Basic Structure of My Evidential Arguments

Epistemic Interpretation of Probability
In this article series, when I refer to probability I shall be adopting the epistemic interpretation of probability. The epistemic probability of a statement is a measure of the probability that a statement is true, given some stock of knowledge. In other words, epistemic probability measures a person’s degree of belief in a statement, given some body of evidence. The epistemic probability of a statement can vary from person to person and from time to time (based upon what knowledge a given person had at a given time).[1] For example, the epistemic personal probability that a factory worker Joe will get a pay raise might be different for Joe than it is for Joe’s supervisor, due to differences in their knowledge.

Two Types of Evidence and Corresponding Probabilities
Let us divide the evidence relevant to naturalism and theism into two categories. First, certain items of evidence function as “odd” facts that need to be explained.  Second, other items of evidence are background evidence, which determine the prior probability of rival theories and partially determine their explanatory power.
These two types of evidence have two probabilistic counterparts which are useful for evaluating explanatory hypotheses: (1) the prior probability and (2) the explanatory power of a hypothesis H. (1) is a measure of how likely H is to occur based on background information B alone, whether or not E is true. As for (2), this measures the ability of a hypothesis (combined with background evidence B) to predict (i.e., make probable) an item of evidence. [2]
Let us proceed, then, to defining some basic probabilistic notation.
B: background evidence
E: the evidence to be explained
H: an explanatory hypothesis
Ri: the rival explanatory hypotheses to H
Pr(x): the probability of x
Pr(x | y): the probability of x conditional upon y
Next, let us define the following conditional probabilities.
Pr(H | B) = the prior probability of H with respect to B—a measure of how likely His to occur at all, whether or not E is true.
Pr(Ri | B) = the prior probability of Riwith respect to B—a measure of how likely Ri is to occur at all, whether or not E is true.
Pr(E | H & B) = the explanatory power of H—a measure of the degree to which the hypothesis Hpredicts the data E given B.
Pr(E | Ri & B) = the explanatory power of Ri—a measure of the degree to which Ri predicts E given B.
Pr(H | E& B) = the final probability that His true conditional upon the total evidence Band E.
Bayes’s Theorem
Bayes’s Theorem is a mathematical formula which can be used to represent the effect of new information upon our degree of belief in a hypothesis. In its general form, Bayes’s Theorem may be expressed as follows:

Explanatory Arguments
It follows from Bayes’s Theorem that H will have a high final epistemic probability on the evidence B and E:
just in case it has a greater overall balance of prior probability and explanatory power than do its alternatives collectively:

Using this insight enables us to easily define the following argument schema for the explanatory arguments in this series.
Let F be some fact and H1 and H2 be rival explanatory hypotheses.
1. F is known to be true, i.e., Pr(F) is close to 1.
2. H1 is not intrinsically much more probable than H2, i.e., Pr(H1 | B) is not much more probable than Pr(H2 | B).
3. Pr(F | H2) > Pr(F | H1).
4. Other evidence held equal, H1 is probably false, i.e., Pr(H1 | B& F) < 0.5.[3]
Evaluating Auxiliary Hypotheses
It is often be useful to determine the evidential significance of an auxiliary hypothesis. In the course of considering various explanatory arguments for naturalism and against theism, we will consider several auxiliary hypotheses. Here are four examples.

In the explanatory argument schema described above, it may be possible to defeat premise (3) by showing that an auxiliary hypothesis is evidentially significant in the relevant way, i.e., that A sufficiently raises Pr(F | H1) or lowers Pr(F | H2). In order to assess the evidential significance of such hypotheses, we would need to apply the theorem of the probability calculus known as the theorem of total probability.

If we let A represent some auxiliary hypothesis, then it follows from the theorem of total probability that:
Pr(F | H1) = Pr(A | H1) x Pr(F | A& H1) + Pr(~A | H1) x Pr(F | H1& ~A)
In the context of explanatory arguments, Draper calls that theorem the “weighted average principle” (WAP).[4] As Draper points out, this formula is an average because Pr(A | H1) + Pr(~A | H1) = 1. It is not a simple straight average, however, since those two values may not equal 1/2; that is why it is a weighted average.[5] The higher Pr(A | H1), the closer Pr(F | H1) will be to  Pr(F | A& H1);  similarly, the higher Pr(~A | H1), the closer  Pr(F | H1) will be to Pr(F | H1& ~A).[6]
Cumulative Cases
Suppose that F is partitioned into 3 facts, F1, F2, and F3,such that F is logically equivalent to F1 & F2& F3. One could easily formulate three independent explanatory arguments for H1 over H2, by showing that each item of evidence individually is antecedently more probable on H1 than on H2, i.e., Pr(F1| H1) > Pr(F1 | H2), Pr(F2 | H1) > Pr(F2 | H2), and Pr(F3 | H1) > Pr(F3| H2). But how could one show that the items of evidence collectively are antecedently more probable on H1 than on H2? In other words, how could one give a about cumulative case argument?
Using the chain rule, it is possible to give a formal, mathematical definition for a cumulative case argument. Using the chain rule it follows that :
Pr(F | H & B) = Pr(F1 & F2 & F3 | H & B)= Pr(F1| H & B) x Pr(F2| F1 & H & B) x Pr(F3 | F1& F2 & H & B)
Therefore, F1, F2, and F3are a cumulative case for H1 and against H2 just in case:
Pr(F | H1 & B) = Pr(F1 & F2 & F3 | H1 & B)= Pr(F1| H1 & B) x Pr(F2| F1 & H1 & B) x Pr(F3 | F1& F2 & H1 & B)
is greater than:
Pr(F | H2 & B) = Pr(F1 & F2 & F3 | H2 & B)= Pr(F1| H2 & B) x Pr(F2| F1 & H2 & B) x Pr(F3 | F1& F2 & H2 & B)Another way to compare these two values is to compare the ratio of H1′s final probability to H2′s final probability. The ratio of these two formulas gives us Bayes’s Theorem in its compound odds form:

