#468

April 02, 2016

Deductive Arguments and Probability

Hello, Dr. Craig.

You have often said that a deductive argument is good if it meets two conditions: It is valid, and each premise is more probable than it's denial. Furthermore, in a recent newsletter, you said, "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."

But why would the probability of a premise establish minimal probability of a conclusion? Shouldn't it establish maximal probability?

Imagine we have a valid, three-premise argument, and imagine the first premise is 75% probable. Even if the other two premises are 100% probable, the probability that all three premises are true at the same time would only be 75%. I.e., 0.75 x 1.0 x 1.0 = 0.75.

Now imagine that all three premises are 75% probable. To keep things simple, let's assume that the probabilities of the premises are independent of each other--that the truth value of one premise doesn't influence the truth value of another premise. The probability that all three premises are true at the same time would be 0.75 x 0.75 x 0.75 = ~0.42. So in this case, you have three premises that are much more probable than their denial, but they are probably not all true.

So I don't get it: Why do you say that the probability of a premise establishes the minimal probability of a conclusion rather the maximal probability of that conclusion? And why couldn't a valid deductive argument based on probable premises still be a bad argument? I mean, if it's probably not the case that all of the premises are true at the same time, then it's a bad argument, right?

Dale


United States

This issue came up in my dialogue with Kevin Scharp at Ohio State University, and as there has been some misunderstanding about this on the Internet, I’m grateful for your question, Dale.

The fundamental question concerns the relationship between the probability of the premises of a deductive argument and the probability of the argument’s conclusion. Some people have mistakenly thought that the probability of the conjunction of the premises = the probability of the conclusion. So they have inferred that in order for the conclusion of a deductive argument to be, say, 51%, the probability of the conjunction of the premises must be 51%.

That is mistaken. You can’t calculate the probability of the conclusion of a deductive argument by figuring the cumulative probability of the argument’s premises. As Prof. Timothy McGrew of the University of Western Michigan pointed out to me, the theorem on the accumulation of uncertainty places only a lower bound on the probability of the conclusion.[1] So the probability of the conjunction of the argument’s premises establishes merely a minimum probability of the conclusion. The probability of the argument’s conclusion can’t be any lower than the probability of the conjunction of the argument’s premises. So to use your example, if the probability of a deductive argument’s three premises is ~0.42, then you’re guaranteed that the probability of the argument’s conclusion is no less than ~0.42. It could be much greater, but it can’t fall below that lower bound.

In my dialogue with Dr. Scharp, I pointed out that the probability of the conclusions of my deductive theistic arguments should not be equated with the probability of the arguments’ respective premises. The probability of the arguments’ conclusions is at least as great as the probability of each argument’s premises, and you’ll notice that Dr. Scharp didn’t dispute the point. His complaint was, rather, that a conclusion which is only 51% probable is insufficient to support belief in that conclusion. I then pointed out that I never claimed, nor do I think, that the conclusions of my theistic arguments are only 51% probable. My claim was that they are at least 51% probable and therefore qualify as good arguments.

Now that raises the further question of what qualifies as a “good” deductive argument. I take it that a good argument is one whose conclusion is shown to be more plausible than not. So under what conditions is an argument good? As you note, I have long said that in order for a valid deductive argument to be a good one, it suffices that each individual premise of the argument be more probable (or plausible) than its contradictory. It seemed to me that if you think that each premise of the argument is true and the premises imply the conclusion, then you ought to think that the conclusion is true!

