This paper is concerned with Fisher's tests for index numbers. In particular, uniqueness and inconsistency theorems are proved. Beyond that, Fisher's system of tests is weakened considerably. Without any regularity assumption (such as differentiability or continuity) it is shown that every subset of the system of weakened tests is consistent while the whole system is inconsistent. The question of how far the whole system must be weakened to obtain a consistent set of tests is also considered.