Mundane vs Miraculous claims

I've been thinking lately about a particular criticism of apologist techniques. Apologists will argue for the general reliability of texts using relatively mundane claims which can be verified externally, and then once the reliability has been established, argue that it follows that the miracle claims are then likely to be true. There is a sense I feel that establishing mundane claims in a text would only make other mundane claims likely, and not miracle claims. For example, say a text has a controversial claim that so far can't be verified externally -- something like, Shakespeare had a ghost writer. If you pointed to really mundane claims also in the text, like the sun rose in the east, it rained in March in England on some days, etc... it would do nothing at all to boost ones belief in the ghost writer.

As much as I have an intuition for this, I don't see where it immediately follows from the rules of probability. Here is a first attempt at a toy problem that tries to capture some of this situation.

The Toy Problem

Say we measure the overall reliability of a text with a number from 0 to 1, which we'll call $\theta$ and will represent the probability that any random claim in the text is true. This is like having a biased coin which we flip, heads the claim this true, tails the claim is false. As we externally confirm claims, it will be like having data on previous flips of the coin and we update out estimate of $\theta$. Roughly, the best estimate of $\theta$ will follow Laplace's Rule of Succession,

$$\hat{\theta }=\frac{N_H+1}{N_H+N_T+2}$$

where $N_H$ is the number of externally confirmed claims, $N_T$ is the number of externally disconfirmed claims, and $\hat{\theta }$ is our best estimate of the probability the next claim we look at will be true.

We are interested in the probability a particular miracle, $M$, is true given a claim, $C$, of that miracle in the text, (see another post of mine about testimony to see a similar toy problem)

$$P(M|C)=\frac{P(C|M)P(M)}{P(C|M)P(M)+P(C|\neg M)P(\neg M)}$$ We'll make the following assumptions and definitions,

• the probability a claim is made, given the miracle event occurred is what we mean by reliability: $P(C|M)=\theta$
• the probability a claim is made, given the miracle event did not occur is the opposite of what we mean by reliability: $P(C|\neg M)=1-\theta$
• the prior probability of the miracle is defined for convenience: $P(M)\equiv m$
• there are no instances of any of the claims being disconfirmed externally: $N_T=0$. This, in my view, gives a definite pro-miracle bias to this model
• there is a sequence of $N_H=n$ confirmed claims before we are examining the miracle claim.

So the $n$ confirmed claims will bring the estimate of the reliability, $\hat{\theta }$, closer and closer to 1.

From these assumptions and definitions we rewrite the equation above as,

$$\begin{array}{rl}P(M|C)& =\frac{\hat{\theta }\cdot m}{\hat{\theta }\cdot m+(1-\hat{\theta })\cdot (1-m)}\\ & =\frac{\left(\frac{n+1}{n+2}\right)\cdot m}{\left(\frac{n+1}{n+2}\right)\cdot m+\left(\frac{1}{n+2}\right)\cdot (1-m)}\\ & =\frac{(n+1)\cdot m}{(n+1)\cdot m+(1-m)}\\ & =\frac{(n+1)\cdot m}{nm+1}\end{array}$$ If we want to believe the miracle, then we should have $P(M|C)>0.5$, which leads us to:

$$\begin{array}{rl}\frac{(n+1)\cdot m}{nm+1}& >0.5\\ nm+m& >0.5nm+0.5\\ 0.5nm& >0.5-m\\ n& >\frac{0.5-m}{0.5m}\\ n& >\frac{1-2m}{m}\\ n& >\frac{1}{m}-2\end{array}$$

Interpretation

Through this analysis, I have not made any statements about the value of the prior probability, $m$. As such, we can use this toy example both for miracle claims where $m$ is tiny and for mundane claims where $m$ is moderate and the calculation of $n$ is the number of claims that must be externally verified before the new claim is to be believed.

For example, if the prior is moderate, like $m=0.1$, then $n=8$ and we need only a handful of externally verified claims before we believe this new mundane claim. However if the prior is tiny, like $m=1/100\text{billion}$ -- the optimistic prior for the resurrection of Jesus. This leads to $n=100\text{billion}$ claims that must be externally verified before this new claim is to be believed! No amount of text would be able to reach this threshold.

Caveats

I grant that this is not exactly the pattern for establishing historical reliability, and there are some simplifying assumptions. I also grant that this doesn't solve the entire problem which motivated my inquiry -- my intuition still says that many claims on the order of "the sun rises in the east" shouldn't raise the probability of any claim above the prior for that particular claim. But I think the current analysis does clearly point out one reason why establishing mundane claims in a text can't establish miracle claims in the same text and captures this in a pretty simple and compelling way.

What problems are you interested in? How can I help?

• @bblais
• bblais@bryant.edu
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• Bryant University
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