  Prior and Meredith stated in their 1956 paper that the distribution principle \(\Box(p \rightarrow q) \rightarrow  (\Box p \rightarrow \Box q)\) is a theorem in the absence of any special axioms for \(U\); that \(\Box p \rightarrow p\) is a theorem if axiom 1 is imposed; that axiom 2 gives the S4 principle \(\Box p \rightarrow \Box \Box p\); and that 2 together with 3 give the S5 principle \(\Diamond \Box p \rightarrow \Box p\). Their approach was proof-theoretic in its basic orientation and they offered no completeness results. In (1962a) and (1962b) Prior extended this approach to systems between S4 and S5, and to systems independent of S4 between T and S5.