Wednesday, 8 August 2012

WLC8: Sweet Probability

When I first looked at Craig, I only identified seven fallacious arguments.  Recently, I’ve gone back and listened to some more debates, Craig-Ehrman in particular, and identified another.

I should point out (again) that I am a little wary about counting Ehrman as being a fully-fledged opponent to Craig.  Ehrman says he is an agnostic, specifically not an atheist, and I read that as meaning that Ehrman has stopped believing that humans can know there is a god.  He still seems to think that certain parts of the Bible are reliable theologically, but can’t be relied upon historically.

What I most certainly don’t want to do is crow about a scalp taken from the ranks of New Testament scholars, like the theists like to do with regard to Antony Flew.  I do note, however, that Ehrman is still in full possession of his faculties.

Anyway … probability.

So far, I’ve seen two variations of the probability argument: in Craig-Krauss and in Craig-Ehrman.  For simplicity, I’ll only address the Craig-Ehrman version.  Just note, that Craig is not consistent in his use of this argument.

This Professor Earman (sic) is not a Christian; in fact, he’s an agnostic. He doesn’t even believe God exists. Nevertheless, you see what he thinks of Hume’s argument: it’s not merely a failure, it is an abject failure. That is to say, it is demonstrably, irremediably, hopelessly fallacious.

Let me explain why.

When we talk about the probability of some event or hypothesis A, that probability is always relative to a body of background information B. So we speak of the probability of A on B, or of A with respect to B.

Pr (A/B)

So in order to figure out the probability of the resurrection, let B stand for our background knowledge of the world apart from any evidence for the resurrection. Let E stand for the specific evidence for Jesus’ resurrection: the empty tomb, the post-mortem appearances, and so on. Finally, let R stand for Jesus’ resurrection. Now what we want to figure out is the probability of Jesus’ resurrection given our background knowledge of the world and the specific evidence in this case.

Calculating the Probability of the Resurrection
B = Background knowledge
E = Specific evidence (empty tomb, post-mortem appearances, etc.)
R = Resurrection of Jesus

Pr (R/B & E) = ?

Now probability theorists have developed a very complex formula for calculating probabilities like this, and I’m going to walk you through it one step at a time, so that you’ll be able to get it.

The first factor that we need to consider is the probability of the resurrection on the background knowledge alone:
Pr (R/B) is called the intrinsic probability of the resurrection. It tells how probable the resurrection is given our general knowledge of the world.

Next we multiply that by the probability of the evidence given our background knowledge and the resurrection:
Pr (E/B&R) is called the explanatory power of the resurrection hypothesis. It tells how probable the resurrection makes the evidence of the empty tomb and so forth. These two factors form the numerator of this ratio.

Now below the line, in the denominator, just reproduce the numerator. Just move everything above the line down below the line:
Finally, we add to that the product of two more factors: the intrinsic probability that Jesus did not rise from the dead times the explanatory power of the hypothesis of no resurrection:
Basically, Pr (not-R/B) × Pr (E/B& not-R) represent the intrinsic probability and explanatory power of all the naturalistic alternatives to Jesus’ resurrection.

So the probability of Jesus’ resurrection relative to our background information and the specific evidence is equal to this complicated ratio.

You might want to see the original presentation in the You-Tube video.

So, let’s just take a little look at Bill’s workings, shall we?

Craig is basically rehashing (without attribution) Richard Swinburne’s misuse of “Bayesian inference”, specifically where used to assess the probability of a hypothesis.  Craig introduces the notation Pr(R|B&E) which he describes as meaning “the probability of Jesus’ resurrection given our background knowledge of the world and the specific evidence in this case”.

Now, it should be noted that the ampersand symbol isn’t the most commonly used notation in probability, and for the uninitiated there is room for confusion, but I think it is fair enough to assume that he means the intersection ∩ (since the probability that both A and B occur = P(A and B) = P(A ∩ B)).  He could possibly mean the union U (since the probability that either A or B or both A and B occurs = P(A U B) = P(A) + P(B)), but he’d be really ridiculously incorrect in his use of notation and Craig usually seems very well prepared for his debates.  Let’s give him the benefit of the doubt.  We’re also giving Swinburne the benefit of the doubt.

