Women Visiting Jesus’ Tomb

A number of Christian apologists claim that the women discovering the empty tomb of Jesus brings points in favor of the empty tomb because of the so-called criterion of embarrassment, with women (who were admittedly of a lower social status) being the first witnesses of it. But a person making up this story (not saying it was made up; maybe it was merely legendary) might have included the women just to be realistic. From the Women in the Bible website:

Tombs were visited and watched for three days by family members. On the third day after death, the body was examined. This was to make sure that the person was really dead, for accidental burial of someone still living could occur. On these occasions, the body would be treated by the women of the family with oils and perfumes. The women's visit to the tombs of Jesus and Lazarus are connected with this ritual.

A person knowing this likely would have had the women be the first discoverers of Jesus’ empty tomb.

More Plausible Than Their Denials

Introduction

William Lane Craig has long been a fan of the “more plausible than its denial” approach with premises in pro-theism arguments. In addition to being formally valid (the conclusion logically and inescapably following from the premises), having true premises, being informally valid (making no informal fallacies, like begging the question) Craig seems to think adding “the premises are more plausible than their negations” to this list is enough to make a good argument. Craig’s position on this is problematic however.

The Problem

Take for example this Q & A article. As one might expect, Craig says that among the criteria for a good argument are the argument being deductively valid (the conclusion follows logically and inescapably from the premises) and true premises. (In logic, a deductively valid argument with true premises is called a sound argument.) Craig however also says this:

I’ve argued that what is needed is that the premisses be not only true but more plausible than their opposites or negations. If it is more plausible that a premiss is, in light of the evidence, true rather than false, then we should believe the premiss.

Should we? Consider this hypothetical scenario. I am very sick, and there is a potion that has a 51% chance of curing me if I drink it, but also has a 49% chance of killing me if I drink it. If I don’t take the potion, I will recover in a few days, albeit it will be a rather unpleasant few days, akin to having the flu. Now, if I knew the potion would cure me, the rational thing for me would be to take it. And yet, the rational thing for me to do is to not accept “This potion will cure me” as something I know to be true. We have a claim that is more plausible than its negation, but I’m pretty sure the uncertainty level is great enough that I should withhold my belief in it. So this argument isn’t a good one even though each premise is more plausible than its negation:

1. If drinking the potion would cure me, then I should drink it.
2. Drinking the potion would cure me.
3. Therefore, I should drink it.

Individually each premise is more plausible than its negation, and yet the conclusion isn’t quite true.

Or to use a less dramatic example to illustrate the point, suppose a random number generator has displayed an integer 1 through 20, but I haven’t yet looked at the screen yet to know which number it picked. The claim, “It picked a number less than 12” is more plausible than its negation; it has a 55% of being true, but I don’t really know it to be true until I look at the screen. I would be quite rational in withholding my belief about whether “It picked a number less than 12” is in fact true.

Suppose our random number generator has three trials, with each event being probabilistically independent of the other. Now consider the following deductive argument:

4. The first trial picked a number less than 13.
5. The second trial picked a number less than 13.
6. The third trial picked a number less than 13.
7. Therefore, the first, second, and third trials each picked a number less than 13.

Each premise has a 60% chance of being true, and thus each premise is more plausible than its denial. And yet, the probability that we have a sound argument is only 21.6%, and the probability that the conclusion is false is 78.4%. Yet if we were to follow Craig’s logic, we should believe all three premises (since each is more plausible than its negation), and as a consequent we should believe the conclusion (on pain of inconsistency, since the conclusion follows from the premises) even though we know the conclusion has a 78.4% chance of being false!

Craig seems kind of aware of this problem in the Q & A article I linked to, but he never seems to quite address it. Craig does say, “It’s logically fallacious to multiply the probabilities of the premisses to try to calculate the probability of the conclusion.” But whether that is true will depend on the circumstances. In the case of the random number generator argument, multiplying the probability of the premises to calculate the probability of the conclusion works just fine. Here’s an example where it doesn’t work, where I roll a fair six-sided die but didn’t see the die come up:

8. The six-sided fair die I rolled is four or less.
9. The six-sided fair die I rolled is five or less.
10. Therefore, the six-sided die I rolled is (a) four or less; and (b) five or less.

