On defining necessity in terms of entailment

Studia Logica 38 (2):95 - 104 (1979)
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 C 1.C2 .... Cnp, where each C iis equivalent in E to either pp or ((pp)p)p.
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DOI 10.1007/BF00370435
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