I prove that the Boolean Prime Ideal Theorem is equivalent, under some weak set-theoretic assumptions, to
what I will call the Cut-for-Formulas to Cut-for-Sets Theorem: for a set F and a binary relation |- on Power(F),
if |- is finitary, monotonic, and satisfies cut for formulas, then it also satisfies cut for sets. I deduce the CF/CS
Theorem from the Ultrafilter Theorem twice; each proof uses a different order-theoretic variant of the Tukey-
Teichmüller Lemma. I then discuss relationships between various cut-conditions in the absence of finitariness
or of monotonicity.