Multiplying the probabilities gives us about 55.6% probability for the conclusion, when the conclusion’s probability is closer to 66.7%. The reason multiplying probabilities doesn’t work here is that the premises aren’t probabilistically independent of each other.

Still, the random number generator argument I gave (lines 4 through 7 above) still provides an apparent counterexample to Craig’s claim regarding what makes a good argument, because here we have an argument that meets Craig’s criteria and yet we know the argument is probably unsound. We also have reason to doubt Craig’s claim that, “If it is more plausible that a premiss is, in light of the evidence, true rather than false, then we should believe the premiss” due to the argument in lines 1 through 3 above.

A Caveat

I don’t want to go too far and assert that “Each premise is more plausible than its negation” is a worthless criterion. Often times if a deductively valid argument were to meet that criterion, the argument would provide at least some degree of rational support for the conclusion. To illustrate, suppose we knew that each premise of the following argument is more plausible than its denial:

11. If God does not exist, then objective moral values and duties do not exist.
12. Objective moral values and duties do exist.
13. Therefore, God exists.

The fact (if it were so) that each premise is more plausible than its denial would make the atheist intellectually uncomfortable, because each premise being more plausible than its denial means that the argument provides at least some support for the conclusion, and it would just be a question of how much support that argument provides. (For those interested in seeing me argue that it is not the case that each premise is more plausible than its negation, see my article The Moral Argument and William Lane Craig.)

Consider again the case of the random number generator: the argument in lines 4-7. There the argument’s premises are (individually) more plausible than their denial such that we at least know the probability of the conclusion is not less than 21.6%. So each premise being more plausible than its denial lends some support for the conclusion—21.6% is better than 0% after all—it’s just that the degree of rational support is not enough to make the conclusion’s probability more than 50%.

And I think this sort of thing is an often overlooked weakness in pro-theism arguments. It’s possible for a pro-theism argument to provide some but only a little support for the conclusion. Indeed, that was the approach I took when rebutting the Leibnizian cosmological argument where I said this (albeit after bestowing a number of objections to the argument):

One way an argument can fail to be convincing is if it provides no rational support for its conclusion, but we should not make the mistake of thinking that’s the only way an argument can fail to be convincing. Another way is if the argument provided nonzero but nonetheless too little support for its conclusion. So we can accept that ceteris paribus a worldview that explains e.g. why there is something rather than nothing is better than one that doesn’t, but given the plausibility of physical reality existing eternally without an external cause, the degree of evidential support this provides is rather small.

And if the probability of God’s existence is much less than 50% (due to say the argument from evil), the Leibnizian cosmological argument would do very little to remedy that problem for theism.

Beneficial Gene Duplication

I have noticed a number of creationists bringing up this creationist drivel:

No known mutation has ever produced a form of life having greater complexity and viability than its ancestors.

If true, this would indeed be a notable objection, since in the evolution from single-celled organisms to homo sapiens surely some increase in complexity happened along the way. Evolution requires genetic changes that increase genetic information, and at least some of these changes need to be beneficial so that they are chosen by natural selection. What sort of mutation could do that?

Answer: gene duplications. Gene duplications allow for an increase in genetic information, and science has observed beneficial gene duplications in real life. From Evolution after Gene Duplication:

The beneficial impact of gene duplication has been shown for several classes of genes. Perhaps the clearest example of an adaptive increase of gene dosage through a gene is that of the amylase gene in humans. Amylase is secreted in the pancreas and saliva, and it starts digestion in the course of chewing food with a significant portion of starch hydrolysis occurring before food reaches the stomach….the number of copies of the amylase gene was found to be significantly larger among populations with a high-starch diet. In addition, the frequency of individuals with more than six copies was two times higher in high-starch diet populations. Most important, there is a clear interdependence between the number of gene copies and the amount of amylase in saliva.[1]

This is a beneficial gene duplication in human beings. The book also says that, “insecticide resistance through gene duplication has been recognized as a major force by many others”[2] though to avoid giving a false impression I should also quote them saying that an “important aspect of insecticide resistance through gene duplication is that at least some of these duplications are actually deleterious in an environment without the pesticide.”[3] Sometimes in evolution it’s a “you win some, you lose some” sort of situation. A mutation that is on the whole beneficial for the environment you’re in might still come at a price, such as the loss of some beneficial traits. That’s why the theory of evolution isn’t necessarily committed to us retaining all the beneficial traits of our ancestors, such as superior physical strength akin to apes.

