This is a Q & A blog post by our Visiting Scholar in Philosophy, William Lane Craig.
Dear Dr. Craig,
I have a question about how (or if) multiplied probabilities are to be applied to the premises of an argument.
For example, for the Kalam Cosmological Argument, let's say that I find the first premise to be “fairly convincing.” We might even say I find it 60% subjectively convincing. Same for the second premise. In finding both premises 60% convincing am I logically required to find the conclusion 60% true?
It seems to me that if the first premise has a 60% chance of being true, and the second premise has a 60% chance of being true, then it would follow that for both premises to be true would only be about 36% due to multiplied probabilities, and I would be logically required to reject the conclusion.
This would be sad, because it would mean that (1) for the Kalam, we would need for each premise to be at least 70% true, and (2) that for arguments in general, the more premises an argument had, the less likely it would be true in general.
I'm sure this is not the way arguments work, but I can't see where I've gone wrong, and this seems to undercut the power of all these wonderful arguments we have for God's existence.
William Lane Craig's Response
You’re right, Charles, that that is not how deductive arguments work! In a deductive argument (one in which the conclusion is deduced from the premisses using the logical rules of inference), you can’t correctly compute the probability of the conclusion by multiplying the probabilities of the premisses together. Computing the probability of the conjunction of the premisses only sets a lower limit on the probability of the conclusion. So in your example, in which each premise has a probability of 60%, the probability of the conclusion of the argument can be no lower than 36%, but could be much higher. So the sad result you mention does not follow. To repeat: you can’t determine the probability of the conclusion of a deductive argument by computing the probability of the conjunction of its premisses. The lesson, I think, is that we should focus on whether the premisses of the argument are true, because if they are, then necessarily, the conclusion is true.
In any case, in the deductive arguments that I defend for God’s existence, I don’t think that this is an issue. For I think that the premises are sufficiently probable to guarantee that the conclusion is more probable than not. If the probability of the conjunction of the premisses of a valid deductive argument is >50%, then the argument’s conclusion is guaranteed to be more probable than not, since 50% is just the lower bound. Take, for example, the kalām cosmological argument:
1. Whatever begins to exist has a cause.
2. The universe began to exist.
3. Therefore, the universe has a cause.
Despite the skeptics, I think that (1) is virtually certain, having therefore a probability of around 100%. So the lower bound on the probability of (3) is determined entirely by the probability of (2), which I’d say is way more than 50%. So whatever the exact probability of (3) might be, it must be way more than 50%.
Notice, too, that if you have multiple, independent arguments for God, then even if the conclusions of those arguments have a probability <50% respectively, still the cumulative probability of all of them together may be much higher than 50%. Just as in a trial in a court of law no single piece of evidence may be sufficient to convict the accused, but the cumulative force of all the evidence taken together makes the guilt of the accused beyond reasonable doubt, so also multiple, independent arguments for God’s existence can make God’s existence much more probable than not, even if each argument considered individually is not sufficient to prove God’s existence.
- William Lane Craig