Simpson's Paradox: A Singularity of Statistical and Inductive Inference

The occurrence of Simpson's paradox (SP) in $2\times 2$ contingency tables has been well studied in the literature. The first contribution of the present work is to comprehensively revisit this problem. We provide a threadbare analysis of SP in $2\times 2$ contingency tables and present new results, detailed proofs of previous results and a unifying view of the important examples of SP that have been reported in the literature. The second contribution of the paper suggests a new perspective on the surprise element of SP, raises some critical questions for the causal analysis of SP and provides a broad perspective on logic, probability and statistics with SP at its centre.