The widely claimed replicability crisis in science may lead to revised standards of significance. The customary frequentist confidence intervals, calibrated through hypothetical repetitions of the experiment that is supposed to have produced the data at hand, rely on a feeble concept of replicability. In particular, contradictory conclusions may be reached when a substantial enlargement of the study is undertaken. To redefine statistical confidence in such a way that inferential conclusions are non-contradictory, with large enough probability, under enlargements of the sample, we give a new reading of a proposal dating back to the 60's, namely Robbins' confidence sequences. Directly bounding the probability of reaching, in the future, conclusions that contradict the current ones, Robbins' confidence sequences ensure a clear-cut form of replicability when inference is performed on accumulating data. Their main frequentist property is easy to understand and to prove. We show that Robbins' confidence sequences may be justified under various views of inference: they are likelihood-based, can incorporate prior information, and obey the strong likelihood principle. They are easy to compute, even when inference is on a parameter of interest, especially using a closed-form approximation from normal asymptotic theory.