This paper concerns the elimination of higher type quantifiers and gives two theorems. The
first theorem shows that quantifiers in formulae of a specific form can be eliminated. The
second theorem shows that quantifiers in formulae of a similar form cannot be eliminated,
that is, such formulae do not have an equivalent first-order formula. The proof is based on
the Ehrenfeucht game. These theorems are important for design of an interpreter of a ν act,
which is a representation of mathematical action. Moreover, even if the universe is assumed
to be finite, these theorems hold.