Published ‘positive’ results of epidemiological studies on possible associations (descriptive or causal) are ever more commonly ‘false positives’ and, thus, false warrants for claiming discovery. More common examination of a multitude of possible associations is widely seen to be the principal cause of this trend. I dispute this explanation and take the principal basis for the trend to be the ever decreasing prior plausibility of the associations that are reported on; and publication bias leading to missing ‘negatives’ in the published results exacerbates the appearance of the problem. The problem is, however, eminently remediable. We epidemiologists, as a collective of researchers, should leave behind the decision-oriented, inference-denying cult of statistical ‘significance’ adduced by Neyman and Pearson, and in its stead we should embrace the Fisherian culture of focusing on the production of statistical evidence, for use in inference by our readers. I recommend a simple, objective measure of evidence, suitable for readers’ Bayesian-type inferences about the existence of an association.