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 ∈ ∏i=1ⁿ
D〈1,a_i〉. 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].