Resolving the induction problem: Can we state with complete confidence via induction that the sun rises forever?

Abstract

Induction is a form of reasoning that starts with a particular example and generalizes to a rule, namely, a hypothesis. However, establishing the truth of a hypothesis is problematic due to the potential occurrence of conflicting events, also known as the induction problem. The sunrise problem, first introduced by Laplace (1814), is a quintessential example of the probability-based induction. In his solution, a zero probability is always assigned to the hypothesis that the sun rises forever, regardless of the number of observations made. This is a symptom of fundamental deficiency of probability-based induction: A hypothesis can never be accepted via the Bayes-Laplace approach. Alternative priors have been proposed to address this issue, but they have failed to fully overcome the deficiency. We investigate why this occurs and demonstrate that the confidence does not exhibit such a deficiency, as it is not a probability and therefore does not adhere to Bayes' rule. The confidence is neither a likelihood to allow not only a reconciliation between epistemic and aleatory interpretations of probability but also a resolution in agreement with the evidence by enabling us to accept a hypothesis with complete confidence as a rational decision.