The Argument from Scale (AS) Revisited, Part 2

Originally published on 14-Nov-11; updated 20-Nov-11

In part 1 of my series on the evidential Argument from Scale (AS), I concluded that Everitt’s formulation of AS is unsuccessful. At the same time, however, I said that there is something about the AS I find intuitive and so I wanted to try revising AS as a Bayesian argument to see if I could make a stronger version. The purpose of this post is to attempt to do just that.


Let us organize the relevant evidence into B, the relevant background evidence, and E, the evidence to be explained. E represents the evidence which functions as puzzling facts that need to be explained. B represents evidence which influences the explanatory power of rival theories — that is, the probability of rival theories given the evidence to be explained.[1]

B: The Relevant Background Evidence

1. A physical universe, which operates according to natural laws and which supports the possibility of intelligent life, exists.

2. Human beings are a type of intelligent life and exist only on Earth. Furthermore, human beings are moral agents.

E: The Evidence to be Explained

1. Temporal Scale: Humans appeared in the universe long after the beginning of the universe. For more than 99.999% of the history of the universe, humans have been absent from it. Even if we only consider earth history, for more than 99.99% of the history of the earth, humans have been absent from the earth.[2]

2. Spatial Scale: The total universe is many orders of magnitude greater than the size of the earth. The greater part of the universe is not accessible to human exploration.[3]

H: Rival Explanatory Hypotheses

Finally, let us consider two rival explanatory hypotheses which might be used to explain this data.

theism (T): the hypothesis that there exists an omnipotent, omniscient, and morally perfect person (God) who created the universe.

metaphysical naturalism (N): the hypothesis that the universe is a closed system, which means that nothing that is not part of the natural world affects it.

Basic Strategy

According to Bayes’ Theorem, the final probability of a hypothesis, Pr(H | B & E), is determined by multiplying its prior probability, Pr(H | B), by its explanatory power, Pr(E | H & B). If we have two rival hypotheses, the relative odds form of Bayes’ Theorem enables us to measure the ratio of these values in order to determine which hypothesis has the greatest overall balance of prior probability and explanatory power:

Pr(H1| B & E)   Pr(H1 | B)   Pr(E | H1 & B)
------------- = ---------- X --------------
Pr(H2| B & E) Pr(H2 | B) Pr(E | H2 & B)

If E is evidence favoring H1 over H2, then the ratio on the left hand side of the equation will be greater than one. If that ratio is less than one, E favors H2 over H1. And if the ratio equals one, then E is evidentially neutral: it favors neither H1 nor H2 over the other.

One strategy for showing that E favors H1 over H2 is to show that both of the ratios on the right-hand side of the equation are greater than one. Another method would be to show that one of the ratios on the right-hand side of the equation is greater than one, while the other ratio on the right-hand side of the equation is greater than or equal to one. It is the latter strategy which I shall pursue here.

Bayesian Argument — Version 1

Let’s begin by applying the relative odds form of Bayes’ Theorem to our rival explanatory hypotheses N and T:

Pr(N| B & E)   Pr(N | B)   Pr(E | N & B)
------------ = --------- X --------------
Pr(T| B & E) Pr(T | B) Pr(E | T & B)

Again, my strategy will be to determine if there is a way to show that one of the ratios on the right-hand side of the equation is greater than one, while the other ratio on the right-hand side of the equation is greater than or equal to one.

The Ratio of Prior Probabilities

Let’s consider the first ratio on the right-hand side of the equation. This asks us to compare the prior probability of N (i.e., the probability of N conditional upon B) to the prior probability of T (i.e., the probability of T conditional upon B). Here I am going to argue that Pr(N | B) > Pr(T | B). This statement is supported by a comparison of the two hypotheses’ scope and simplicity.

First, let’s consider scope, which may be defined as how much a theory purports to tell us about (the contingent features of) the world.[4] Roughly speaking, the greater scope of a theory, the more it says that might be false and the more likely it is to say something that is false.[5] N entails that natural entities all lack supernatural causes, whereas T says they all possess supernatural causes. Moreover, T also makes very specific claims about the attributes of this supernatural cause; T says the supernatural cause is omnipotent, omniscient, morally perfect, etc. Thus, N does not have greater scope than T.[6]

Second, let’s turn to simplicity, which may be defined as “a measure of the degree of (objective) uniformity that the theory attributes to the world.”[7] Following Paul Draper, I am inclined to believe that simplicity is intrinsically more probable than variety or change, since to believe otherwise would make it hard to have a non-circular justification for our reliance on inductive reasoning.[8] Turning to N and T, it seems obvious that N is simpler than T. T entails that reality includes at least one radically different kind of entity (i.e., a supernatural person) which N denies. Thus, N attributes greater uniformity to reality than T does. Therefore, N is ontologically simpler than T.[9]

I conclude, therefore, that Pr(N | B) > Pr(T | B). In other words, the prior probability of naturalism is higher than the prior probability of theism.

