(Note: the revised version of this post begins in the third paragraph after the section title, “The Ratio of Explanatory Powers.”)
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.
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.
3. Humans are the supreme knowledge-possessing species as far as we know. (221)
4. There is no evidence that the universe is teeming with intelligent life, comparable to or even greater than humans. (221)
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. 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.
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.
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. Roughly speaking, the greater the scope of a theory, the more it says that might be false and the more likely it is to say something that is false. 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.
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." 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. 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.
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. 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. In other words, Pr(E | N & B) = Pr(E | T & B).
Remember from part 1 that Everitt appeals to the idea that if classical theism is true, then humans are the “jewel of creation.” Since theism does not entail that humans are the jewel of creation, we may treat the hypothesis that humans are the jewel of creation as an auxiliary hypothesis. Let us first define the “Jewel of creation” hypothesis (J) as the doctrine that human beings are the most valuable things in the physical universe. Then, according to the theorem of total probability,
Pr(E | T & B) = Pr(J | T) x Pr(E | J & T & B) + Pr(~J | T) x Pr(E | ~J & T & B)
Now since the whole point of conjoining J with T is to try to increase the value of Pr(E | T & B), we may effectively ignore the second half of the right-hand side of that equation and focus on the first half: Pr(J | T) x Pr(E | J & T & B). What reason is there to think Pr(J | T) is greater than Pr(~J | T)? As I read him, Everitt provides three reasons: (i) T & E1 & B3 “entail” that God regards humans “as especially valuable; (ii) T & E2 & B4 is “committed to” J; and (iii) theists have traditionally believed that J is true.
Let us consider each of these reasons in turn. Regarding (i), Everitt overlooks two crucial pieces of background information: (B5) humans are the result of evolution; and (B6) the sheer probable quantity of habitable planets in the history of the universe. Once we take into account B5 and B6, it then becomes highly probable that intelligent life has evolved (and will evolve in the future) on numerous planets in the universe. Thus, while we have no direct, observational evidence of intelligent life outside of earth, our background information makes it very highly probable that such life has existed, does exist, and will exist. This undercuts, not supports, J.
(ii) suffers from a similar defect. T & E2 & B5 & B6 make it highly probable that intelligent life, including life at least as intelligent as humans, exists elsewhere in the universe. But that entails that humans are not uniquely the most valuable things in the universe. Again, this undercuts J.
Finally, even if (iii) is historically accurate, it is evidentially irrelevant. What matters is whether classical theists had any good antecedent reason on classical theism to believe J. Furthermore, it seems to me that classical theism provides an antecedent reason to deny J. In a discussion of the multiverse objection to an argument from evil, Paul Draper provides a fascinating argument for the conclusion that a multiverse is highly probable on theism. Here is Draper:
God, if she existed, would be very likely to create vast numbers of good worlds. Indeed, we can transcend our anthropomorphism just for a moment, the idea that an all-powerful, all-knowing, and morally perfect being would create just our world and no others borders on the absurd. What a colossal waste of omnipotence and omniscience that would be! Surely a perfectly good God of limitless creative resources would create vastly many worlds, including magnificent worlds of great perfection as well as good but essentially flawed worlds that are more in need of special providence.
To be sure, Draper himself acknowledges that this argument makes “some very controversial axiological assumptions,” which he defends. While a discussion of those assumptions is beyond the scope of this paper, I think it is safe to say that Draper’s argument provides a prima facie reason, at least, to deny J. In short, on the assumption that theism is true, God probably did create creatures which are more impressive than humans, in other parts of our universe and in other universes.
The upshot of this discussion is that reasons (i)-(iii) are not good reasons for believing that J is antecedently more probable than ~J on the assumption that T is true. But this entails that (i)-(iii) are not good reasons for believing that E is more probable on theism (conjoined with J) than on naturalism. In other words, (i)-(iii) are not good reasons for concluding that Pr(E | T & B) is greater than Pr(E | N & B). I conclude, therefore, that this version of AS fails.
In part 3 of this series, I will consider another Bayesian version of AS.
Series on the Argument from Scale
 I owe this way of explaining the distinction between B and E to Robert Greg Cavin.
 Nicholas Everitt, The Non-Existence of God (New York: Routledge, 2004), 216, 218.
 Everitt 2004, 217.
 In this paragraph, I have borrowed heavily from Paul Draper, "Evil and Evolution," unpublished paper.
 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."
 Cf. Everitt 2004, 221: “theism is committed to saying that humans are the most valuable things in creation.”
 This is suggested, but not explicitly stated, by Everitt’s reference to “the purposes traditionally ascribed” to God, combined with his discussion of what I call J. See Everitt 2004, 225.
 Paul Draper, “Cosmic Fine-Tuning and Terrestrial Suffering: Parallel Problems for Naturalism and Theism.”