So, how do we use this ∩ thing?  First … a Venn diagram.

Say we have a jar in front of us containing 100 jellybeans.  The jellybeans were all made from three batches of “jellybean stuff”, one red, one blue and one yellow.  The jellybeans can be pure red, pure blue, pure yellow, a combination of two of them (making orange, green and purple) or all three (making white).  Note that there are no jellybeans in the jar which are not made of “jellybean stuff”, such jellybeans would (chromatically speaking) be black and (ontologically speaking) non-existent.

Records show that the relevant numbers were as follows:
Primary colours only – 20 each – total of 60
Combinations of two colours – 10 each – total of 30
Combination of three colours – 10 each – total of 10
Grand total – 100
This distribution is illustrated below.

We can ask a few different questions about these jellybeans.  We would mostly likely want to ask “can we have some?” to which the answer is “no, I am in the middle of using them to prove a point”.  Once we get over the disappointment, other questions follow:

1.      How many jellybeans were made from red jellybean stuff?

2.      How many jellybeans were made from blue and yellow jellybean stuff?

If you answered “50” and “I’m not quite sure what you mean by ‘and’ in that question”, well done, you may take a couple of black jellybeans.

The red jellybean stuff is found in not only the red jellybeans, but also in the purple, orange and white ones.  If I put a blindfold on and dipped into the jar to extract a single jellybean at random, what would be the probability that I would extract one with red jellybean stuff?  This would be:

Pr(Red)+Pr(Purple)+Pr(Orange)+Pr(White) = 20/100+10/100+10/100+10/100 = 50/100 = 0.5

Now to the second question, what did I mean by ‘and’?  I could have meant “how many jellybeans would I have if I took all those that were made from blue jellybean stuff and all those that were made from yellow jellybean stuff?” which would be the 80 that are not pure red.  I could have meant “how many jellybeans would I have if I took only those that contain both blue jellybean stuff and yellow jellybean stuff?” which would be 20 (10 white and 10 green).  Or I could have meant “how many jellybeans would I have if I took only those that were made from only blue jellybean stuff together with yellow jellybean stuff?” which would be the 10 green ones. 

You have to be careful with ‘and’.  This is where the notation comes in:

Option 1 = Pr(Contains Blue U Contains Yellow) = 80/100 = 0.8

Option 2 = Pr(Contains Blue ∩ Contains Yellow) = 20/100 = 0.2

Option 3 = Pr(Contains Blue ∩ Contains Yellow ∩ Does Not Contain Red) = 10/100 = 0.1

In other words, Pr(Contains Blue ∩ Contains Yellow) is the probability that a jellybean contains blue jellybean stuff simultaneously while containing yellow jellybean stuff.

Note that Pr(Contains Blue) and Pr(Contains Yellow) have the same value as Pr(Contains Red) at 50/100.  If I randomly select a jellybean, there is a 50/100 chance of it containing red, a 50% chance of it containing blue and a probability of 0.5 of it containing yellow.  The chances of it containing blue and yellow is less than the chance of it containing just blue.

Okay, part of the way there.

Craig uses the format Pr(R|B&E) which should be Pr(R|B∩E).  Now he’s been a little inaccurate here, he should use some brackets to make clear precisely what he means.  From his wording, he really means Pr(B|(E∩B)).

Let us look at what the notation Pr(X|Y) means.  The notation refers to conditional probability and means “the probability of X given Y”.

Looking again at our jar of jellybeans, we could ask “what is the probability of a randomly selected jellybean containing blue jellybean stuff given the jellybean selected contains yellow jellybean stuff?”

We know that Pr(Contains Blue) = 50/100 = 0.5 and Pr(Contains Yellow) = 50/100 = 0.5.  However, we’ve trimmed the selection down to only jellybeans that contain yellow.  Conceptually, we’ve emptied the jar of all jellybeans that don’t contain yellow, so we only have 50 of them remaining.  How many of the 50 remaining contain blue?  That would be the green and white jellybeans, so 20.

