The March Newsletter from Reasonable Faith just came out, and it includes a brief lesson in logic from William Lane Craig. However, the lesson presents a point that is clearly and obviously WRONG, and it promotes bad reasoning that could be used to rationalize UNREASONABLE beliefs.  It appears that WLC is himself in need of some basic lessons in logic.

William Craig recently debated a professor of philosophy named Kevin Scharp at Ohio State University, and in the current Reasonable Faith Newsletter, Craig criticizes what he takes to be Scharp’s main objection to Craig’s apologetic arguments:

What was odd about Prof. Sharp’s [correct spelling: Scharp] fundamental critique was that, apart from the moral argument, he did not attack any of the premises of my arguments. Rather his claim was that all the arguments suffer from what he called “weakness.” For even if the arguments are cogent, he says, they only establish that God’s existence is more probable than not (say, 51% probable), and this is not enough for belief in God. 

Why did he think that the arguments are so weak? Because I claim that in order for a deductive argument to be a good one, it must be logically valid and its premises must be more probable than their opposites. Prof. Sharp [sic] apparently thought that that is all I’m claiming for my arguments. But in our dialogue, I explained to him that that was a mistake on his part. My criteria were meant to set only a minimum threshold for an argument to be a good one. I myself think that my arguments far exceed this minimum threshold and provide adequate warrant for belief in God. I set the minimum threshold so low in order to help sceptics like him get into the Kingdom! 

This reply makes a fair point.  Establishing a minimum threshold for an argument to be considered “good” does not imply that no good arguments have premises that exceed this minimum.  Thus, when Craig claims that his deductive arguments for God’s existence are “good” arguments, he is NOT saying that the premises in these arguments each have a probability of only .51.

But then Craig goes further and provides this short lesson in logic (or lesson in illogic, as I shall argue):

Besides, I pointed out, in a deductive argument the probability of the premises establishes only a minimum probability of the conclusion: even if the premises are only 51% probable, that doesn’t imply that the conclusion is only 51% probable. It implies that the conclusion is at least 51% probable. Besides all this, why can’t a person believe something based on 51% probability? The claim that he can’t seems to me just a matter of personal psychology, which varies from person to person and circumstance to circumstance.

Thus, Prof. Sharp’s [sic] fundamental criticism was quite misconceived, and since he never attacked the arguments themselves, he did nothing to show that the arguments I defended are, in fact, weak.

Craig’s claim that “even if the premises [in a deductive argument] are only 51% probable” this “implies that the concusion is at least 51% probable” is clearly and obviously false.  This is, for me, a jaw-dropping mistaken understanding of how deductive arguments work.

First of all, deductive arguments can have multiple premises.  If multiple premises in a deductive argument each have a probability of only .51, then it is OBVIOUSLY possible for such arguments to FAIL to establish that the conclusion has a probability of “at least” .51.  For example, consider the following valid deductive argument form:

1. P

2. Q

3. IF P & Q, THEN R

THERFORE:

4. R

Suppose that the probability of P is .51 and that the probability of Q (given that P is the case) is also .51.  Suppose that we know premise (3) with certainty.  What is the probability conferred on the conclusion by this argument?   In order for this deductive argument to confer any probability to the conclusion, BOTH P and Q must be true.  Thus it only takes ONE false premise to ruin the argument.  The probability of the conclusion would NOT be .51 but would, rather, be .51 x .51 = .2601  or about .26.   This is a simple and obvious counter-example to Craig’s claim.

Another problem is that there is almost always other relevant information that could impact the probability of the conclusion of an argument.  So, one might well be able to construct additional relevant deductive arguments AGAINST the conclusion in question.

Suppose that X implies that R is not the case, and Y implies that R is not the case, and Z implies that R is not the case.  Then we could construct three additional deductive arguments against R:

5. X

6. IF X, THEN it is not the case that R.

THEREFORE:

7. It is not the case that R.

===============

8. Y

9. IF Y, THEN it is not the case that R.

THEREFORE:

7. It is not the case that R.

===============

10. Z

11. IF Z, THEN it is not the case that R.

THEREFORE:

7. It is not the case that R.

Suppose that the probability of X is .9, and the probability of Y is  .9, and the probability of Z  is .9.   Suppose that the truth of X, Y, and Z are independent of each other.  Suppose that the conditional premises in each of the above arguments is known with certainty.  In this case, what probability is conferred on the conclusion that “It is not the case that R”?

Let’s (temporarily) ignore the prevous deductive argument in support of R, and imagine that X, Y, and Z are the only relevant facts that we have regarding the truth or falsehood of R.  Each of these three valid deductive arguments would, then, individually confer a probability of .9 on the conclusion that “It is not the case that R”.  Therefore, if we combine the force of these three arguments, they will confer a probabilty that is GREATER THAN .9 on the conclusion that “It is not the case that R”.  All we need is for ONE of the premises (X, Y, or Z) to be true, in order for the negative conclusion to be secured, and each of the three premises is very likely to be true.

We can analyze the probabilty calculation into three cases in which at least one of the three premises is true:

I. X is true  (probability = .9)

II. X is not true, but Y is true  (probability = .1 x .9 =  .09)

III. X is not true, and Y is not true, but Z is true (probability = .1 x .1 x .9 = .009)

Add the probabilities of these three cases together to get the total probability conferred on the negative conclusion:

.9 + .09 + .009 = .999

Thus, the combined force of these three deductive arguments would make it nearly certain that “It is not the case that R”, assuming that these three arguments encompassed ALL of the relevant evidence.

But we also have the posititive evidence of P and Q to consider, which will, presumably increase the probability that R is the case and reduce the probability of the negative conclusion that “It is not the case that R”.

Adding in this additional relevant evidence, however, could make the overall probability calculation significantly more complex.  It all depends on whether the truth of P is independent of the truth of X, Y, and Z, and whether the truth of Q is independent of the truth of X, Y, and Z, and whether the truth of the conjunction “P and Q” is independent of the truth of X, Y, and Z.  If there are dependencies between the truth of these claims, then that will rquire additional complexity in the probability calculation.

If for the sake of simplicity, we assume that the truth of P is independent of the truth of X, Y, and Z, and the truth of Q is independent of X, Y, and Z, and the truth of “P and Q” is independent as well, we can at least conclude (without needing to do any calculations) that the overall probability of R will be greater than .001 and less than .2601, in which case Craig’s claim that the probability of the conclusion must be “at least 51%” is clearly false in this case, in part because of additional relevant evidence against the conclusion.

Thus, there are two major, and fairly obvious, problems with WLC’s claim: (1) deductive arguments with multiple premises can confer a probability on the conclusion that is LESS than the probability of any particular premise in the argument, and (2) there is almost always OTHER relevant information/data that impacts the probability of the conclusion of a particular deductive argument (which has premises that are only probable), and consideration of this additional evidence might very well lower the all-things-considered probability of the conclusion.

These two points are fundamental to understanding the logic of deductive arguments for the existence of God, so Craig’s apparent confusion about, or ignorance of, these points is shocking.