The Ratio of Explanatory Powers

Let us now turn to the second ratio on the right-hand side of the equation. This asks us to compare the explanatory power of N to the explanatory power of T. Everyone would agree, I think, that N provides us with no antecedent reason to expect E, i.e., N & B do not provide us with a reason to expect E.

But what about T? Based solely on the statements I have explicitly identified as our relevant background information, B1 & B2, I think it’s clear that T also provides us with no antecedent reason to expect E. Some skeptics have argued that T does provide us with an antecedent reason to expect E, namely, that the non-human scale of the universe suggests that humans were created by a very wasteful process.[10] In my opinion, however, this argument fails. It assumes that, if God exists, God’s sole purpose for creating the universe was for the use and benefit of human beings. That assumption strikes me as just that: an assumption with no evidence or reason offered in support. Furthermore, if we assume that God’s purpose(s) include the creation of embodied intelligent life, then it seems probable that He would have created intelligent life many times throughout the universe, not just on Earth. If that is the case, however, then it would be extremely unlikely that God created the universe solely for the use and benefit of human beings.

I conclude that, relative to the background information B, both N and T have equal explanatory power with respect to E. Therefore, this version of AS fails.

In my next post, I will consider another Bayesian version of AS.

Series on the Argument from Scale


[1] I owe this way of explaining the distinction between B and E to Robert Greg Cavin.
[2] Nicholas Everitt, The Non-Existence of God (New York: Routledge, 2004), 216, 218.
[3] Everitt 2004, 217.
[4] In this paragraph, I have borrowed heavily from Paul Draper, “Evil and Evolution,” unpublished paper.
[5] Ibid.
[6] Ibid.
[7] Ibid.
[8] Ibid.
[9] Ibid.
[10] As Dr. Ellie Arroway (played by Jodie Foster) put it in the movie Contact, “If we are alone in the universe, it sure seems like an awful waste of space.”

Religious Experience – Recognizing God
Evolution vs. The Argument from Providence
Adam Lee: The Argument from Locality
Rape them Atheists!
About Jeffery Jay Lowder

Jeffery Jay Lowder is President Emeritus of Internet Infidels, Inc., which he co-founded in 1995. He is also co-editor of the book, The Empty Tomb: Jesus Beyond the Grave.

  • Wes

    Maybe you should just assume, for the sake of argument, that the prior probabilities are the same (50/50), otherwise the argument doesn't technically fail, but the explanatory power is irrelevant.

    You only note, though, that neither N nor T doesn't give us reason to expect E, but you must also indicate that neither gives us reason to not expect E.

    This is where I think your intuition about this argument (as well as mine) comes into play–i.e. the expanse of the universe seems odd given T + some common background assumptions about God's interest in humanity. Of course, it doesn't work without the background assumptions, but, with them, it does seem a little surprising that the universe is as expansive as it is.

    If E is surprising given T, but not surprising given N, this would mean that Pr(N|E) > Pr(T|E). Even holding your prior probabilities equal, then, would result in the argument giving some evidence for naturalism over theism.

    I still don't think it will make a particularly powerful argument, but I agree with you that it seems that something is there.

  • Angra Mainyu

    @Jeffery Jay Lowder
    I've got a minor quibble and a question, if you don't mind:

    Minor quibble: When computing the final probability, it seems to me you forgot to divide by P(E | B) (assuming P(E | B) > 0) – or you meant to leave it implicit?

    In any case. when you compute the relative odds, the terms in both equations cancel each other out, so that does not affect your argument.

    Do you have any definition of the terms "natural" and "supernatural", or are you using them intuitively?
    I know that most terms need no definition, but I have some doubts about the "natural" and "supernatural" being precise enough for the purpose of this argument, and generally philosophical arguments.

  • Jeffery Jay Lowder

    Wes — I would say the argument fails as an explanatory argument, in the sense that the evidence of the scale of the universe does not raise the ratio of the probability of naturalism to the probability of theism.

    I agree with your second paragraph.

