The unavoidability of misleading evidence

Three definitional assumptions:

1. E is only evidence if there is some hypothesis H to which E makes an evidential difference, i.e., P(H|E)≠P(H).

2. E is incomplete if and only if it is evidence such that there is a hypothesis H such that 0 < P(H|E)<1, i.e., E doesn’t make everything certain.

3. E is misleading with respect to a hypothesis H if and only if either H is true and E is evidence against H (i.e., P(H|E)<P(H)) or H is false and E is evidence for H (i.e., P(H|E)>P(H)).

Then:

4. Every piece of incomplete evidence is misleading (with respect to some hypothesis).

[Proof: Suppose E is incomplete evidence. Either E is or is not true. If it is not true, it is misleading, since it lowers its own probability to zero. So, suppose that E is true. Let H₁ be a hypothesis such that 0 < P(H₁|E)<1. Replacing H₁ by its negation if necessary, we can assume H₁ is true. Note that the fact that E is evidence implies that 0 < P(E)<1. Let H be the disjunctive hypothesis: ∼E or (H₁&E). This is true as the second disjunct is true. Now, note that P(H₁&E)<P(E) as P(H₁|E)<1. Thus, (1 − P(E))P(H₁&E)<(1 − P(E))P(E). Thus, P(H₁&E)<P(E)P(H₁&E)+(1 − P(E))P(E). Thus: P(H₁|E)=P(H₁&E)/P(E)<P(H₁&E)+(1 − P(E)) = P(H₁&E)+P(∼E)=P(H). Thus, E is evidence against H even though H is true.]

In particular, we should not take misleadingness of evidence to be an evil. Misleadingness of evidence is a normal part of reasoning with incomplete information.