But as a result of reading Timothy McGrew and John Depoe’s interesting article, “Natural Theology and the Uses of Argument,” Philosophia Christi 15/2 (2013) 299-309, as well as personal correspondence with McGrew last summer, I came to see that my condition for a good deductive argument was inadequate. My changing attitude was first signaled in my Defenders III lectures on natural theology last August [http://www.reasonablefaith.org/defenders-3-podcast/transcript/excursus-on-natural-theology-part-1]. McGrew helped me to see that there can be cases in which each individual premise is more probable than not and yet it would be irrational to believe the conjunction of the premises. He explained,

The problem here is one of closure – specifically, closure under conjunction. There is a literature on this, and one of the key papers in that literature is Henry Kyburg, “Conjunctivitis,” in M. Swain, ed., Induction, Acceptance, and Rational Belief (1970). I take it that the chief lesson of that literature is that there are cases where it is rational to believe P and rational to believe Q without its being rational to believe the conjunction (P&Q). Lotteries provide very intuitive examples of this, since in a simple fair finite lottery with exactly one winner to be drawn, it is reasonable to believe that ticket 1 is a loser, reasonable to believe that ticket 2 is a loser, ... all the way up to the last number. But obviously it is unreasonable to believe the conjunction of these statements; the conjunction would contradict the very terms of the lottery since (taken with our background knowledge) it would entail that there is no winner.[2]

So while a deductive argument may well be a good one if it meets the condition I laid down, that’s not enough to guarantee that it is a good one. In order to guarantee that the conclusion is more probable than not, the conjunction of the premises must be more probable than not. McGrew proceeds,

The only way to guarantee that a conclusion deductively drawn from a set of premises is more plausible than not is to use premises the conjunction of which is more plausible than not. This statement is a little tricky, since there is a possible misunderstanding. What we are looking for is a general principle that stipulates conditions under which, for any logical consequence C of the set of premises, C is more plausible than not. Particular consequences will vary in their plausibility. But the objective is to state a condition under which this feature is guaranteed, no matter which consequence one draws. And satisfaction of that condition is necessary to do the job.

This claim can be proved in a fairly direct manner. Suppose, for example, that we are dealing with the following set of premises:

{P, Q, R}

And suppose that the big conjunction ((P & Q) & R) fails to be more plausible than not. Then there is at least one logical consequence of the set that fails to be more plausible than not – namely, that very conjunction. So it is a necessary condition for the argument to “preserve plausibility” (I am coining this phrase to mean “guarantee, from information about the plausibility of the premises alone, that any conclusion drawn from those premises by deductive inference is itself more plausible than not”) that the conjunction of the premises be more plausible than not.

So in order guarantee that the argument’s conclusion be at least 51% probable, the conjunction of its premises must be 51% probable. Recall that the probability of the conclusion is not equal to the probability of the conjunction of the premises, but it is guaranteed to be at least that probable.

So in my dialogue with Kevin Scharp I emphasized that my theistic arguments meet even the condition stated by McGrew and so qualify as good arguments. Indeed, in some cases, the premises are so few and one of them so certain that the lower bound of the argument’s conclusion just is the probability of the argument’s key premise.[3] So I think that the conjunctions of the premises of my respective theistic arguments are in each case more probable than not, thereby guaranteeing that the conclusions are more probable than not, so that the arguments are good ones.

A final important point raised by McGrew and Depoe which I neglected to mention in my dialogue with Scharp is that even if the conclusion of an argument is not at least 51%, that doesn’t imply that the argument is no good. For even if various arguments establish their conclusions to have lower bounds less than 50%, the cumulative force of the arguments may well raise the lower bound to greater than 50%. Given the plethora of theistic arguments mentioned by Scharp that I have defended the cumulative case for theism may be very strong even if each individual argument is not esteemed to be “good.”

Notes

[1] See also Timothy McGrew and John Depoe, “Natural Theology and the Uses of Argument,” Philosophia Christi 15/2 (2013): 3012-305. They further reference Ernest W. Adams, A Primer of Probability Logic (Stanford: CSLI, 1998), pp. 31-34.

[2] Timothy McGrew to William Lane Craig, August, 2015.

[3] For example, in the kalam cosmological argument, the causal premise is so nearly certain that the probability of the conjunction of the two premises is effectively the probability of the premise “The universe began to exist.” The case of the argument from fine-tuning is similar.