A Probabilistic Critique of Evidentialism

Here, I use probabilistic arguments to rebut evidentialism (on a plausible interpretation).*

[ *Based on: “A Probabilistic Critique of Evidentialism,” pp. 199-222 in Believing in Accordance with the Evidence: New Essays on Evidentialism, ed. Kevin McCain (Springer, 2018). ]

1. What Is Evidentialism?

In epistemology, evidentialism is the view that all (epistemic) justification for belief derives from evidence. Unfortunately, evidentialists don’t always tell us what they think “evidence” means. Here are five accounts:

a)     They’re just using “evidence” in the ordinary English sense.

• Comment: No, they’re not. Many evidentialists, e.g., claim that all of your evidence consists of mental states. This rules out, e.g., that fingerprints at a crime scene or the murder weapon might count as evidence, that someone might fabricate evidence, or introduce evidence at a trial, or destroy evidence, etc.

b)     Evidence = that which confers epistemic justification.

• Comment: This renders the thesis trivial & uninteresting.

c)     E is evidence for H iff P(H|E) is high.

• Comment: No, because E could just be a little evidence for H, and P(H|E) could still be low.

d)     “Evidence for H” just means reasons to believe H.

• Comment: That sounds fine, as long as we add that we’re only talking about epistemic or evidential reasons, as opposed to pragmatic or moral reasons. But then we need to hear more about what makes something an evidential reason to believe H. Perhaps that would be answered by the next account …

e)     The Bayesian account: “Evidence” = facts that raise the probability of a proposition above its initial probability. I.e., e is evidence for h iff P(h|e) > P(h).

In what follows, I’ll assume the Bayesian account.

2. The Argument Against Evidentialism

My main argument:

1. Some propositions are justified because they have high initial probablities. Why:

   1. A proposition with a (sufficiently) high initial probability is thereby justified.

   2. There are propositions with (arbitrarily) high initial probabilities.

2. This kind of justification is not evidential. Why:

   1. Evidence for h is something that raises the probability of h above its initial probability.

   2. h’s having a high initial probability does not depend on something that raises the probability of h above its initial probability.

3. Therefore, not all justification is evidential.

Discussion:

Premise 1a: This sounds close to analytic (assuming we take an epistemic or logical interpretation of probability). Note that I leave open that the threshold needed for justification could vary depending on the particular proposition; I also leave open that you can set the threshold as high as you like (even up to probability 1 if you’re a real stickler).

Premise 1b: This is just required by probability theory, as long as you have a rich enough set of propositions. E.g., suppose you’re assigning initial probabilities to the possible colors of a book cover, and assume color is a continuum. Then there are propositions with arbitrarily small probabilities (just take arbitrarily small regions of the color space); the negations of these propositions have arbitrarily high probabilities. In the extreme case, a single point in the possibility space will have initial probability 0, and its complement (or the proposition that you don’t hit that point) has initial probability 1.

Btw, this does not depend on the Principle of Indifference. It just depends on having a continuum of possibilities. There is no coherent probability distribution (whether uniform or not) that would assign nonzero probabilities to every one of a continuum of possibilities.

Premise 2a: I take this to be definitional.

Premise 2b: This is self-evident. The initial probability of a proposition, by definition, does not depend on evidence. That’s what makes it “initial”.

So it looks like conclusion 3 is inevitable.

3. Three Objections

3.1. A Priori Priors?

Objection: Some people reject the notion of a priori prior probabilities; they think all probability assessments depend on evidence. In support of this, note how difficult it is to say what the completely a priori probability is of any interesting theory, say, the Theory of Evolution, or Special Relativity, or the existence of God. No one seems to have any idea what their ultimate priors are.

Reply: According to Bayes’ Theorem,

P(h|e) = P(h)*P(e|h) / P(e).

This gives a mathematical relationship between the prior probability of h and its probability in the light of evidence. It is just a mathematical fact that, if you hold the other quantities fixed, and you vary P(h) from 0 to 1, then P(h|e) must vary through all possible values from 0 to 1. This means that if the value of P(h) is completely unknown or indeterminate, then the value of P(h|e) must also be completely unknown or indeterminate.

And that means that if you deny that h has any prior probability, then you must deny that it has a probability in the light of e either. Since “e” could be defined to include all the evidence that you have relevant to h, this means that if there is no a priori prior probability of h, then there is no probability of h in the light of any evidence either.

(For more, see my “There Is No Pure Empirical Reasoning”.)

3.2. Lottery cases

Objection: Suppose you have a ticket in a large lottery where you know that exactly one ticket will win. Some people intuit that you cannot know in advance that your ticket will lose, no matter how unlikely it is that it will win. Some would even deny that you are justified in believing “This ticket will lose”; perhaps you are only justified in believing “This ticket will probably lose” or something like that. Therefore, high probability doesn’t suffice for justification.

