Learning without change in beliefs

There are books of weird mathematical things (e.g., functions with strange properties) to draw on for the sake of generating counterexamples to claims. This post is in the spirit of an entry in one of these books, but it’s philosophy, not mathematics.

Surprising fact: You can learn something without gaining or losing any beliefs.

For suppose proposition q in fact follows from proposition p, and at t₁ you have an intellectual experience as of seeing q to follow from p. On the basis of that experience you form the justified and true belief that q follows from p. This belief would be knowledge, but alas the intellectual experience came from a chemical imbalance in the brain rather than from one’s mastery of logic. So you don’t know that q follows from p.

Years later, you consider q and p again, and you once again have an experience of q following from p. This time, however, the experience does come from your mastery of logic. This time you see, and not just think you see, that q follows from p. Your belief is now overdetermined: there is a Gettiered path to it and a new non-Gettiered path to it. The new path makes the belief be knowledge. But to gain knowledge is to learn.

But this gain of knowledge need not be accompanied by the loss of any beliefs. For instance, the new experience of q following from p doesn’t yield a belief that your previous experience was flawed. Nor need there by any gain of beliefs. For while you might form the second order belief that you see q following from p, you need not. You might just see that q follows from p, and form merely the belief that q follows from p, without forming any belief about your inner state. After all, this is surely more the rule than the exception in the case of sensory perception. When I see my colleague in the hallway, I will often form the belief that she is in the hallway rather than the self-regarding belief that I see her in the hallway. (Indeed, likely, small children and most non-human animals never form the “I see” belief.) And surely this phenomenon is not confined to the case of sensory perception. At least, it is possible to have intellectual perceptions where we form only the first-order belief, and form no any self-regarding second-order belief.

So, it is possible to learn something without gaining or losing beliefs.

In fact, plausibly, the original flawed experience could have been so clear that we were fully certain that q follows from p. In that case, the new experience not only need not change any of our beliefs, but need not even change our credences. The credence was 1 before, and it can’t go up from there.

OK, so we have a counterexample. Can we learn anything from it?

Well, here are two things. One might use the story to buttress the idea that even knowledge of important matters—after all, the relation between q and p might be important—is of little value. For it seems of very little value to gain knowledge when it doesn’t change how one thinks about anything. One might also use it to argue that either understanding doesn’t require knowledge or that understanding doesn’t have much value. For if understanding does require knowledge, then one could set up a story where by learning that q follows from p one gains understanding—without that learning resulting in any change in how one thinks about things. Such a change seems of little worth, and hence the understanding gained is of little worth.