Here’s a puzzle for the E=K (i.e. all and only knowledge is evidence) theory.
bq. Jack is inspecting a new kind of balance made by Acme Corporation. He thoroughly inspects the first 10 (out of a batch of 1,000,000) that come off the assembly line. And each of them passes the inspection with flying colours. Each of them is more accurate than any balance Jack had tested before that day. So Acme is making good balances. He knows, by observation, that the first 10 balances are reliable. He also knows, by induction, that the next balance will be reliable. It’s not obvious that he knows the next one will be phenomenal, like the ones he has tested, but he knows it will be good enough for its intended usage. But he doesn’t know they all will be that good. Surprisingly, it will turn out that every balance made in this assembly line will be reliable. But you’d expect, given what we know about assembly lines, for there to be a badly made machine turn up somewhere along the way.
bq. So Jack knows the first 11 are reliable, and doesn’t know the first 1,000,000 are reliable. Let _n_ be the largest number such that Jack knows the first _n_ are reliable. (I’m assuming such an _n_ exists; those who want to hold on to E=K by giving up the least number theorem are free to ignore everything that follows.) For any _x_, let _R(x)_ be the proposition that the first _x_ are reliable. So Jack knows _R(n)_. Hence by E=K _R(x)_ is part of his evidence. But he doesn’t know _R(x+1)_. This is extremely odd. After all, _R(x)_ is excellent evidence for _R(x+1)_, assuming it is part of his evidence. And _R(x+1)_ is true. Indeed, by many measures it is safely true. So why doesn’t Jack know it?
It seems to me there is a mystery here that, given E=K, we can’t explain. If we have a more restrictive theory of evidence, then it is easy to explain what’s going on. If, for instance, evidence is perceptual knowledge, then Jack’s evidence is simply _R(10)_. And it might well be true, given the correct theory of what hypotheses are supported by what evidence, that _R(10)_ supports _R(84)_ but not _R(85)_. That explanation isn’t available to the E=K theorist. And we might well wonder what explanation could be available.
I have one idea that saves the letter of E=K, though at some cost I think to the spirit of it. Let’s say that evidence can be of better or worse _quality_. If you don’t know _p_, then _p_ is of no evidential use to you. But even if you do know it, how much evidential use is might depend on how you know it. For instance, if you infallibly know _p_, then _p_ is extremely useful evidence. More relevantly for today’s purposes, if you have *sensitive* knowledge that _p_, then _p_ is more useful than if you have *insensitive* knowledge that _p_.
Let’s go through how this plays out in Jack’s case. Although he knows _R(11)_, this knowledge is insensitive. If _R(11)_ were false, he would still believe it. Had the production system malfunctioned when making the 11th balance, for instance, then the 11th machine would have been unreliable, but Jack would have still believed it. The only sensitive evidence he has is _R(11)_. By the time he gets to _R(n)_, his knowledge is extremely insensitive. There are all sorts of ways that _R(n)_ could have been false, in many fairly near worlds, and yet he would still have believed it.
So here’s a hypothesis. The more insensitive your evidence is, the less inductive knowledge it grounds. If Jack had sensitive knowledge that _R(n)_, he would be in a position to infer, and thereby know _R(n+1)_. The reason he can’t know _R(n+1)_ is not that he doesn’t have enough evidence, but rather that the evidence he has is not of a high enough quality. That’s an explanation for why Jack can’t infer _R(n+1)_ that neither leads to inductive scepticism, nor violates the letter of E=K. I’m not sure that E=K was meant to go along with the view that how you know something is evidentially relevant, not just whether you know it, so I don’t think this keeps the spirit of E=K. But perhaps the letter of E=K is more defensible than the spirit of it.