Wolfgang Schwarz

Can evidence be inadmissible?

First, a quick reminder of history. David Lewis once proposed a principle (the 'Principal Principle') linking rational credence and objective chance. It says (or rather, entails) that your rational credence in any proposition A, on the assumption that the objective chance of A is x, should also be x, no matter what (further) evidence E you have:

OP: P(A | ch(A)=x & E) = x.

This principle, the 'Old Principle', is widely taken to suffer from two defects. First, suppose your evidence E includes ~A. Then probability theory ensures that P(A | ch(A)=x & E) = 0, irrespective of x. Lewis responded by restricting OP to cases where E is 'admissible'. He suggested that a (true) proposition is admissible iff it is entailed by the history of the world up to now together with the laws of nature.

We can easily get rid of this restriction. Let E* be the admissible part of E -- whatever that means --, and let ch=f be the proposition that the function f currently plays the chance role. By (a slight generalisation of) the restricted version of OP,

P(AE | ch=f & E*) = f(AE), and
P(E | ch=f & E*) = f(E).

Since P(A|EX) = P(AE|X)/P(E|X), we have

P(A | E & ch=f & E*) = f(AE)/f(E) = f(A|E).

And since E entails E*, this reduces to

IP: P(A | ch=f & E) = f(A|E).

IP is the 'Intermediate Principle'. It fixes one of the two problems with the Old Principle: the restriction to admissible evidence. IP needs no such restrictions. (Note that IP reduces to the OP if f(A|E) = f(A), which gives us a new handle on admissibility: E is admissible for the evaluation of A iff the conditional chance of A given E equals the unconditional chance of A.)

Here is another, perhaps more intuitive explanation of where OP went wrong. We want to treat chance as an expert whose judgements we endorse; but we shouldn't endorse an expert's judgements about A if we have independently found out that ~A -- unless we know that the expert has found that out as well. To trust an expert is to endorse her conditional judgements, conditional on whatever information we have that she might lack. IP takes this into account, OP doesn't.

However, IP doesn't fully take it into account. For suppose we are ignorant of both A and B and wonder how likely A is given B. As before, we want to align our credence with the expert's judgement. Clearly the relevant judgement here is not the expert's unconditional credence in A, nor her credence in A conditional on our total evidence E, but rather her credence in A conditional on E and B. Now let B = the proposition that ch=f and you see that IP makes exactly this mistake. Conditional on ch=f, we shouldn't align our credence with the expert's credence in A given E, but with her credence in A given E and ch=f. Fixing this second problem yields the 'New Principle', defended in Lewis 1994 and Hall 1994 (though in a formulation that is, I think, due to Hall 2004):

NP: P(A | ch=f & E) = f(A | ch=f & E).

Perhaps the second fix is redundant. If chances are 'self-aware' in the sense that for any candidate chance function f, f(ch=f)=1, then NP reduces to IP. The assumption that ch=f is then no news to the expert, so we don't need to tell her. Humean chances are notoriously not self-aware, which caused the undermining problems for Humean Supervenience.

Let's not assume that chances are self-aware, but that E is admissible. Then NP reduces to a different intermediate principle, one that fixes only the second problem:

IP2: P(A | ch=f & E) = f(A | ch=f).

Now here is my question: can evidence ever be inadmissible? If not, then IP2 is good enough.

There can certainly be inadmissible assumptions. For instance, I might consider the possibility of a certain coin landing tails tomorrow conditional on the assumption that it will land tails. But this doesn't violate IP2 if E is restricted to our evidence. (So we might still want NP to handle cases where E goes beyond our evidence. My question is if we really need NP only for such cases.)

Lewis and others suggested that we might have genuine evidence about the outcome of the coin toss. For instance, there might be an oracle (or a time-traveler) predicting that the coin will land heads. If the oracle has a sufficiently good track record, this should affect our credence in heads. I agree, but we only get a counterexample to IP2 if the chance of heads is still 1/2. And I'm not sure how it could be. Consider these three kinds of worlds:

In type-1 worlds, the laws of nature guarantee that the outcome of tomorrow's chance process is correctly predicted by the oracle today.

In type-2 worlds, the laws of nature made it very probable, earlier today, that the oracle will correctly predict the outcome.

In type-3 worlds, there are no laws of either kind. In some of these worlds, the oracle's track record is a matter of sheer luck. In others, the oracle is guaranteed to get things right only under special conditions, and these conditions have obtained until now, but don't obtain any more.

Trusting the oracle means giving sufficient credence to worlds of type 1 and 2. If all your credence went to type-3 worlds, your credence in heads should not significantly rise upon hearing the oracle's prediction.

But what is the objective chance that the coin will land heads in type-1 worlds, after the prediction has been made? In all those worlds, the chance of a 'heads' prediction is now 1, and there is a non-chancy law ensuring that given this prediction, the coin will land heads. (This doesn't mean that the prediction causes the outcome. The causation may well go in the other direction.) So the chance of heads in type-1 worlds is now 1. By parallel reasoning, the chance of heads in type-2 worlds is also greater than 1/2.

If that is right, then oracles and time-travelers can bring us news about the future, and thereby affect our credence, but not because they provide us with inadmissible evidence. Instead, they merely reveal something about the objective chances. The oracle's prediction tells us that there is a high objective chance of heads. A prediction of tails would have told us that there is a high chance of tails.

So can we forget about inadmissible evidence and return to IP2?

Update: I forgot to mention another reason why evidence is always admissible -- it is entailed by the history of the world up to now.