Set aside, for now, E = K. Focus on E → K.
I’m fond of saying that we don’t need to know p in order for p to be in our evidence.
However, consider this argument which I’ve adapted from Williamson (he uses it for a related but different argument, an argument of the form p is evidence for h only if p entails ~h).
[Set aside, for now, my assertion that you know you’ve lost the lottery when you have the true belief that you have based on the odds.]
So for some suitably large n, you’ve observed (on video, all the draws have already occurred, including the last draw) n balls taken out of a bag and all have been red.
From this you reason that the n+1 ball which was drawn was red.
Surely that the n + 1 ball was red is not part of your evidence.
Yet it may be as probable as you please.
Now bring back in E = K.
That entails that you don’t know that the n + 1 ball was red.
This contradicts my assertion that we do know the results of good, solid inductions.
Suppose I’m right.
Then either E = K is false or that the n + 1 ball was red is part of my evidence after all.
I think I can make plausible the latter claim. Here’s a try.
You ask me if Earl’s going to be at the party. I think he’s more likely to be there if Rich is going to be there, and I’m pretty sure Rich is going to be there, so I answer that I think Earl will probably be there.
I’ve clearly conditioned on the proposition that Rich is going to be there even though I don’t know he is, it’s only quite probable for me.
I think this *establishes* that it’s OK to use the unknown as evidence.
I think the reason we are less likely to say this in the case of the case of enumerative induction is that the mathematical framework invokes a way of thinking which makes us want to “wait and see.”
After all, that next draw is coming up, why not just wait and see? We are hesitant to use something as evidence if we think it’s status might change soon or if it’s status can be fixed quite easily.
This is a very natural and rational way to *use* evidence, but it shouldn’t affect our *analysis* of evidence.
So if I’ve succeeded thus far, either we do *know* that the next draw was red (assuming it was) and furthermore we know that we’ve lost the lottery (assuming we have) or E = K is false.
Either way, Williamson is mad and I’m happy. :-)~