## Thursday, August 29, 2019

### 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:

1. 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.

#### 4 comments:

William said...

Blind men and an elephant.

Feryll said...
This comment has been removed by the author.
Feryll said...

Interesting. But there are two typos, and it was difficult to follow the proof's logic regardless. The evidential difference condition should read "P(H|E)≠P(H)," not "P(H|E)≠P(E)," and conditional probability means "P(H1|E)=P(H1&E)/P(E)" and not "P(H1|E)=P(H1&E)P(E)."

Alexander R Pruss said...

Feryll:

It was worse than that: most of the proof was wrong. I revised it. Is it better?