**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:

- If drinking the potion would cure me, then I should drink it.
- Drinking the potion would cure me.
- 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:

- The first trial picked a number less than 13.
- The second trial picked a number less than 13.
- The third trial picked a number less than 13.
- 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:

- The six-sided fair die I rolled is four or less.
- The six-sided fair die I rolled is five or less.
- 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:

- If God does not exist, then objective moral values and duties do not exist.
- Objective moral values and duties do exist.
- 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 providesAnd 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.norational support for its conclusion, but we should not make the mistake of thinking that’s theonlyway an argument can fail to be convincing. Another way is if the argument provided nonzero but nonethelesstoo littlesupport for its conclusion. So we can accept thatceteris paribusa 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.

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