In the March Reasonable Faith Newsletter William Craig asserted this 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.

There are a variety of natural tendencies that people have to reason poorly and illogically when it comes to reasoning about evidence and probability.  So, 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.

One way that Craig’s 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.  This means that the premises of a valid deductive argument can each have probabilities of .51 or greater while the conclusion has a probability of LESS THAN .51.  Examples of such arguments were given in Part 2 of this series of posts.

Another way that Craig’s principle can FAIL is based on situations where one or more premises of a valid deductive argument have dependencies with other premises in the argument.

Here is an example of a valid deductive argument with a premise that has a dependency on another premise :

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

2. I will get tails on the first 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 tails on the first random toss of this fair coin.

The probability of (1) is .5, and the probability of (2) is also .5 (considered on its own).  However, these two premises are mutually exclusive.  If (1) is true, then (2) must be false, and if (2) is true, then (1) must be false.  Thus, the conclusion (3) asserts a logical contradiction, and thus the probability that (3) is true is 0.   In the case of this argument, we cannot simply multiply the probability of (1) , considered by itself, times the probability of (2), considered by itself, in order to determine the probability of the CONJUNCTION of (1) and (2).

We have to multiply the probability of (1) times the probability of (2) GIVEN THAT (1) is the case.   Because the truth or falsehood of (1) impacts the probability of the truth or falsehood of (2), we cannot use the simple multiplication rule with this argument.  We must use the general multiplication rule:

The probability of the conjunction of A and B is equal to the product of the probability of A and the probability of B given that A is the case.

Here is the mathematical formula for the general multiplication rule of probability:

P(A & B) =  P(A) x P(B|A)

NOTE: The general multiplication rule can be used whether or not there is a dependency relationship between the premises of an argument.  If there is no dependency relationship between A and B, then the probability of B given that A is the case will be the SAME as the probability of B considered by itself.

Since the truth of (1) clearly excludes the possibility of the truth of (2), the probability of (2) GIVEN THAT (1) is the case is 0.  The probability of the conjunction of (1) and (2) is thus equal to:  .5   x  0  =  0.  So, the probability of the conclusion (3) is 0, even though the probability of (1) is .5.

This demonstrates how the probability of the conclusion of a valid deductive argument can be LESS THAN the probability of a premise in the argument (considered by itself).  The main reason why the probability of (3) is 0 is that there is a logical incompatability between premise (1) and premise (2) which rules out the possibility of it being the case that BOTH premises are true.  The truth or falsehood of (1) has an impact on the probability of the truth or falsehood of (2), so there is a dependency between the truth or falsehood of these premises.

Considered by itself, premise (2) has a probability of .5, but for the argument to work, both premises have to be true, and the probability of (2) can be impacted by whether (1) is true or false, so we need to assess the probablity of (2) on the assumption that (1) is true, and when we do so, the probability of (2) in that scenario is reduced from .5 down to 0.  Therefore, it is this dependency relationship between (2) and (1) that results in the conclusion having a probability that is extremely low, as low as probabilities can get: 0.

The same mathematical relationship holds when the probability of an 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 roll a six on the first 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 roll a six on the first random roll of this fair die.

The probability of (4) considered by itself is 5/6 or about .83, and the probability of (5) considered by itself is 1/6 or about .17.  However, these two premises are mutually exclusive. If (4) is true, then (5) must be false, and if (5) is true, then (4) must be false. Thus, the conclusion (6) asserts a logical contradiction, and thus the probability that (6) is true is 0. In the case of this argument, we cannot simply multiply the probability of (4) considered by itself, times the probability of (5) considered by itself, in order to determine the probability of the CONJUNCTION of (4) and (5).

Because there is a dependency relationship between (4) and (5), we must use the general multiplication rule to determine the probability of the conclusion.  The probability of the conjunction of (4) and (5) is equal to the product of the probability of (4) and the probability of (5) given that (4) is the case:

P[(4) & (5)] =  P[(4)]  x  P[(5)|(4)]

=  5/6  x   0 =  0

Thus, because of the dependency relationship between (4) and (5), the probability of the conclusion is reduced to 0, even though the probability of premise (4) considered by itself is 5/6 or about .83, which is GREATER THAN .51.  This argument is therefore another counterexample to Craig’s principle.  It is a valid deductive argument which has a probable premise with a probability GREATER THAN .51 but where the probability of the conclusion is LESS THAN .51.

The dependency relationship between premises need not be as strong as in the above examples. So long as the truth or falsehood of one premise impacts the probability of some other premise in the argument, Craig’s principle about valid deductive arguments can  FAIL.

Here is a counterexample against Craig’s principle that involves a dependency relationship that is weaker than in the above examples (something less than being mutually exclusive):

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

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

THEREFORE:

12. I will not select a heart card on the first randomly selected card from this standard deck, and I will not select a diamond card on the first randomly selected card from this standard deck.

The probability of (10) considered by itself is .75, and the probability of (11) considered by itself is .75.  However, there are dependency relationships between these premises which make the conjunction of the premises less probable than if we simply multiplied these probabilities of each premise considered by itself.

If we ignored the dependency then the probability of the conjunction of the three premises would be calculated this way: .75  x  .75  = .5625 or about .56.  But to properly determine the probability of the conjunction of the three premises, we need to use the following equation (based on the general multiplication rule):

P[(10) & (11)] =  P[(10)]  x  P[(11)|(10)]  

=  3/4   x   2/3    =   6/12  =  1/2  =  .50

Thus, the probability of the conclusion of this argument is .50, which is LESS THAN .51.

The probability of premise (10) considered by itself is 3/4 or .75, and the probability of (11) is 3/4 considered by itself, which is GREATER THAN .51, and the probability of (11) given that (10) is the case is 2/3 or about .67, which is still GREATER THAN .51, but the probability of the conclusion of this argument is LESS THAN .51, so this argument is a counterexample to Craig’s principle, and part of the reason why the probability of the conclusion is so low is that there is a depenedency relationship between the premises.

Here is a final counterexample based (in part) on there being a dependency between premises:

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

15. I will not roll a five on the first random roll of this fair die. 

16. I will not roll a four on the first random roll of this fair die.

THEREFORE:

17. I will not roll a six on the first random roll of this fair die, and I will not roll a five on the first random roll of this fair die, and I will not roll a four on the first random roll of this fair die.

Each of the premises in this argument has a probability of 5/6 or about .83 when considered by itself.  If we ignored the dependency relationship between these premises, then we would calculate the probability of the conjunction of premises (14), (15), and (16) simply by multiplying these probabilities:  5/6  x  5/6  x  5/6   =  125/216   which approximately equals .5787 or about .58.  However, because there are dependencies between these premises, we must use the general multiplication rule.  Here is a formula for this argument that is based on the general multiplication rule:

P[(14) & (15) & (16)] =  

P[(14)]  x  P[(15)|(14)]  x  P[(16)|[(14) & (15)]]  

= 5/6  x  4/5  x  3/4  =   60/120  =  1/2  =  .50

Thus, the probability of the conclusion (17) is 1/2 or .50 which is LESS THAN .51.

So, the probability of each premise (considered by itself) is greater than .51, and the probability of premise (16) given that all the other premises are true is 3/4 or  .75, which is still greater than .51, but the probability of the conclusion (17) is LESS THAN .51, so Craig’s principle FAILS in this case, and thus Craig’s principle is shown to be FALSE.