Do extraordinary claims require extraordinary evidence? (Part 2)

Summary

Last time I introduced the topic, claiming that miracles do not need miracle-level evidence to support them.  The reason: you cannot assume your conclusion before the argument begins!  That is the mistake that atheists make (perhaps not realizing it) when they state that extraordinary claims require extraordinary evidence.  But like I said last time, you cannot a priori assume that miracles cannot happen; if you do this, there is no reason to have the discussion. 

Below, I show this rather rigorously using Bayes' Theorem, in a combination with a very simple analysis of the prior probability of a miracle happening.  I also put forth the Resurrection as a concrete example.  Some skeptical readers may be disgruntled with some of the numbers I put to things.  But even so, keep in mind that I'm not after some mathematical proof  beyond a shadow of a doubt that a given miracle did occur. No, I am simply showing that the statement "extraordinary claims require extraordinary evidence" is false.  Indeed, you'll find that, even with very conservative assumptions, miracles can be plausibly supported by ordinary evidence.

But readers be forewarned: there is a lot of math below!

The mathematics of Bayes' Theorem

To put things mathematically, Bayes theorem tells you how to evaluate P(A | B): the probability of a proposition, "A", given (i.e., in light of) another proposition, "B".  To see this, note that the definition of such a conditional probability is:

P(A | B) = P(A & B)/P(B).

If we multiply through by "P(B)" then we get:

P(A | B)*P(B) = P(A & B).

Since "A" and "B" are interchangeable in the right hand side of this equation, you can also write:

P(A | B)*P(B) = P(A & B) = P(B | A)*P(A).

Moving P(B) back to the other side:

P(A | B) = P(B | A)*P(A)/P(B).

Let's put some concreteness to this.  Say "A" is the proposition that some miracle happened (for simplicity, call it "M" instead of "A").  Now say "B" is the quite ordinary evidence for said miracle (and call it "E" for evidence).  Now:

P(M | E) = P(E | M)*P(M)/P(E).

In other words, the probability that a miracle happened, given you have evidence for that miracle, is how well the miracle can explain the evidence, weighted by how likely you think the miracle and evidence should occur on their own.  These are called prior probabilities. That is, P(M) is, "What do you think is the probability of a miracle happening before you consider the evidence?"  And P(E) is, "What do you think the probability of the evidence happening is, without considering a miracle happened?"  Since the evidence is "ordinary", P(E) is probably not too small.

But it seems we're stuck with this nasty prior probability, P(M).  Surely the probability of a miracle happening is so small (by definition) that you must then conclude that P(M | E) is never going to be large enough to convince someone, right?  That is, unless P(E) is super-duper small.  Unless the evidence is also extraordinary.  Hence, the conclusion that to prove a miracle you need miraculous evidence.

This is where you have to question your presuppositions. This is where you have to take a second look at what goes into the prior, P(M).

Why you don't need miraculous evidence to prove a miracle.

The probability of any proposition, A, can be split up into two parts contingent on another proposition, B:

P(A) = P(A | B)*P(B) + P(A | ~B)*P(~B),

(where "|" means "given," and "~" is a negation).  In our case, what's P(M), the prior probability of a miracle, M?  It can be split up into two parts; the probability of a miracle happening given that God exists (G) and the probability of a miracle happening given God does not exist (~G):

P(M) = P(M | G)*P(G) + P(M | ~G)*P(~G).

This is mathematically true, and also philosophically sound, because if you are arguing about whether or not miracles are possible (and usually it's an argument between a theist and an atheist), you cannot a priori assume that God does not exist.  That is the basis of the whole argument.  If someone assumes the other person's position is impossible (probability zero), then there can be no discussion.  So you cannot assume P(G) = 0.  You can assume it is small, but not arbitrarily small (as Dawkins attempts to do).

So let's look at the terms that make up P(M).  I argue (and so would the atheist) that P(M | ~G) is essentially zero.  A miracle is not going to happen if God does not exist, because by definition it is an extraordinarily rare or impossible event.  So the second term disappears, and we're left with:

P(M) = P(M | G)*P(G).

I think we can successfully argue that P(M | G) (ie, the probability that the miracle in question would happen assuming God exists) is not vanishingly small. (If we are specifically saying G = God of the Bible, and M = the Resurrection, then I would say P(M | G) = 1.)  So the only way you end up with a zero prior for a miracle happening is if you assume that God cannot exist (ie, P(G) = 0).

Note that this is different from believing God does not exist.  Atheists believe God does not exist, but no self-respecting atheist believes God cannot exist.  If you ask someone whether they think P(G) = 0 and they say yes, the conversation is over.  That person believes that God cannot exist and therefore nothing you say can convince them.  You always must leave room for your beliefs to be falsified.