Reply: Surely you’d at least agree that you have some justification for thinking the ticket will lose (even if it’s not enough to call it a justified belief). So just modify my argument so that (1) says, “Some propositions are justified to some degree because they have high initial probabilities.”

3.3. Unknown probabilities

Objection: According to standard probability theory, every necessary truth has probability 1. But we’re not automatically justified in believing every necessary truth, since we might not be able to see that it’s true (usually, you need a proof, and the proof may not be obvious). E.g., you’re not automatically justified in believing the Banach-Tarski Paradox (without having seen a proof nor heard any testimony about it), merely because it is in fact a necessary truth.* Therefore, a proposition’s having a high probability is not sufficient for it to be justified.

[*The Banach-Tarski Theorem says that it’s possible to divide a sphere into finitely many pieces, rearrange the pieces and reassemble them into two spheres, each the same size as the original. This depends on the Axiom of Choice.]

Reply: Fair enough: to be justified in believing a highly probable proposition, its high probability needs to be accessible to you, you need to see why it’s likely, or something like that. So just modify my argument to include that condition (however you want to formulate it). E.g., you could modify (1) to:

1. Some propositions are justified to some degree because they have high initial probabilities and we can see that.

The argument can still proceed, because there are in fact cases in which a proposition has a high initial probability and we can see that.

4. Questions About Other Views

4.1. Evidentialism with room for self-evidence

Suppose we redefine evidentialism as the view that all epistemic justification is either based on evidence for the target proposition or grounded in “self-evidence”. (Contrary to what the name implies, I assume that “self-evident” propositions aren’t evidence for themselves but instead are propositions that do not need evidence because you can immediately see them to be true.) This is a reasonable modification since the original formulation of evidentialism sounds like it rules out foundationalism, yet most evidentialists are foundationalists. Would my argument still apply to this view?

Reply: Yes. People who talk about “self-evident” propositions usually say that they are necessary truths that one can see to be true just on the basis of understanding the propositions. My propositions with high a priori prior probabilities do not sound like examples of “self-evident truths” as traditionally understood. E.g., the proposition [x is not exactly 6 feet tall] has a high prior probability (actually, 100%), yet it is not a necessary truth, and it’s not really the sort of thing you’d say you could see to be true just by understanding the proposition.

4.2. Subjective Bayesianism

I’m an objective Bayesian; I think there are objective constraints on the “correct” prior probabilities.

But subjective Bayesianism is probably more popular. Subjective Bayesians hold that any set of prior probabilities that is coherent (consistent with the Kolmogorov axioms) is rationally permissible. What should subjectivists say about my argument?

Reply: The subjectivists should reject evidentialism too. They will say one of 3 things:

a. “Your priors are automatically justified as long as they are coherent.” But they’re not based on evidence, so you have a counter-example to evidentialism whenever someone assigns a high prior.

b. “Your priors are neither justified nor unjustified; nevertheless, when you conditionalize on some amount of evidence, starting from those priors, your posterior credences get to be justified.” This is also incompatible with evidentialism, because you could have subjects with different priors wind up with different posterior probabilities, even if they have the same evidence.

c. “Nothing is ever justified.” This would make evidentialism vacuously true. But this is a crazy thing to say, and hardly any evidentialist would be happy with this, so we don’t need to talk about this.

4.3. Maybe states of understanding are evidence

Take the case where I am justified in believing that some person, x, is not exactly 6 feet tall because the prior probability of x’s being exactly 6 feet is extremely low. I suppose that I’m justified in believing [x isn’t exactly 6 feet] based on my general understanding of probability theory and the fact that [x is exactly 6 feet tall] is only one of a continuous infinity of possibilities. Might the evidentialist say that this is my “evidence” for the proposition?

They might say this, but this is a weird example of evidence. (Almost) whenever a person is justified in believing a theory, the person must first understand the theory, but we do not normally include that understanding in listing the “evidence for” the theory. E.g., the evidence for the theory of evolution is stuff like the fossil record, the morphological similarities among different species, etc.; the evidence for the theory of evolution isn’t my understanding of what the theory says. At least, not in the ordinary English sense of the word.

Evidentialists might say they are using a technical sense of “evidence”, in which understanding does count as part of one’s evidence. What might that sense be? I have a hard time seeing how it would be anything other than “evidence = that which confers justification”, which would (again) trivialize the thesis of evidentialism.

Maybe evidentialists would just say, “Yep, our thesis is trivial.” In that case, I would just say

• (i) I think you guys are misusing the word “evidence”.

• (ii) You’ve also been unclear, because up till now, you haven’t made clear that your thesis is merely that all justification comes from wherever it comes from. There wouldn’t have been nearly as much discussion of evidentialism as there has been, if people knew that was all you were saying.