Łukasiewicz’s argument against inductive inference is simple: under any objective interpretation of the probability calculus, the probability that a single hypothesis is true where m is the set of logical consequences of the hypothesis and if n events are taken to possess property P is m / (m+n). Thus, m is always greater than n, so that the probability of the hypothesis cannot be greater than ½. Moreover, since scientific theories are often universal, m approaches infinity and the probability of the scientific theory approaches 0. Carnap rediscovers this argument after attempting to produce an objective inductive logic and is shaken to the core.
Łukasiewicz, Jan. 1909. On Probability of Inductive Conclusions, Przeglad Filozoficzny, 12, 209–210.
Carnap, Rudolph. 1950. The Logical Foundations of Probability. 2nd ed., Chicago: The University of Chicago Press, 572.