NOTES:

---

[1] Katharina Dittmar and David Liberles Evolution after Gene Duplication (Wiley-Blackwell, 2010) pp. 63-64.

[2] Katharina Dittmar and David Liberles Evolution after Gene Duplication (Wiley-Blackwell, 2010) p. 64

[3] Ibid.

Hume, Probability, and Miracles

Hume

Here’s a quote from David Hume in section 10 part 1 of his classic Enquiry Concerning Human Understanding (with the part incased in single quotes being what some call “Hume’s Maxim”):

The plain consequence is (and it is a general maxim worthy of our attention), ‘That no testimony is sufficient to establish a miracle, unless the testimony be of such a kind that its falsehood would be more miraculous, than the fact, which it endeavours to establish: And even in that case there is a mutual destruction of arguments, and the superior only gives us an assurance suitable to that degree of force, which remains, after deducting the inferior.’ When anyone tells me, that he saw a dead man restored to life, I immediately consider with myself, whether it be more probable, that this person should either deceive or be deceived, or that fact, which he relates, should have really happened. I weigh the one miracle against the other; and according to the superiority, which I discover, I pronounce my decision, and always reject the greater miracle. If the falsehood of his testimony would be more miraculous, than the event which relates; then, and not till then, can he pretend to command my belief or opinion.

For some, when talking about whether there is sufficient evidence for miracles it might be tempting to say, “Hume proved there can’t be evidence for it; look it up.” One reason this temptation should be resisted is because the theist can play the same game and say, “Hume’s case against miracles is an abject failure; look it up,” and we will have gotten nowhere.

There’s an annoying maneuver I’ve seen used by theists and atheists alike that I call “the literature toss.” Instead of explaining why they think an opponent’s position is wrong, they say something like, “Read this book.” Don’t get me wrong, reading books is great. Theists should read pro-atheism books and atheists should read pro-theism books to better understand different points of view and think critically. But a literature toss is often not a good substitute for real dialogue.

An atheist can throw a Hume book at the theist, the theist can throw a book at the atheist called Hume’s Abject Failure by philosopher John Earman, and the atheist can throw a book at the theist called A Defense of Hume on Miracles by philosopher Robert J. Fogelin. To those two people I say, “When you’re done throwing books at each other, maybe you can engage in some real conversation like adults.”

But there’s another reason why simply saying “look at Hume” won’t work here; Hume is (to at least some degree) too ambiguous. When Robert J. Fogelin critiques John Earman in chapter two of A Defense of Hume on Miracles he doesn’t critique Earman’s math (and Earman makes extensive use of mathematics to prove various points) but rather how Earman interpreted Hume. The Stanford Encyclopedia of Philosophy notes that the maxim is “open to interpretive disputes” and on the passage as a whole that I quoted, the article says the “interpretive issues are too extensive to summarize.” So instead of saying, “Hume showed we can’t have evidence against miracles” you should instead just put forth the argument itself, because your interpretation of Hume might not be their interpretation.

Actually, some scholars believe Hume didn’t intend to make a proof that we in principle can’t have sufficient testimonial evidence of miracles in his Enquiry, and that rather his position is that no testimony has in fact provided sufficient evidence of miracles. All that said, here is part of what I think Hume is trying to say in the passage above: when we encounter testimony of a miracle, we need to consider which is more likely: that the testimony is false (“this person should either deceive or be deceived”) or that the testimony is true. Even if we judge it to be true, whatever evidence we have that the testimony is false mitigates the evidential force of the pro-miracle testimony (“and the superior [evidence of the testimony] only gives us an assurance suitable to that degree of force, which remains, after deducting the inferior [evidence that the testimony is false]”). Sounds simple enough, and while the theist may say that it is more likely that a person’s testimony is telling the truth, due to a (presumed) reliability of a given witness, there’s some mathematics that throws a wrench into the gears of the theist’s thinking.