If this ratio is greater than 1, then H1′s final probability is greater than H2′s final probability; if the ratio is less than 1, then vice versa; if the ratio equals one, then H1 and H2 have equal final probabilities.This insight gives us one way to show that one hypothesis has a higher final probability than the other hypothesis. For example, suppose want to show that H2 has a higher final probability than H1. In other words, we want to show that:

One way to show that H2 has a higher final probability than H1 is to show that the ratio of each multiplicand on the right-hand side of Bayes’s Theorem in its compound odds form is also less than 1, i.e.,




This enables us to define the following schema for cumulative case arguments.

1. Pr(F1 | H2 & B) > Pr(F1 | H1 & B). [from formula x]
2. Pr(F2 | F1& H2 & B) > Pr(F2 | F1& H1 & B). [from formula y]
3. Pr(F3 | F1& F2 & H2 & B) > Pr(F3 | F1 & F2& H1 & B). [from formula z]
4. Some fact F is known to be true, i.e., Pr(F | B) is close to 1.
5. H2 is intrinsically more probable than H1, i.e., Pr(H2 | B) > Pr(H1 | B). [from formula w]
6. Other evidence held equal, H1 is probably false, i.e., Pr(H1 | B& E) < 0.5. [By Bayes's Theorem]
This schema can obviously be expanded as needed to incorporate the desired number of cumulative lines of evidence.
[1] Brian Skyrms, Choice & Chance: An Introduction to Inductive Logic (4th ed., Belmont: Wadsworth, 2000), 23.
[2] I owe these definitions to Robert Greg Cavin in private correspondence.
[3] I owe this schema to Paul Draper.
[4] Paul Draper, “Pain and Pleasure: An Evidential Problem for TheistsNoús 23 (1989): 331-50 at 340.
[5] Draper 1989.
[6] Draper 1989.

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Great Critique of Presuppositional Apologetics by a Christian
Evolution vs. The Argument from Providence
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

    I just linked this on Facebook. I'm looking forward to future additions to your index.

    This would be a great couple of posts to permanently link somewhere easy to find on the site.

    When Draper finally publishes his paper on intrinsic probability (i.e. the one I think he's currently titled "Simplicity" and has been presenting at conferences), that would be a good one to add as well. I have a copy of it, but he doesn't want it distributed until it is published. I'm not sure how he'd feel about having it summarized online.

    Also, it would be nice to see some of the responses you and Keith have made to some of the arguments purporting to be evidence in favor of theism over naturalism linked here in some way as well.

    It's really surprising that something like this hasn't been done before. The closest anyone has come is Draper in his debate with Craig. One has to piece together a lot of work to come up with an index like yours. This is definitely a worthwhile project and an excellent start to it. It is also a good way to frame the debate.

    Thanks for starting this. I hope more atheists come to appreciate the value of this kind of argumentation (I've found that a lot of them, especially internet atheists with little background in philosophy, overlook the significance of this kind of cumulative case). Kudos!

  • Hiero5ant

    I hate to sound like a broken record, but how is the "my" in the title not superfluous?

  • Jeffery Jay Lowder

    Hierosant — Please say more. I'm not following you.

  • Jeffery Jay Lowder

    Update: I just expanded the section on cumulative cases by adding a discussion of the compound odds form of BT, in order to illustrate how the argument schema for cumulative cases satisfies the pattern of probability relations required by BT.

  • Rob

    Great post! I can’t see the pictures where you supply some formulas. For example, after you say “One way to show that H2 has a higher final probability than H1 is to show that the ratio of each multiplicand on the right-hand side of Bayes’s Theorem in its compound odds form is also less than 1, i.e.,”…you supply 4 formulas and I can’t see them. Can you send me the text form of what all the pics say please? I’d really appreciate it. Thank you.

    • Rob

      There are others which I can’t see, btw. In the “explanatory arguments” section, there are 2 pictures; 1 in “Bayes’s Theorem”; and 6 in “cumulative cases” including the ones I mentioned above.