore truth of a statement is equal to whether it is provable.
As a consequence, antirealism is claimed to identify realm of facts with facts understood in a certain epistemic position. Since the only way to reach the truth is to prove our statements, ‘truth’ will depend on our provability schemes. It is therefore also not surprising that antirealists are often finitists. Since logical structures of provability determines us to know about finite masses, antirealists must think that facts constitute a finite set.
Provability, according to verificationist antirealism, is a matter of verification. A statement is provable (or disprovable) if there can be given all the relevant conditions of its verification. It is not surprising, I think, that one of the fathers of probabilistic logic, Rudolf Carnap claimed in his Inductive Logic and Probability that conditional probability is the degree of being confirmed. Bostrom and Cirkovic claims “By ‘probability’ we here mean rational subjective credence”. What makes our ‘subjective credence’ rational is but nothing else than some factual evidence. There is no problem, of course, of applying probability theories to any questions. The problem, for a non-verificationist, is that if truth is expressed in terms of probability, then either truth depends on our epistemic position which is unacceptable or, more radically, there is nothing in such theories which could be called truth at all.



The divergence of the structures of probabilistic logic and realm of facts is obvious if we see how ineffective the application of probability can be in some particular cases. Suppose a simple case introduced as Self-Sampling Assumption by Bostrom.

“Imagine a universe that consists of one hundred cubicles. In each cubicle, there is one person. Ninety of the cubicles are painted blue on the outside and the other ten are painted red. Each person is asked to guess whether she is in a blue or a red cubicle. (And everybody knows all this.)”

Bostrom’s claim is if you are in one of these cubicles, your chances to be in a blue cubicle is 90%, and, therefore, you are strongly advised to believe that you are living in a blue cubicle. This argument is undubiously right. However, consider what will happen if you become to know that actually you lived in a red cubicle. It has not too much but also not negligible probability. You have decided rationally, so rationally as you can, but you have failed. Probability helps to 90% but leaves 10% in the lurch.

Suppose a case when probability theory is not only fallible but straightforwardly useless. This example is also not an original, and even not a complicated, one. Suppose again that our universe is divided into 100 cubicle. What is the prior probability, consider only the simple fact that I am in one of them, that I am in the cubicle in which I am? As far as I know, there is no probabily theory which could give the correct and quite evident answer that it is sure. More strangely, it is almost totally improbable that in this certain time I am exactly at this certain place, speaking about this certain issue and some of you are just thinking that ‘the probability of this situation is almost zero’. However, none of us would be doubtful about it (reasonably, at least).

It seems to be therefore, that in some cases, probability theories are not only insecure but more insecure that some pure intuition. It also seems to be that there are some cases when it is not simply the case that the more probability a hypotheses has, the more reasonable is to believe it. Probability, if it is a “rational subjective credence”, it should be applied only via some criteria which make the application reasonable. Statements about facts are either true or false, whereas statements about probability of facts are gradual. There is no clear vedge between ‘reasonably true’ and ‘reasonably false’ probable statements; it depends on the certain application of the formulae. As my second example shows, sometimes it is much more reasonable to believe in quite improbable hypotheses (like I am existing in 2004, I am now at CEU in Budapest, etc).
Statements about facts are either true or false; DA works with probabilistic terms. If we directly apply DA for reality, as DC does, we have to accept the antirealist claim that principle of bivalence is false. It is not the case that either it is the case or it is not the case. It can be more or less probable; however, if it meant that there were any direct implications of this probability to reality itself, then reality would also be probabilistic, not binary coded, as we mostly think.



I do not want to claim after all, that probability operations cannot be applied for the doomsday scenarios. What I try to show is simply that it is not necessarily the case that if probabilistic logic regards something as highly probable, then it is reasonable to think, in every single case, that it is really so. There is, it seems to be after all, a difference between yes/no questions of facts and gradual questions of probability.
What is then the case with DC? My argumentation shows only that it grounds on some typical antirealist presuppositions like 1) finitism, 2) supposition of truth depends on our epistemic position, 3) supposition that truth can be fully explained in terms of verification (like probability or provability), and 4) principle of bivalence is false.
These presuppositions can, of course, be accepted. They are actually accepted by some verificationist antirealists. But if someone claimed that any of them should be abandoned, it is questionable whether she can hold DC consistently. On the other hand, if someone claimed that DC should be abandoned, as most of us did, then she should simply argue against its verificationist backgrounds from an ordinary realist or non-verificationist antirealist point of view.
My own opinion is that DA yields some consequences (in its ordinary interpretations like DC, at least), which are in themselves unacceptable. I simply regard such a strong determinism as counterintuitive. There are but, fortunately, some objections which DC should reply. There is no evidence that the number of human beings is finite, there is no evidence that our epistemic position is ‘objective’ enough to judge our epistemic position itself, there is no evidence that terms of verification make it clearer what the terms ‘true’ and ‘false’ means, and, last but not least, there is no evidence that it can be the case that something neither is, nor is not the case. Some of these suppositions can be true. But whether they are or not, is questionable. Without proving them, DC cannot be conclusive.
On the other hand, even if DA was right, every single newly born human being seems to be a counterexample of it since even if doomsday is coming, it has not still come. DC itself, according to probability operations, is less and less probable. According to its own principles, at a supposed ideal end, DC can, with certain probability, be falsified. If there will be an observer which can speak factually about DC, DC must be wrong. Claiming that it will have turned out that DA is right, there must be an observer in the future who will have verified it. But if there were any, DC would be false. Hence, it cannot be claimed to be true but only probable. And, because of its own presuppositions, if it cannot be claimed to be true, it cannot be true at all.
Nevertheless, if it is formally right but factually wrong, there is some problem of its presuppositions. So if DC can do anything important, it is to yield some arguments against its own presuppositions, which are some fundamental points of some versions of verificationist antirealism. Therefore I think, we have some good reasons to believe that the doomsday of verificationist antirealism is coming soon.