So P(G) > 0.  If you are talking about any God, then P(G) should be higher than 1/2, since more than half of all people believe in a god of one form or another (and if we're just basing our priors off of low-shelf statistics, which is what you normally do).  Along those lines, if you're talking about the God of the Bible, then P(G) is more like 1/3.  But even if you are speaking to a hardened atheist, such as Richard Dawkins, then P(G) is as high as 1 - 6.9/7 = 0.0143.  I would be willing to make that allowance.

You can evaluate P(M | G) in various ways.  If the god you are talking about has a holy text that claims the particular miracle in question (as in, the Bible and the Resurrection), then I would say P(M | G) = 1 (for all intents and purposes). On the other hand, if you are just attributing some random "miracle" to some random god, then it gets tricky.  For the sake of argument, let's say P(M | G) is a conservative one out of ten.  This means P(M) is rather high: one out of a thousand!  We take insurance policies out against probabilities more remote than that.  We play the lottery on the hope of winning big with probabilities far more remote than that.

If you are still disgruntled at my analysis of P(M), keep in mind that my argument for P(G) is essentially unassailable.  If you deny that argument, you are entering into the land of logical fallacy.  You may question my choice for P(M | G), however.  In that case, let's just leave P(M) equal to "some small number" p.  That way, we can see later what Bayes' Theorem does, and try to agree on a value for "p" later.  (Just remember: you can't make p = 0 a priori.)

Finishing the analysis

Now that we are armed with the knowledge that p can't be equal to zero, let's go ahead and complete our analysis using Bayes' Theorem.  We last left Bayes Theorem as:

P(M | E) = P(E | M)*p/P(E).

We can split P(E) up in the same way that we split up P(M), but this time we will condition it on "M":

P(E) = P(E | M)*P(M) + P(E | ~M)*P(~M),

or,

P(E) = P(E | M)*p + P(E | ~M)*(1 - p).

Putting this back into the equation for P(M | E):

P(M | E) = P(E | M)*p / (P(E | M)*p + P(E | ~M)*(1 - p)).

If we rearrange the P(E | M) factors:

P(M | E) = p / (p + (1-p)*P(E | ~M)/P(E | M)).

Now we see there are two operative variables here: "p" (the prior probability of a miracle; remember, this cannot equal zero), and the following ratio:

R = P(E | ~M)/P(E | M).

This last term is the explanatory power of the miracle.  How much more likely is it that the evidence would have happened if miracle did not occur over if the miracle had occurred?  If the miracle happening makes the evidence more likely to occur, then R < 1.  If the miracle happening makes the evidence less likely to occur, then R > 1.  Obviously, if you are marshalling evidence for a miracle, let's hope you've chosen something that favors the miracle happening, so then R < 1.  How much less than one, of course, depends on the particular miracle you are investigating and what evidence you have for it.

At any rate, this gives us a very simply formula for the probability of a miracle happening given the evidence at hand:

P(M | E) = p / (p + (1-p)*R).

To make things concrete, let's choose a specific example.

A concrete example: the Resurrection 

Let's now suppose the miracle, M, we're investigating is the Resurrection.  Of course Habermas and Licona have presented five "minimal facts" that most biblical scholars, skeptical and conservative, agree upon in regards to the events surrounding Jesus' death.  However, since this is an internet blog and not a scholarly work, and since that means there are many hyper-skeptics out there who might visit and comment on this blog, let's just stick to one piece of evidence: the conversion and martyrdom of the early church fathers.  And, let's just stick with two church fathers: Paul and James.

In this case, the probability P(M | E), what we wish to find, is the probability that the Resurrection occurred, given the (ordinary historical) evidence that Paul and James were both converted from hostility/skepticism and were both martyred for their beliefs (never renouncing the claim that they saw the risen Lord).

The probability p = P(M) = P(M | G)*P(G), where P(G) is the probability that the God of the Bible exists, and P(M | G) = 1.  So your prior p is essentially how likely you think it is that the God of the Bible exists.  Remember: the world's most famous atheist says this is about 0.01 (more or less. I admit that if he were interviewed, he might say the particular God of the Bible is even less likely to exist than any random god, but then why does he spend so much time focused on Christianity?).

So the question boils down to: what's R?  In words, R is the probability that Paul and James would die for their beliefs given the Resurrection did not happen, divided by the probability that the they would die for their beliefs given the Resurrection did happen.  It seems extremely, extremely, extremely unlikely to me that, if the Resurrection did not happen, then both Paul and James would be converted and go to their deaths proclaiming Christ is King.  But if it is even just 10 times more likely for the Resurrection to make more sense about either man dying for Jesus, then R = 0.01, and we get:

P(M | G) = 0.01 / (0.01 + 0.99*0.01) > 0.5

Now, you may not agree with me that Paul and James actually did convert and actually did die for their beliefs.  You may not even agree with me that the Resurrection makes more sense of them dying for their beliefs over the Resurrection not happening.  But what you are now forced to accept is that it does not take much in the way of "extraordinary" evidence for even a miracle to be plausible.  Remember, I took the position of the most die-hard atheist and a pitiful collection of historical evidence (compared to the wealth of historical evidence that we do have), and a conservative estimate of what that might mean, and I still am forced to arrive at the conclusion that the resurrection is more than 50% likely.