Surprising Math

To illustrate the general problem, consider the following scenario I'll call the “Taxicab Hit.” Suppose we have two taxicab companies, one which uses taxis painted blue and another which uses taxis painted green. On the roads, 85% of the taxicabs are green, and 15% are blue. One night, a taxi hit another car and drove off, with the eyewitness saying it was a blue cab. Under the conditions like those on the night of the car accident, the witness correctly identifies the color 80% of the time (and thus failing 20% of the time). The question: what is the probability that the witness is correctly identifying the color of the taxicab in the Taxicab Hit scenario?

Many would say that the probability is 80%, but that turns out to be a mathematical mistake. To understand why I’ll quickly introduce a bit of math, starting with some basic symbolism you might already be familiar with:

• P(A) = the probability that A is true.
• P(A|B) = the probability that A is true given that B is true.

In the case where events B and C are mutually exclusive (i.e. both couldn’t be true) and exhaustive (i.e. one of them had to have happened), the following equation is true due to something called Bayes’ theorem:

P(B|A) =	P(B) × P(A|B)
	P(B) × P(A|B) + P(C) × P(A|C)

Let’s use the following symbols:

• W = the witness identifies the taxi was blue.
• B = the taxi that hit the car is blue.
• G = the taxi that hit the car is green.
• P(G) = the prior probability that the taxi was green (i.e. the probability that the taxi was green prior to our consideration of the eyewitness evidence). Thus, P(G) = 0.85
• P(B) = the prior probability that the taxi was blue (i.e. the probability that the taxi was blue prior to our consideration of the eyewitness evidence). Thus, P(B) = 0.15.
• P(B|W) = the probability that the taxi was blue given that the witness reports that it is blue.
• P(W|G) = the probability that the witness reports the color as blue when the color was green.

Remember: the witness correctly identifies a taxi’s color 80% of the time; regardless of whether the taxi’s actual color is green or blue, the witness gets it right 80% of the time and thus wrong 20% of the time. That means that P(W|B) is 80% (0.8) and P(W|G) is 20% (0.2) So our equation is this due to Bayes’ theorem:

P(B|W) =	P(B) × P(W|B)
	P(B) × P(W|B) + P(G) × P(W|G)

And plugging in our values gives us this:

P(B|W) =	0.15 × 0.8	≈ 0.41
	0.15 × 0.80 + 0.85 × 0.2

Strange but true, even with the evidence of the witness’s testimony the taxicab is probably not blue. If you made this mistake of thinking the taxicab was probably blue in light of the evidence of the testimony, you’re not alone. This taxicab question was famously asked to people by Amos Tversky and Daniel Kahneman. The thing that throws people off is the base rates of the green and blue taxis (85% of the green taxis, and 15% of the blue taxis), which in math terms are the “prior probabilities.”

If you want a more concrete way to look at it, suppose there are 100 taxicabs, 85 of them are green and 15 are blue. Now suppose our eyewitness correctly identifies the color 80% of the time, which means he incorrectly identifies them 20% of the time. Since there are 85 green cabs and he incorrectly identifies them 20% of the time, this means he incorrectly reports 17 green taxicabs as blue (since 85 × 0.20 = 17) even though there are only 15 genuinely blue taxicabs! To make matters worse, of the 15 real blue taxicabs, he correctly identifies only 12 of them (since 15 × 0.80 = 12). So among the taxicabs he reports as blue, 17 of them are not blue and only 12 of them are actually blue. Because so few of the taxicabs are actually blue, an 80% success rate is not enough to overcome the base rate of blue taxicabs.

Probability and Miracles

The same principle holds for miracles; when the prior probability is low enough, even a highly reliable witness might not be enough to overcome the prior probability. To give a concrete example, let us optimistically assume that the prior probability of a particular person miraculously rising from the dead is one in a million. That’s absurdly optimistic when you consider that billions of people die without miraculously rising from the dead, but let’s go with that absurdly high prior probability for now. Now suppose we have good old reliable Pete who is always 99.9% reliable (which means he’s wrong 0.1% of the time). Now suppose we use the following symbols:

• W = Pete reports as an (alleged) eyewitness of the miracle having ocurred.
• M = the miracle occurred.
• ¬M = the miracle did not occur.
• P(M) = the prior probability of the miracle; thus P(M) = 0.000001.
• P(¬M) = the prior probability of the miracle not having occurred; thus P(¬M) = 0.999999.
• P(W|M) = the probability that Pete reports the miracle given that the miracle did occur; thus P(P|M) = 0.999.
• P(W|¬M) = the probability that Pete reports the miracle given that the miracle did not occur; thus P(P|¬M) = 0.001.

