We clarify the theoretical foundations of partisan fairness standards for district-based democratic electoral systems, including essential assumptions and definitions not previously recognized, formalized, or in some cases even discussed. We also offer extensive empirical evidence for assumptions with observable implications. We cover partisan symmetry, the most commonly accepted fairness standard, and other perspectives. Throughout, we follow a fundamental principle of statistical inference too often ignored in this literature—defining the quantity of interest separately so its measures can be proven wrong, evaluated, and improved. This enables us to prove which of the many newly proposed fairness measures are statistically appropriate and which are biased, limited, or not measures of the theoretical quantity they seek to estimate at all. Because real-world redistricting and gerrymandering involve complicated politics with numerous participants and conflicting goals, measures biased for partisan fairness sometimes still provide useful descriptions of other aspects of electoral systems.