Michael Byrd

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 (p→p)→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 Lp→p, ⊢EL(p→q)→(Lp→Lq), ⊢E p→Lp} if and only if its has the form C1→.C2→ ...→. Cn→p, where each Ciis equivalent in E to either p→p or ((p→p)→p)→p.