On defining necessity in terms of entailment

In their book Entailment, Anderson and Belnap investigate the consequences of defining Lp (it is necessary that p) in system E as (pp)p. Since not all theorems are equivalent in E, this raises the question of whether there are reasonable alternative definitions of necessity in E. In this paper, it is shown that a definition of necessity in E satisfies the conditions {_E Lpp, _EL(pq)(LpLq), _E pLp} if and only if its has the form C₁.C₂ .... C_np, where each C_iis equivalent in E to either pp or ((pp)p)p.