when i make an argument that the presence of a necessary condition is evidence for its sufficient conditions, i am often countered with the fact that im making an logical fallacy called the inverse error. when in actuality im making a completely different logical argument (claim 2).
note: each probability that is conditioned on an event occurring assumes the probability of that event is greater than 0. (P(A|B) → P(B)>0)
claim 0: if P(A|B)>P(A) then P(B)<1 and P(A)>0
proof by contradiction:
if P(B)=1 then P(A|B)=P(A)!
if P(A)=0 then P(A|~B)<0 which is an invalid probability!
claim 1: if A is evidence for B, P(A|B)>P(A), then B is evidence for A, P(B|A)>P(B)
→ P(B|A)P(A)/P(B)>P(A) by applying bayes theorem to the LHS
∴ P(B|A) >P(B)
definition: A is a necessary condition for B iff P(A|B)=1 (if B then A )
definition: A is a trivial necessary condition for B iff P(A|B)= 1 = P(A)
note: a trivial necessary condition is not evidence since P(A|B)= P(A)
and that A and B are independent events. independence is sufficient to show that events are not evidence for each other.
claim 2: if E is a non-trivial necessary condition for H, or equivalently P(E|H) = 1> P(E), then E is evidence for H, P(H|E)>P(H)
since P(E|H) = 1> P(E)
by definition, H is evidence for E.
∴ E is evidence for H by claim 1
If E was an absence of genuine perceptual evidence of aliens (not sensing aliens when we observed) and H were the hypothesis that aliens do not exist, we can see that P(E|H) = 1 > P(E). also if aliens existed we would at least expect P(E|~H)<P(E)<1 since P(E)=P(E|H)*P(H) + P(E|~H)*P(~H) and therefore P(E|H) > P(E) >P(E|~H) since it is a weighted average of the two probabilities.
so therefore the presence of a non-trivial necessary condition is evidence for its sufficient conditions.
some people foolishly argue that a non-trivial necessary condition is “compatible” with the negation of its sufficient condition. the term compatible in this context has a formal definition. an event E is compatible with with a hypothesis H iff P(E|H)>0. but this argument omits critical information about the relationships between E,H, and ~H, namely, the likelihood of seeing an event under the sufficient condition is higher than its negation, and therefore the event is evidence for the sufficient condition and evidence against the negation (see my last video on absence of evidence)
this reasoning provides some of the foundation behind things like inductive evidence, statistical testing, and rational belief systems.