Weak Bayesian Coherentism

Here, I partly take back what I said in an earlier article and argue that the Coherence Theory of Justification is not really refuted by probability theory.* (Warning: somewhat technical.)

[*Based on: “Weak Bayesian Coherentism,” Synthese 157 (2007): 337-46.]

1. The Coherence Theory

The Coherence Theory of Justification (“Coherentism”) holds that beliefs are ultimately justified solely by coherence, i.e., the way a system of beliefs fit together, supporting and explaining each other. It’s important for coherentists that there be no foundational justification, i.e., no belief has any justification independent of support by other beliefs.

Imagine police interviewing a collection of witnesses to a crime, then comparing the witness testimonies. If the witnesses independently tell stories that fit together, the police will conclude that something like what the witnesses are saying must have happened, since, if the witnesses were unreliable, it would be an incredible coincidence if they happened to agree with each other (without collusion). It is said that you could draw this conclusion even if you initially had no belief in the reliability of any individual witness. Any individual witness claim will then be credible if it fits with the rest of the witness testimonies, and will be doubted or rejected if it clashes with other testimonies. That’s supposed to show how you can have justification based solely on coherence, without foundations.

2. Objections

2.1. Huemer

In earlier work, I argued that the probabilistic reasoning here is wrong: if you attach no initial credibility to any individual witness assertion, then you should also attach no credibility to any assertion no matter how many witnesses agree on it. The proof of this assumes Conditional Independence, i.e., that, if you fix the truth about A, then the probability of one witness asserting A is independent of what any other witness asserts. See:

Probability & Coherence Justification

2.2. Olsson, Bovens & Hartmann

There’s another probability-theory objection, from Erik Olsson, Luc Bovens, and Stephen Hartmann. Assume:

i) The degree of coherence of a set of witness assertions is a function of the probability distribution over those assertions;

ii) The individual assertions are probabilistically independent of each other, conditional on the truth about the thing they’re talking about. (Conditional Independence)

Then it can be proven that there is no possible measure of “degree of coherence” such that increased coherence in general contributes to likelihood of truth. I.e., there will always be cases where you can hold other things equal, increase the “degree of coherence” of a set of assertions, and the probability of them all being true goes down.

3. The Coherentist’s Defense

3.1. A Probability Model

Let A be some proposition about the world, let W1 be the proposition that Witness #1 asserts A, and let W2 be the proposition that Witness #2 asserts A. The diagram below shows a possible probability distribution.

Probabilities are represented by areas; the size of the whole square is 1, the size of the “A” region is 1/4, etc. Note that you can also read off conditional probabilities, e.g., P(A|W1) is the proportion of the W1 region that overlaps with A.

You can see from the diagram that the following probabilities obtain:

P(A) = 1/4

P(A|W1) = 1/4

P(A|W2) = 1/4

P(A|W1&W2) = 1

So that’s a model of what the coherentist was talking about: each witness has no individual credibility, yet the two have strong credibility when combined.

3.2. Rejecting Conditional Independence

How is this consistent with the theorem reported above? Well, this probability distribution violates Conditional Independence. P(W2|W1,A) = 1, whereas P(W2|~W1,A) = 0, so, given A, W1 and W2 are probabilistically dependent.

Is this a problem for the coherentist? Actually … not really. The reason for stipulating conditional independence was that, in describing the witness scenario (sec. 1 above), coherentists normally say that the witnesses “independently” tell the same or similar stories. I (and other people in this literature) interpreted that in terms of Conditional Independence.

But the coherentist doesn’t have to say that. The purpose of the “independently” qualifier was to rule out the case where one witness asserts A solely because he heard the other witness assert A—in that case, clearly the coherence of their testimonies is no evidence for their truth. Here is a weaker condition that rules out that sort of scenario: if A is false, then witness1’s asserting A would not make witness2 more likely to assert A. That is, P(W2|~A,W1) <= P(W2|~A).* That condition is of course satisfied for the above probability distribution.

*To be more precise, the coherentist needs this condition: P(W2|W1,A) > P(W2|W1,∼A). That condition turns out to be necessary and sufficient for pure coherence to provide justification (i.e., for W1 and W2 together to confirm A, when neither alone does so). There is a proof of that in the paper, but it’s not fun, so I won’t reproduce it here.

3.3. The Other Objection

If we reject Conditional Independence, then the theorem of sec. 2.2 also goes away.

Plus, the objection of section 2.2 assumes that the coherentist must say that coherence always contributes to likelihood of truth. But they need not say that. The coherentist could instead claim only that coherence sometimes enhances a belief system’s likelihood of truth, and that this is what happens for the beliefs of ours that are actually justified.

4. But Is it True?

What I showed above is that there is a possible probability distribution in which the sort of thing the coherentist is saying is true. So the theory is not ruled out by pure probability theory. That doesn’t mean it’s true, though.

The coherentist needs a probability distribution in which W1 is not more likely given A than given ~A, but W1 is more likely given A and W2 than given ~A and W2. (They need the first part to stop W1 from having “independent credibility”, to model lack of foundational justification. They need the second part to enable W1 and W2’s congruent testimonies to provide justification for A.)

That’s not incoherent, but I just can’t see any reason why that would be true. So I still don’t think the Coherence Theory is right, though I can’t prove it’s not right.