Conclusion

In these posts (including last time), I have shown, using careful (and conservative) mathematical and philosophical arguments that it does not take extraordinary evidence to back and extraordinary claim.  This is based on the recognition that you cannot assume your conclusion before you start. (That is the only way that you could ever arrive at the conclusion that extraordinary claims require extraordinary evidence.)  Once you give even a little ground to the possibility that God exists, and you must do so for intellectual honesty, you are forced to accept the fact that, given enough ordinary evidence, such as well-attested to historical facts, or personal testimony, that miracles actually can be plausible.  And while it is true that the evaluation of some of these probabilities are subjective, keep in mind this was not meant as a mathematical proof of any particular miracle, only a demonstration that, even with very conservative assumptions, miracles can be plausibly supported by ordinary evidence.

Do extraordinary claims require extraordinary evidence? (Part 1)

Summary

How many of you have heard the statement, "Extraordinary claims require extraordinary evidence?" This is a common soundbyte from skeptics to try to discredit the existence of miracles.  But more often than not, it's used incorrectly, and that sort of disappoints me, because it shows that folks are not thinking logically.  Those who hold to this claim are essentially saying that they require evidence that is as miraculous as the miracle itself for them to be convinced a miracle happened.  Do you want to convince someone of the Resurrection?  It would take something on the order of lightning in the sky that spelled out, "Jesus rose from the dead," to convince such a skeptic.

It is actually rather simple to show this argument against miracles is false: it's because the argument only holds when you first assume miracles cannot happen.  And if you assume miracles cannot happen, then of course you are going to conclude that no evidence is enough to prove a miracle.  This is the mistake Hume made, the mistake the Jesus Seminar folks made, and the mistake that Bart Ehrman makes.

But you cannot a priori assume that miracles cannot happen; if you do this, there is no reason to have the discussion.  It is a form of circular reasoning.  Therefore, you must instead make an allowance (however small) that they can happen.  If they can happen, then any sort of everyday evidence could be evidence in favor of a miracle.  The question would then become whether the evidence is sufficiently in favor of a miracle to make someone believe, not whether something as commonplace as eyewitness testimony could be evidence of a miracle.

This can be shown rather rigorously using Bayes' Theorem, which I will do next time.  (Be forewarned: it can be a bit technical, but if you like math, or even if you don't, but you don't mind getting out a pencil and paper and following along, it should be pretty straightforward.)  Below I just give a brief, intuitive introduction to Bayes Theorem.

The intuition of Bayes Theorem

First of all, I love Bayes Theorem.  I've talked about it several times on this blog, mostly because it's really good to show that indeed, miracles can happen.  It's also super-easy to use; you just have to know a few things about probability, which are all taught in high school (I think).

Bayes Theorem is what helps us answer questions about what might be true (i.e., the converse of what we know to be true). Here's what I mean.  Let's say we know that for sure, the statement "if A then B" is true.  This means that if we know that A is true, then B must be true. But this does not mean that if B is true, then A must be true. But it seems like "B" being true adds some weight to "A" being true.  Bayes Theorem is how we figure out just how much weight that is.

Here's an example in which "B" being true definitely does not imply that "A" is true.  Let's say the statement "If I am swimming a race against Ryan Lochte (A), I will lose the race (B)" is true. Then let's say I swim in some race, and later tell my friend that I lost (B).  Can my friend infer that I must have been swimming against Ryan Lochte (A)?  Of course not.  There are many other swimmers out there that I would lose to.  In this case, B does not even come close to implying A.

On the other hand, here's an example where it seems B almost implies A: If you stand out in the rain (A), you will get wet (B).  Here, A implies B.  Does that mean if someone is wet, they stood outside in the rain?  No, but what happens if you add further evidence to the scenario?  You are in a shop, and someone comes in from the outside and they are dripping wet.  You'd probably conclude that it is indeed raining outside.  Adding further evidence, what if everyone who came in either had a wet umbrella, a wet raincoat, or was just wet (on their clothes)?  Now add the fact that when you went into the shop, you noticed dark clouds in the sky.  So yeah, you don't need to actually see the rain to be pretty sure it's raining.  (Thanks to pleaseconvinceme.com for inspiring this example.)

That's a brief, intuitive example of how Bayes' theorem works.  The more evidence that's in favor of an event (A) above and beyond other explanations, the more you can be certain that, given said evidence (B), A is true.