How did Rudolf Carnap define the degree of confirmation that evidence gives a hypothesis?

Rudolf Carnap approached the problem of confirmation by seeking to define a "degree of confirmation" as a purely logical, quantitative measure of the support a body of evidence ($E$) provides to a hypothesis ($H$).12

In his work during the 1940s and 1950s, Carnap aimed to emulate the precision of deductive logic by analyzing artificial, simplified logical languages. He argued that statements ascribing an inductive probability to a hypothesis are analytic, as they express the strength of support provided by the evidence. His approach was probabilistic, utilizing Bayesian methods where the degree of confirmation is identified with the conditional probability $P(H/E)$.123

To define these probabilities on a purely logical basis, Carnap assigned a priori probabilities to "state-descriptions" (maximal consistent sets of atomic sentences) within his formal systems. He later developed the "$\lambda$-continuum" of inductive methods, where $\lambda$ serves as an index of caution, determining how quickly an inquirer updates their beliefs based on observed relative frequencies versus a priori probabilities. While Carnap acknowledged that his specific project faced limitations—such as the existence of infinitely many admissible functions—he maintained that confirmation is fundamentally connected to the axioms of the probability calculus.31