Binary connectives functionally complete by themselves in S5 modal logic

This paper answers affirmatively the open question of Massey [1] concerning the existence of binary connectives functionally complete by themselves in two-valued truth tabular logic, i.e. in the modal theory S5. Since {∼, ⊃, ◊} is a functionally complete set of connectives (Massey [1, § 4]), the following definitions show that the binary operator ф, the semantics of which is given below, is functionally complete by itself: It is left to the reader to verify, by means of complete sets of truth tables (see Massey [1, §§ 1 and 3]), that the foregoing definitions are correct.