Our equation is this:

P(M|W) =	P(M) × P(W|M)
	P(M) × P(W|M) + P(¬M) × P(W|¬M)

And plugging in our values gives us this:

P(M|W) =	0.000001 × 0.999	≈ 0.000998
	0.000001 × 0.999 + 0.999999 × 0.001

So the probability that the miracle actually occurred, even given the absurdly high prior probability, is still only about 0.1%, so we can be about 99.9% sure that the “miracle” is baloney. Of course, the base rate for miraculous resurrections is much, much lower than one in a million. You have my permission to come up with your own miracle scenarios and do some math yourself.

Mutant Humans With Enhanced Abilities

Many biblical creationists argue that there are no beneficial mutations, that mutations are always harmful, never an improvement.

Numerous counterexamples could be given, but the creationist could ask, what about humans? Yes, there is sickle-cell disease that makes people have a higher resistance to a certain disease, but that clearly comes at a cost. Why don’t we see beneficial mutations that don’t have such drawbacks? What about mutations in humans that (for example) make them stronger or give them bones that are harder to break?

Actually, we have examples of both sorts of mutations. There is a family in Connecticut that displays very strong bones due to a mutation, comparing them to the hero in Unbreakable (a movie character who had unbreakable bones).

As for increased strength, there is something called myostatin-related muscle hypertrophy. From this scientific organization:

Myostatin-related muscle hypertrophy is a rare condition characterized by reduced body fat and increased muscle size. Affected individuals have up to twice the usual amount of muscle mass in their bodies. They also tend to have increased muscle strength. Myostatin-related muscle hypertrophy is not known to cause any medical problems, and affected individuals are intellectually normal.

Myostatin affects skeletal muscle but it doesn’t affect cardiac muscle (the heart) so cardiac muscle hypertrophy doesn’t result from this.

Yet another example of a beneficial mutation in humans is a sort of mutation found in women allows them to perceive more colors, at least in the sense of having greater perception of different shades of colors.

I don’t want to give a false impression though; not all religious people deny the existence of beneficial mutations in humans. While doing some research I stumbled upon this Ratio Christi article that concedes their existence.

Outsourcing Child Abuse

Imagine if parents told their children this:

If you always agree with our views, you will live a very pleasant life. If you disagree with our views however, we will torture you for as long as you exist.

This would seem like some form of child abuse, right? Then if we are to be consistent, should we not say the same thing about religious parents who tell their child the same thing but outsource the reward/torturing job to a deity?

Then of course there’s the problems with the existence of a supposedly morally good deity behaving in a manner akin to child abuse. Should God behave this way towards children—or for that matter people of any age? Would a good deity torture people forever simply because they weren’t convinced he was real?

I should add as a caveat that these problems don’t apply to all religions or religious families, but for those it does apply, it does seem to be a criticism worth considering.

Radiometric Dating and the Bill Nye Debate

In the Bill Nye versus young earth creationist Ken Ham debate, Ken gives this response to radiometric dating:

Unfortunately, Bill Nye did not have a satisfactory response to this. Is there one? Yep. Notice from the above video it is apparent that a creationist took the samples and then sent them off to the laboratory for dating (Potassium-Argon for the basalt and radiocarbon for the wood). There is a method of radiometric dating called isochron dating that has a built-in check for most forms of contamination. However, Chris Stassen of TalkOrigins notes that

some dating methods (e.g., K-Ar and carbon-14) do not have a built-in check for contamination, and if there has been contamination these methods will produce a meaningless age. For this reason, the results of such dating methods are not treated with as much confidence.

Bingo: a creationist sent in some stuff for dating when the dating method in question had no built-in check for contamination! Potassium-Argon and radiocarbon dating is not necessarily useless, but one does have to be careful about not sending in contaminated samples, and I am willing to bet the creationists who took the samples did not do that.