I admit it.  I enjoyed pointing out that William Lane Craig had made a major blunder in his recent discussion of the logic of deductive arguments (with premises that are probable rather than certain).

However, there are a variety of natural tendencies that people have to reason poorly and illogically when it comes to reasoning about evidence and probability.  The fact that a sharp philosopher who is very experienced in presenting and analyzing arguments could make such a goof just goes to show that it is easy for people to make logical mistakes and to reason illogically, especially when reasoning about evidence and probability.

So, I think it is worth taking a little time to carefully review Craig’s mistake in order to LEARN from his mistake, and to understand how the logic really works in this case, so that we can avoid making the same mistake ourselves, and so that we can more readily notice and identify when others make similar mistakes in their reasoning.

In the March Reasonable Faith Newsletter Craig asserted a FALSE principle about valid deductive arguments that have premises that are probable:

… 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.

One way that this principle can fail is because of the fact that a valid deductive argument can have multiple premises and because standard valid forms of deductive inferences/arguments require that ALL premises be true in order to work, in order to logically imply the conclusion.  In the case of a valid deductive argument with multiple premises that are probable rather than certain, it is usually the case that ALL of the premises must be true in order for the argument to logically imply the conclusion.

If the probable premises of such an argument are independent from each other (so that the truth or falsehood of one premise has no impact on the probability of the truth or falsehood of other premises in the argument), then the simple multiplication rule of probability applies, because what matters in this case is that the CONJUNCTION of all of the probable premises is true, and the probability of the conjunction of the premises of such an argument is equal to the product of the individual probabilities of each of the probable premises:

P

Q

THEREFORE:

P and Q

If the probability of P is .5, and the probability of Q (given that P is the case) is .5, then the probability of the conjunction “P and Q” is .25..  Here is an example of such a valid deductive argument:

1. I will get heads on the first random toss of this fair coin.

2. I will get heads on the second random toss of this fair coin. 

THEREFORE:

3. I will get heads on the first random toss of this fair coin, and I will get heads on the second random toss of this fair coin.

The probability of (1) is .5, and the probability of (2) given that (1) is the case is also .5 (because these two events are independent–what comes up on the first toss has no impact on what comes up on the second toss), so the probability of the conjunction of (1) and (2) is .25.  Thus, the probability of (3) is .25.  This example shows that the probability conferred on the conclusion of such an argument can be LESS THAN the probability of any individual premise of the argument.  This is because when you multiply one number that is greater than zero but less than 1.0 by another number that is greater than zero but less than 1.0, the product is LESS THAN either of those factors.

The same mathematical relationship holds when the probability of the individual probable premises is greater than .5:

4. I will not roll a six on the first random roll of this fair die.

5. I will not roll a six on the second random roll of this fair die.

THEREFORE:

6. I will not roll a six on the first random roll of this fair die, and I will not roll a six on the second random roll of this fair die.

The probability of (4) is 5/6 or about .83, and the probability of (5) given that (4) is the case is also 5/6 or about .83 (because these events are independent).  Since both premises have to be true in order to logically imply the conclusion, the multiplication rule applies in this case, so the probability of the CONJUNCTION of (4) and (5) is equal to the product of the probabilities of each individual premise:  .83 x .83 = .6889  or about .69, which is LESS THAN the probability of each of the individual premises.

Based on this sort of mathematical relationship, we can devise an example on which Craig’s principle will FAIL:

7. I will not roll a six or a five on the first random roll of this fair die.

8. I will not roll a six or a five on the second random roll of this fair die.

THEREFORE:

9. I will not roll a six or a five on the first random roll of this fair die, and I will not roll a six or a five on the second random roll of this fair die.

The probability of (7) is 4/6 or about .67, and the probability of (8) given that (7) is the case is also 4/6 or about .67 (because these are independent events).  The probability of the conjunction of (7) and (8) is equal to the product of their individual probabilities: .67 x .67 = .4489 or about .45.  To be more exact the probability of the conjunction of (7) and (8) is equal to: 4/6  x 4/6 = 16/36 = 4/9 = .44444444…  Thus, although the probability of each premise is greater than .51, the probability of the conclusion (9) is less than .51.  Therefore, Craig’s principle FAILS in this case.  Thus, his principle is FALSE.

Here is one more similar counterexample against Craig’s principle:

10. I will not select a heart card on the first randomly selected card from this standard deck.

11. I will not select a heart card on the second randomly selected card from this standard deck (after replacement of the first card back into the deck).

12. I will not select a heart card on the third randomly selected card from this standard deck (after replacement of the first and second cards back into the deck).

THEREFORE:

13. I will not select a heart card on the first randomly selected card from this standard deck, and I will not select a heart card on the second randomly selected card from this standard deck (after replacement of the first card back into the deck), and I will not select a heart card on the third randomly selected card from this standard deck (after replacement of the first and second cards back into the deck).

The probability of (10) is .75, and the probability of (11) given (10) is .75, and the probability of (12) given both (10) and (11) is also .75.  The probability of the conjunction of these three premises equals:  .75 x .75 x .75 = .421875 or about .42. Thus, the probability of the conclusion (13) is .421875 or about .42, which is LESS THAN .51, even though each of the premises has a probability that is GREATER THAN .51.

Here is my final counterexample based on the multiplication rule:

14. I will not roll a six on the first random roll of this fair die.

15. I will not roll a six on the second random roll of this fair die. 

16. I will not roll a six on the third random roll of this fair die.

17. I will not roll a six on the fourth random roll of this fair die.

THEREFORE:

18. I will not roll a six on the first random roll of this fair die, and I will not roll a six on the second random roll of this fair die, and I will not roll a six on the third random roll of this fair die, and I will not roll a six on the fourth random roll of this fair die.

Each of the premises in this argument has a probability of 5/6 or about .83.  The events referenced in the premises are independent from each other, so the probability of the conjunction of premises (14), (15), (16), and (17) is equal to:  

5/6  x  5/6  x  5/6  x  5/6 =  625/1,296 = .4822530864…  or about .48.  So, the probability of each premise is greater than .51, but the probability of the conclusion (18) is less than .51, so Craig’s principle FAILS in this case, and thus Craig’s principle is shown to be FALSE.

There is another way that Craig’s principle can FAIL, and that is because one probable premise in a valid deductive argument can have a dependency on another probable premise in the argument, and this can result in conferring a probability on the conclusion that is less than the probability of the individual premises.  I will explore this second issue with Craig’s principle in the next installment.