I had planned to discuss counterexamples (to Craig’s principle) that were based on dependencies existing between the premises in some valid deductive arguments.  But I am putting that off for a later post, in order to present a brief analysis of some key concepts.

It seems to me that an important part of understanding the relationship between valid deductive arguments and probability is keeping in mind the distincition between necessary conditions and sufficient conditions. So, I’m going to do a brief analysis of this relationship.

SUFFICIENT CONDITIONS ESTABLISH A MINIMUM PROBABILITY

1. IF P, THEN Q.

Claim (1) asserts that P is a SUFFICIENT CONDITION for Q.

Assuming that (1) is true, the probability of P establishes a MINIMUM probability for Q.

If the probability of P was .60, then assuming that (1) is true, the minimum probability for Q would also be .60, because whenever P is true, so is Q.

However, (1) is compatible with Q being true even if P is false. There could be some OTHER reason for Q being true:

2. IF R, THEN Q.

If (2) is also true, and if R has some chance of being true even when P is false, then the probability of Q would be GREATER THAN the probability of P.  In this scenario the probability of Q would be GREATER THAN .60.

Suppose that the truth of R is independent of the truth of P. Suppose that the probability of R is .80. We can divide this scenario into two cases:

Case I. P is true.

Case II. It is not the case that P is true.

There is a probability of .60 that case I applies, and if it does apply, then Q is true. This gives us a minimum baseline probability of .60 for Q.

But we must add to this probability any additional probability for Q being true from case II.

There is a probability of .40 that case two applies, and if it does apply then there is a .80 probability that R is true (since the probability of R is not impacted by the truth or falsehood of P).  Since R implies Q, there is (in this second case) a probability of at least .80 that Q is true. So, we multiply the probability that case II applies times the probability of Q given that case II applies to get the (minimal) additional probability: .40 x .80 = .32.

To get the overall minimal probability of Q, we add the probability of Q from case I to the (minimal) probability of Q from case II: .60 + .32 = .92 or about .9.

NOTE: The actual probability of Q might be higher than .92, if there is some chance that Q was true even if both P and R were false.

NECESSARY CONDITIONS ESTABLISH A MAXIMUM PROBABILITY

3. IF Q, THEN P.

Claim (3) asserts that P is a NECESSARY CONDITION for Q.

Assuming that (3) is true, the probability of P establishes a MAXIMUM probability for Q.

If the probability of P is .60, then assuming that (3) is true, the maximum probability of Q would be .60, because whenever P is false, Q must also be false.

However, (3) is compatible with Q being false even when P is true. There could be some OTHER reason why Q is false:

4. IF Q, THEN S.

If (4) is also true, and if S has some chance of being false even when P is true, then the probability of Q would be LESS THAN the probability of P. In this scenario, the probability of Q would be LESS THAN .60.

Suppose that the truth of S is independent of the truth of P. Suppose that the probability of S is .20.  We can immediatly infer that the maximum probability of Q is .20, because the truth of S is a necessary condition for Q.  However, the combination of (3) and (4) reduces the maximum probability of Q even further.

We can divide this scenario into two cases:

Case I. P is true.

Case II. It is not the case that P is true.

Let’s consider case II first.  There is a probability of .40 that case II applies (because there is a probability of .60 that case I applies and the combined probabilities of both cases = 1.0), and if it does apply, then Q would be false (because P is a necessary condition of Q).  This establishes a baseline minimum probability of .40 for the falsehood of Q.

But we must add to this probability any additional probability for Q being false from case I.

There is a probability of .60 that case I applies, and if it does apply, then there is a .20 probability that S is true (because the probability of S is not impacted by the truth or falsehood of P), thus if case I applies, then there is a probability of .80 that S is false, and thus a minimum probability of .80 that Q is false (because S is a necessary condition of Q).  We meed to multiply the probability that case I applies times the (minimal) probability that Q is false given that case I applies:   .60 x .80 = .48.

Now we must add the probability of the falsehood of Q from case II with the (minimum) probability of the falsehood of Q from case I to get the overall minimum probablilty of the falsehood of Q:  .40 + .48 = .88.  The overall minimum probability of the falsehood of Q is .88, and this implies that the overall MAXIMUM probability of Q is .12.

NOTE: The actual probability of Q could be lower than the maximum probability, if there is some chance that Q was false even if both P and S were true.