The Argument from Scale (AS) Revisited, Part 3

In part 2 of my series on the evidential Argument from Scale (AS), I concluded that neither metaphysical naturalism nor theism explain the evidence regarding the scale of the universe, if we restrict our background knowledge to the two propositions I identified as B1 and B2. In this post, I want to explore the effect of adding a new statement (B3) to our background knowledge:

B3. God’s purpose(s) include the creation of embodied moral agents.

I want to emphasize that I don’t claim theism entails B3 or even (merely) makes B3 probable. Rather, I am treating it as an assumption, one which is necessary, I think, for (most?) fine-tuning arguments for God’s existence to get off the ground. If we don’t have any good antecedent reason on theism to think God wants embodied moral agents, it’s hard to see why we have any good antecedent reason on theism to think God would want to design a physical universe in such a way that it would support the existence of embodied moral agents, such as humans.[1] What I want to do is explore what happens if we include B3 in our background knowledge and then assess a Bayesian version of the AS.


For convenience, allow me to state B3 as part of a review of the relevant background evidence, evidence to be explained, and rival explanatory hypotheses.

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. God’s purpose(s) include the creation of embodied moral agents. [assumption]

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

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.

Bayesian Argument — Version 2

Again, I’m going to apply 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)

My strategy will be to determine if there is a way to show that both of the ratios on the right-hand side of the equation are greater than one. I’m going to treat the first ratio on the right-hand side of the equation, the ratio of prior probabilities, as unchanged.  Let’s move directly to the second ratio on the right-hand side of the equation, the ratio of explanatory powers.

The Ratio of Explanatory Powers

If we assume that, if God exists, He would want to create embodied moral agents, then this makes the existence of embodied human moral agents slightly more probable on theism than on naturalism. In other words Pr(E | N & B) > Pr(E | T & B). While T & B does not entail that the universe would be created on a human scale, a human scale of the universe is slightly more likely on T than on N. This can be shown using a theorem of the probability calculus.

If O represents an observation, H is our explanatory hypothesis, and A is an auxiliary hypothesis, then we can measure the “effect” of combining H with A by using the Weighted Average Principle (WAP):

Pr(O|H) = Pr(A|H) x Pr(O|H&A;) + Pr(~A|H) x Pr(O|H&~A)

WAP tells us that as Pr(A|H) increases, the closer Pr(O|H) will be to Pr(O|A&H;). Similarly, as Pr(A|H) decreases, the closer Pr(O|H) will be to Pr(O|H &~A).[2]

Let M represent the hypothesis that God’s purpose(s) include the existence of embodied moral agents. Let S represent the auxiliary hypothesis–that is, auxiliary to B3–that God created the physical universe on a human scale, i.e. the temporal and spatial scale of the universe is favorable to human beings. According to WAP:

Pr(E|M) = Pr(S|M) x Pr(E|M&S;) + Pr(~S|M) x P(E|M&~S)

Pr(S|M) is slightly greater than Pr(~S|M) for two, related reasons. First, the truth of M represents one possible state of affairs in which S is also true, a state of affairs which would not obtain if M is false. Second, the falsity of M does not represent a different, extra state of affairs in which S could be true, in a way that is different from the one represented by the truth of M.

WAP tells us that if Pr(S|M) > Pr(~S|M), then Pr(E|M) will be closer to Pr(E|M&S;) than to Pr(Pr(E|M&~S). But that entails that Pr(E | N & B) > Pr(E | T & B).

Notice that the argument is not that N makes E probable , i.e., Pr(E | N&B;) > 0.5, or that T makes E improbable, i.e., Pr(E | T&B;) < 0.5. Rather, the argument simply compares the explanatory power of N to the explanatory power of T and states that the former is slightly greater than the latter. That claim is not in any way undermined by the fact that Pr(E | N&B;) < 0.5 and Pr(E | T&B;) < 0.5.

The Bayesian Version of AS Formulated

We are now in a position to formally state the Bayesian version of AS.

(1) E is known to be true.
(2) Pr(E/N&B;) > Pr(E/T&B;).
(3) Pr(N/B) > Pr(T/B).
(4) Therefore, Pr(N|E&B;) > Pr(T/E&B;). [From (2) and (3)]
(5) N entails that T is false.
(6) Therefore, other evidence held equal, Pr(T|E&B;) < 0.5. [From (4) and (5)]

Series on the Argument from Scale


[1] Cf. Richard Swinburne, The Existence of God (2nd ed.), who explicitly argues that Pr(B3 | T) > Pr(~B3 | T).
[2] Paul Draper, “Pain and Pleasure: An Evidential Problem for Theists.”