Logical consistency and objectivity are cornerstones of science that distinguish it from cult and dogma. Scientists’ concern with objectivity has led to the dominance of associative statistics, which define the basic concept of independence on observations. The same concern with avoiding subjective beliefs has led many scientific journals to favor frequentist over Bayesian statistics. Our analysis here reveals that to infer causes of a binary outcome, (1) the associative definition of independence results in a logical inconsistency -- even for data from an ideal experiment -- for both frequentist and Bayesian statistics, and (2) removing the logical error requires defining independence on counterfactual causal events. The logically coherent causal definition is the one intuitively adopted by humans. Our findings have direct implications for more consistent and generalizable causal discoveries in medicine and other sciences.