Some Very Rough and Incomplete Thoughts on the Prior Improbability of the Alleged Resurrection of Jesus

Bayes Theorem

Source: Matt Buck; License: CC BY-SA 3.0

DISCLAIMER (5 May 2023): I consider this a partial blog post on the prior improbability of the alleged resurrection of Jesus. I use the word "partial" because I'm not presently motivated to write what I would consider to be the entire blog post. If my level of motivation changes in the future, I will update the blog post.

1. Definitions

In this post, I will use the word "hypothesis" to mean "any proposition we do not know with certainty to be true or false."

1.1 "Worldview" Terminology

Following Paul Draper, I will start with the following non-standard definitions.[1]

• Source physicalism (SP) = the hypothesis that the physical world exists and, if a mental world exists, the physical world caused the mental world to exist.
• Source idealism (SI) = the hypothesis that the mental world exists and, if a physical world exists, the mental world caused the physical world to exist.
• Otherism (O) = the hypothesis that neither SP nor SI is true.

Notice that SP & SI & O are a mutually exclusive and jointly exhaustive set. In other words, exactly one of the options (SP or SI or O) must be true. Again, following Draper, I consider "theism" to be a subset or version of SI. I shall define it as follows.

• Theism (T) = The physical world was created by "God," a loving supernatural person who is perfect in power ("omnipotent"), perfect in knowledge ("omniscient"), and perfect in moral goodness ("omnibenevolent").[2]

1.2 "Resurrection" Terminology

Following Robert Greg Cavin and Carlos Colombetti, I will next adopt the following distinctions and definitions.[3]

• Revivification (R) = the hypothesis that the body of Jesus changed from being dead to being alive in some way
• Resuscitation (Rs) = the hypothesis that the body of Jesus changed from being dead to being alive again in its premortem state
• Resurrection (Rr) = the hypothesis that the body of Jesus changed from being dead to being alive as a glorious and imperishable soma pneumatikon that cannot be injured, become ill, age, or die, and can move instantaneously from place to place by sheer force of will

As Cavin and Colombetti point out, notice that R, as well as Rs and Rr, are neutral with respect to the cause of the change in state that happened to the body of Jesus.[4] We may differentiate between causes which are consistent with source physicalism vs causes which are not. For brevity, I will refer to "causes which are consistent with source physicalism" as "naturalistic." For example:

• Naturalistic Revivification theory (Rn): the hypothesis that natural causes changed the body of Jesus from being dead to being alive again in its premortem state
• Theistic Revivification theory (Rt): the hypothesis that God changed the body of Jesus from being dead to being alive again in its premortem state

Although Rt describes a necessary condition for the Christian belief in the Resurrection, it does not describe a sufficient condition. To describe the necessary and sufficient conditions, we need to define what we may call the "theistic resurrection theory".

• Theistic Resurrection theory (Rrt): the hypothesis that God supernaturally changed the body of Jesus from being dead to being alive as a glorious and imperishable soma pneumatikon that cannot be injured, become ill, age, or die, and can move instantaneously from place to place by sheer force of will.

While this foregoing discussion may seem like stating the obvious or even pedantic, it is necessary to address common confusions, such as "Well, actually, a naturalistic resurrection isn't totally impossible."  The topic under consideration is the theistic resurrection theory, not its naturalistic counterpart.

1.3. Philosophical Theories of Probability

Here is an incomplete survey of philosophical theories of probability.

• Classical theory of probability: To calculate the probability of a statement P being true, first identify a classical set for P. S is a classical set for p if and only if: (i) The members of the set are mutually exclusive and jointly exhaustive on what we know (i.e., what we know entails that exactly one of them is true). (ii) Each member of the set entails either P or ~P.  (iii) The members of the set are equally possible. Once a set S has been identified, then the classical probability of P is f/n, where f is the number of members of S that entail P and n is the total number of members of S. Thus, given a fair, standard six-sided die, the probability that any one side will land is 1/6. In notation, I would write this as ch(any one side) = 1/6.
• Frequency theory of probability: this theory defines probability as the frequency with which an outcome appears in a long series of similar events. A frequentist might say that if we would watch a long series of rolls of a fair die, we would observe that in roughly one-sixth of the rolls of the die, the side with a 6 is going to land. In notation, I would write this as fr("The side with a 6 is going to land") = 1/6.
• Bayesian theory of probability: a measure of the probability that a proposition is true, conditional upon other propositions known by a person to be true. In other words, personal probability measures a person’s level of confidence in a proposition. (Some writers will define personal probability as “degree of belief in a proposition,” but belief, like pregnancy, seems to be an all-or-nothing affair: you either have a belief or you don’t.) If I think a standard six-sided die is fair, my level of confidence in the proposition "The side with a 6 is going to land" is 1/6. In notation, I would write this as clp-Lowder("The side with a 6 is going to land") = 1/6. In contrast, someone who was trying to cheat me in a dice game might know that the die is not fair. Suppose they have somehow modified the die so that the side with a 6 lands only 1 time out of 20. I would write this as clp-cheater("The side with a 6 is going to land") = 1/20.[5]