    Regarding your paragraphs 3-5, see part 3 of my series on AS. There I add a background assumption regarding God's purpose(s), which I think does justify the claim that the evidence of the scale of the universe is slightly more probable on naturalism than on theism.

    Angra — The reason Pr(E|B) does not appear in the relative odds-form of Bayes' Theorem is that it cancels out.

    Regarding the definitions of "natural" and "supernatural," my answer is suggested by my definition of metaphysical naturalism, which I got from Paul Draper. A supernatural person is a person who is neither part nor a product of the natural world, but is nonetheless somehow able to causally interact with it. The natural world is roughly the same thing as the physical world. And notice that metaphysical naturalism, so defined, says nothing about the possibility of 'non-natural' objects such as abstract objects, which by definition have no causal powers.

  • Angra Mainyu


    Yes, it cancels out, so it does not affect your argument as long as the probability is greater than zero.
    I was asking about the part in which you explained how the final probability of a hypothesis is determined.
    Anyway, it was only a minor quibble, so never mind.

    Regarding the definition, thanks.
    That leads us to the concepts of "natural world" and – considering the "roughly" part -, of "physical world".
    Are you using those terms intuitively, or do you have a definition of them?

  • Jeffery Jay Lowder

    Angra — I am using the terms "natural world" and "physical world" intuitively.


    Well at leats Im not th eonly one who wuestions the terms…

    …but isnt yoru intuitive meanign tiself culturally based?

  • Jeffery Jay Lowder

    FYI: I just revised this post by expanding the discussion in the "Ratio of Explanatory Powers" section. It now discusses the idea that a non-human scale of the universe suggests a wasteful process for creating human beings.

  • Jason Pratt


    I'm working up some critical remarks for a Cadre Journal post, but I thought I'd ping what looks like a trivial composition error first:

    "One strategy for showing that E favors H2 over H1 is to show that both of the ratios on the right-hand side of the equation are greater than one."

    I think you've accidentally switched H1 and H2 in that sentence. Currently you have H1 high side on the formula and H2 lowside. E would only register as favoring H2 if both of the ratios on the right side are between zero and 1, not greater than one. (If both are greater than one, that would favor H1, the hypothesis in the numerator.)

    Easy to fix if so. {g}


  • Jeffery Jay Lowder

    Good catch JRP. You're right.

    One point I need to make somewhere in this series is this. I didnt write an entire series of posts on AS because I think it's a strong evidential argument for naturalism. It's not. The reason I wrote the series was partly to give it a formal analysis and partly because it was a great way for me to test out some ideas in Bayesian confirmation theory. I ended up using many of the same techniques in my other posts on other naturalistic arguments.

  • Jason Pratt


    I'd say you indicated pretty clearly in part 1 that you were only kind of ambivalently in its favor (so to speak), and that you were testing to see if you could juice it up any through Bayesian analysis. And I can see you're quite self-critical about results in part 2 and part 3. (I haven't gotten to Part 4 yet.)

    You could 'spoil the ending' I suppose by indicating from the start that, in hindsight, you currently still haven't found a way to create a strong evidential argument by this particular topic, but I thought you qualified yourself very well already.

    So don't worry, I'm not critiquing this argument as though you yourself think it's awesome. If it wasn't for (what I think are) some preliminary methodological problems, I'm sure I'd've been long past this entry already with a few comments here and there–possibly even with some advice for strengthening the argument! (I try to look for ways for opponents to strengthen their arguments as a self-critical disciplinary exercise. But I'd be unfair if I saw a way to improve an oppositional argument and I didn't mention it publicly. {g} I think you can sympathize; you seem to work the same way yourself.)

    Also, despite what I regard as a very serious underlying methodology problem, I think you end up mostly (maybe entirely, I'm not sure yet) neutralizing the main preliminary problem in how you proceed anyway.

    (That's meant to be kind-of reassuring. {g!})

    So from a practical standpoint my preliminary negative critiques (where I think you're making some mistakes) look like they'll end up being rather trivial in a way. My preliminary positive critiques (complimenting you and your procedure) will probably be rather more important in the long run.


  • Jason Pratt

    PS: fixing the sentence "One strategy for showing that E favors H2 over H1 is to show that both of the ratios on the right-hand side of the equation are greater than one", and the following sentence, wouldn't involve flipping H2 for H1, but would be to rewrite so that you're showing both the ratios on the right hand side are less than one. (Or of course one is 1 and the other less than 1.)