Therefore Pr(Contains Blue | Contains Yellow) = 20/50 = 0.4

Using probability theory Pr(X|Y) can be calculated as follows:

Pr(X|Y) = Pr(X ∩ Y) / Pr(Y)


Pr(Green | Contains Yellow) = Pr(Green ∩ Contains Yellow) / Pr(Contains Yellow)

Pr(Contains Blue | Contains Yellow) = Pr(Contains Blue ∩ Contains Yellow) / Pr(Contains Yellow)

Since Pr(Contains Yellow) = 50/100, this gives:

Pr(Green | Contains Yellow) = (10/100)/(50/100) = 10/50 = 0.2

Pr(Contains Blue | Contains Yellow) = (20/100)/(50/100) = 20/50 = 0.4

This is just what we thought, so we can put a big tick next to the Bayesian inference.

We are almost on the home straight.  Remember that Craig used the notation Pr(R|B&E) which should have been Pr(B|(E∩B)) and that this means:

Pr(R|(B∩E)) = Pr(R∩(B∩E)) / Pr(B∩E)

I know, it looks horrible, but let us return to our jellybeans and say that R=Contains Red, B=Contains Blue and E=Contains yEllow.

Pr(Contains Red | (Contains Blue ∩ Contains yEllow))

= Pr(Contains Red ∩ (Contains Blue ∩ Contains yEllow)) / Pr(Contains Blue ∩ Contains yEllow)

Okay, it still looks quite horrible.  The term Pr(Contains Red ∩ (Contains Blue ∩ Contains yEllow)) just means “the probability that a randomly selected jellybean contains red and also contains both blue and yellow”.  In other words, what is the probability of selecting a white jellybean? 10/100

The term Pr(Contains Blue ∩ Contains yEllow) means “the probability that a randomly selected jellybean contains both blue and yellow”.  In other words, what is the chance of selecting a jellybean which is green or white?  20/100

So …

 Pr(Contains Red | (Contains Blue ∩ Contains yEllow))

= Pr(Contains Red ∩ (Contains Blue ∩ Contains yEllow)) / Pr(Contains Blue ∩ Contains yEllow)

= (10/100) / (20/100) = 10/20 = 0.5

This is the same probability for randomly selecting a jellybean that contains red as if you were picking from the full 100 – but you had just removed all the jellybeans which did not contain both blue and yellow including 40 that contained red!

The trick that Craig is playing here is that he doesn’t want to select the equivalent of “contains red”, he wants to pick the equivalent of white and if he manoeuvred like this, he’d increase his chances (conceptually) from 0.1 to 0.5.

But Craig doesn’t play that fair.

Let’s look at his specific argument.

Pr(Resurrection of Jesus | (Background Information ∩ Specific Evidence))

What Craig is asking is “if we assume that both the background information is true and the specific evidence is true, what is the likelihood that the resurrection of Jesus is true?”

The “background information” is fine, that’s what we know about the world.  We can safely assume that it is very highly possible that what we know of the world is true.  Let’s be foolishly optimistic and assign it a probability of 1.0.  This is equivalent to background information stating that there is a jar full of jellybeans – yep, thanks, we can see it.

The “specific evidence” is questionable.  Personally, I take it to be extremely unlikely that all the things reported with regard the resurrection happened as reported.  Note that this is precisely the evidence that Craig wants us to use: the crucifixion, Joseph of Magrathea (or something like that) put Jesus in the tomb, the tomb was empty three days later and people later reported having seen him.  To be generous, I’ll say that the probability that this is true is non-zero.  This is equivalent to saying that only ten out of the jellybeans in the whole jar (representing all the evidence available) are of the right sort – the rest of the jellybeans (and any evidence which is inconvenient) will be removed.

Now, final step.  Craig euphorically (and metaphorically) reaches into the jar, and removes a white jellybean – ten times in a row!  Amazing, what he has proved without a doubt, is that if you remove all the jellybeans apart from the ones you want, then you will have only the ones you want left!

Or rather, if you eliminate all the evidence except that which supports the resurrection – irrespective of how tenuous that evidence is – then the only evidence you will have left is that which supports resurrection!

And now, I am going to go eat something sweet.


If you want another take on this, I suggest a nice piece by Josh, the Honest Searcher.


Edited: 16 May 2017 to fix discussion about the ampersand (&) as a logical symbol

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