Tuesday, August 28, 2007

Back-to-school Post! Knowledge and Evidence

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. :-)~

1 Comment(s):

  • Based on our discussion at the party last night:

    So tell me what you think of this move.

    Jon: Is Earl coming to the party?
    Trent: Well, he probably will if Rich is, and I think Rich is, so I think he will be [let’s bracket dwindling probabilities].
    Jon: What makes you think Rich will be here?
    Trent: I heard him say he’d probably come.

    So I said that in my first comment I (appropriately) condition on Rich’s presence so it’s evidence, but it’s not knowledge, thus it’s not the case that E=K.

    Jon said that my real evidence is that I heard him say he’d come, and this is known.

    My total evidence does include that knowledge, but it’s just a fact that I didn’t condition on it in my first comment. The rule behind my first comment was something like this Pnew(H)=Pold(H/E)*Pnew(E) which is called “Probability Kinematics” a form of Bayesian learning theory. In this case H = Earl will be at the party and E = Rich will be at the party.

    According to my second comment in the dialog—letting A say that I heard Rich say he’d probably be there—P(E)=P(E/A)*P(A). P(A) is knowledge. To hear is to know. So A does play a role, P(E) is a function of it and Pr(H) is in turn a function of P(E). So I don’t dispute for a second that knowledge will always be operative in my reasoning in some sense and it might be that all my “basic evidence” is knowledge (I’d have to think about whether that is true and about whether it bothers me (at the moment I’m inclined to think that it is true and doesn’t bother me, it makes me wonder of by “evidence” W. doesn’t just mean “basic evidence” I want to go look now).

    But even if “basic evidence” is knowledge it doesn’t change the fact that my “superstructural evidence” for H was E, not A. This does not seem to me to be effected by the fact that if you keep questioning my sub-knowledge evidence “But why do you think *that*?” I eventually ground out in some factive mental state, some knowledge.

    By Blogger Trent_Dougherty, at 8/31/2007 11:46 AM