Why could Rudolf Carnap’s early inductive logic not handle strict generalizations in an infinite universe?

In the early forms of Rudolf Carnap’s inductive logic, the system could not handle strict generalizations in an infinite universe because it assigned an a priori probability of zero to such generalizations. Within the members of his $\lambda$-continuum, this probability remained zero and could not be increased by any amount of evidence.1

This limitation arose because Carnap’s initial approach relied on assigning equal a priori probabilities to state-descriptions—maximal consistent sets of atomic sentences and their negations—within a fixed finite domain. Because these early models did not account for the specific structure of infinite domains in a way that allowed for non-zero probability of universal laws, they failed to support inductive generalization.1

This defect was later addressed by thinkers such as Jaakko Hintikka, who proposed assigning non-zero a priori probabilities to "constituents"—sentences that specify which cells of a contingency table are empty and which are not—thereby allowing for a more robust treatment of inductive inference in broader contexts.1