Modality and Probability

On p. 51 of Where the Conflict Really Lies, Plantinga says: “[T]he probability of a contingent proposition on a necessary falsehood is 1.”

So, where C = a contingent proposition and F = a necessary falsehood, Plantinga is saying P(C|F)=1.

This seems false. How are we getting our values? If we understand conditional probability to be defined as:

and we grant that a necessary falsehood has the probability of 0, then P(C|F) ≠ 1, rather, it is undefined. Shoenberg, “If P(A) = 0, then P(B|A) is undefined, just as division by zero is undefined in arithmetic. This makes sense, since if event A never happens, then it does not make much sense to discuss the frequency with which event B happens given that A also happens” (Introduction to Probability with Texas Hold’em Examples, Chapman and Hall, 2011, p.40). After some further investigation, I noticed that Tyler Wunder gives a similar objection here.

However, the above is too quick.

One might take conditional probabilities as primitive so as to avoid divide by zero problems. Such is contentious, but leave that aside. To understand what Plantinga is saying here let’s quote the broader context in which the above quote appears. In responding to Paul Draper’s argument from evil, Plantinga notes that Draper claims that, where E = evolution is true, and N = the thesis of naturalism, and T = the thesis of theism, and ‘>+’ = much greater than, then

1. P(E|N) >+ P(E|T).

To which Plantinga responded:

Suppose…that theism is noncontingent: necessarily true or necessarily false. If so, 1 doesn’t imply that naturalism is more likely than theism; instead 1 obviously entails that theism is true. For if theism is noncontingent and false, then it is necessarily false; the probability of a contingent proposition on a necessary falsehood is 1; hence P(E|T) is 1. But if, as Draper claims, P(E|N) is greater than P(E|T), then P(E|T) is less than 1, hence T is not necessarily false. If T is not necessarily false, however, then (given that it is noncontingent) it is necessarily true. So if theism is noncontingent, and 1 is true, then theism is true, and indeed necessarily true.

What could Plantinga mean by saying “the probability of a contingent proposition on a necessary falsehood is 1.” On the above understanding, his claim is flat-out false, being an instance of division by 0. It turns out, Plantinga means that on a conditional probability claim, P(B|A), B plays the role of the conclusion of an argument and A plays the role of the argument’s premises. But if A is a necessary falsehood, then just ‘anything goes,’ and that includes B. In other words: assume a contradictory world (i.e., A, the necessary falsehood). You could regard just anything as true in that world, including any arbitrary B.

The reasoning here may elude some. Let’s think about logical consequence or valid arguments for a moment. It turns out that it is helpful to define this relation as something like this, working in the language of classical predicate logic: A well formed formula (wff), Φ, is a logical (or semantic) consequence of a set of wffs, Γ, if and only if, for every interpretation ℑ, if V_ℑ(γ)=1 for every γ such that γ ∈ Γ, then V_ℑ(Φ) =1.

Less formally, a conclusion is a logical or semantic consequence of a set of premises if and only if it is impossible for the premises to be true and the conclusion false.

In this case, every argument with a necessarily false premise (or a set of premises that together forms a necessary falsehood, e.g., ‘A & ¬A’) will be a valid argument, for it will be impossible for the conclusion to be false and the premises true—namely, because it will be impossible for the premises to be true. If we represent valid argument according to the above probability calculus, then what P(Φ|Γ)=1 means in this case, is that Φ is a logical or semantic consequence of Γ. And we’ve just seen that, given a technical definition of logical or semantic consequence, or valid argument, if Γ is a necessary falsehood, then P(Φ|Γ) =1.

Now, in the above cited paper, Wunder argues that this is still false, even on the understanding we’ve assigned to Plantinga’s claim. Wunder cites no less an authority than Brian Skyrms, who argues that the result would still be something like ‘undefined’ rather than 1, if we put the above consequence or validity results probabilistically. Outside the domain of probability, within the language of classical logic, it’s true (we’ll agree) that an argument to a conclusion from necessarily false premises is always valid. But when describing the relationship probabilistically, it’s ordinarily the case that if, given the truth of the premises, the conclusion is entailed, that we can assign a 1. But when the premises simply cannot be true, the result isn’t a 1, but something like ‘undefined.’

There are other matters as well. As Daniel Howard-Snyder points out, Draper has in mind epistemic probability. In that domain, the probability of a contingent proposition on a necessary falsehood doesn’t equal 1, where ‘1’ means something like, “are maximally warranted to believe.” Indeed, it does not seem like we want classical logic as our ideal epistemic logic. For example, consider the principle of ex falso quodlibet: from a contradiction anything follows. Consider an agent who believes A and ¬A. If classical logic were our ideal logic, we could prove that the agent believes anything and everything!

So the Plantinga’s claim seems false. But I’d like to offer a further and final criticism. Given Plantinga’s view, I’m not sure why he says: “If so, 1 doesn’t imply that naturalism is more likely than theism; instead 1 obviously entails that theism is true.” Recall, 1 is: P(E|N) >+ P(E|T). How does it entail the truth of theism? Plantinga’s argument can be reconstructed as:

1 P(E|N) >+ P(E|T). (premise)

2. Theism is non-contingent. (premise)

3. Either theism is false or it’s true. (premise)

4. If theism is false, then P(E|T) = 1 (2 and logic?)

5. If true, then necessarily true. (2 and logic)

6. Given 1, theism is not false. (1 and 4)

7. Therefore, theism is true. (3 and 6)

8. Therefore, theism is necessarily true. (5 and 7)

But something is wrong here. Suppose Draper negates the second disjunct in (3). We wouldn’t get “theism is true.” Plantinga thinks that Draper doesn’t want to negate the second disjunct, since that would render (1) false. Aside from the fact that Draper isn’t using probability in the way Plantinga is, what does affirming “theism is false, so P(E|T)=1,” really amount to? Just this: Since theism is non-contingent and false, and so necessarily false, then there exists a valid argument from theism to evolution. That doesn’t entail that theism is true. So Draper should just respond, “Yes, that is true, and there exists a valid argument from theism to the contingent proposition that “the moon is made of green cheese.” So, negating the second disjunct of (3), which is what Draper should do if he thinks theism is necessarily false, neither undercuts his (1) nor entails “theism is true.”