Notice that all of the theories of probability defined above are what we might call "descriptive" theories of probability. In contrast, we may consider the epistemic theory of probability to be a "normative" theory of probability, insofar as it measures what a fully rational agent ought to consider the probability of some proposition.

• Epistemic theory of probability: Relative to some epistemic situation K, some proposition p is epistemically more probable than some proposition q, just in case in any fully rational person in K would have a higher level of confidence in p than q.[6] In notation, I will write Pr(p) to refer to the epistemic probability of p. If we think of K consisting of the conjunction of background information B and evidence to be explained E, then I might also write Pr(p | B & E), which refers to the epistemic probability of p, conditional upon B and E.

1.4. "Bayesian" Terminology and Notation

Bayes' theorem defines the relationship between prior (or "background") and posterior (or "final") probabilities. But what are those? Allow me to explain, relying upon Cavin's helpful definitions.[7] Let us start by dividing the relevant evidence (propositions) into two categories: the background information and the evidence to be explained.

• background information (B): facts that serve as the determinants of the inherent plausibility of the rival explanatory theories (such as the Resurrection, Swoon, and Theft theories) and as the partial determinants of their explanatory power.
• evidence to be explained (E): the unusual facts within the context of this background that need to be explained.

Using the distinction between B and E, it is now possible to define key terms which appear in Bayes' theorem.

• prior or background probability: the probability of some proposition or hypothesis, conditional upon B.
• explanatory power: the probability of E, conditional upon B and the proposition or hypothesis under consideration.
• posterior or final probability: the probability of the hypothesis, conditional upon B and E.

With all of the above definitions and distinctions, then, we are now in a position to clarify the meaning of the "prior probability of the alleged resurrection of Jesus." Let us turn to that topic now.

2. The Prior Probability of the Theistic Resurrection Theory

Again, by "resurrection of Jesus," I mean what I earlier defined as the theistic resurrection theory (Rrt). By "prior probability," I mean the prior epistemic probability. Taken together, then, I shall assume that the "prior probability of the alleged resurrection of Jesus" may be represented in notation as Pr(Rrt | B). 

Remember that our background information (B) is a set of propositions. In order to avoid the misunderstandings and confusions which seem to infect much of what has been written by both sides regarding Pr(Rrt | B), I offer the following modest suggestion. Writers should explicitly state the propositions they are including in B. For example, a source physicalist's background information would include:

• B1: SP is true.

Because SP entails that theism is false whereas Rrt presupposes that theism is true, it follows trivially that:

• Pr(Rrt | B1) = 0.[8]

In contrast, a theist's background information would include:

• B2: T is true.

Unlike B1, B2 has no entailment relation with Rrt. In plain English, if all we know is that some version of T is true, Rrt might or might not be true. So, how, then, should Pr(Rrt | B2) be calculated? 

For now, I will leave that as an exercise for the reader. 

Notes

[1] (PROVIDE DRAPER CITATION HERE)

[2] (PROVIDE DRAPER CITATION HERE)

[3] (PROVIDE CAVIN AND COLOMBETTI CITATION HERE)

[4] (PROVIDE CAVIN AND COLOMBETTI CITATION HERE)

[5] In my notation, "clp" stands for "confidence level-personal" to distinguish it it from "confidence level-group" (clg), which would be used for the intersubjective theory of probability. Additionally, in my notation, I put the name of the person or the group after "clp-" or "clg-", respectively, to explicitly identify the person or group whose level of confidence is represented.

[6] Paul Draper, "Pain and Pleasure: An Evidential Problem for Theists." Nous Vol. 23, No. 3 (June 1989): 331-350 at 349, n. 2.

[7] (PROVIDE CAVIN CITATION HERE)

[8] This is an oversimplification because it assumes that source physicalists know source physicalism to be true with certainty. I don't discuss this point in the main body of the essay because my primary interest in this essay is the prior probability of the theistic resurrection theory, without including SP in the background information.