The paper deals with pp formulas in the language of reduced special groups, and the question of when the validity of a pp formula on each finite subspace of a space of orderings implies its global validity [18]. A large new class of pp formulas is introduced for which this is always the case, assuming the space of orderings in question has finite stability index. The paper also considers pp formulas of the special type $b\in \Pi _{i=1}^{n}\,D\langle 1,a_{i}\rangle $. Formulas (...) of this type with n = 3 are the simplest sort of pp formula not covered by the result, and are also the source of the recent counterexamples in [9] and [19]. (shrink)