β Scribed by Dennis Henry; Michael Byrd
No coin nor oath required. For personal study only.
In their book Entailment, Anderson and Belnap investigate the consequences of defining r (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 { FE/~P -> P, ~E ~ (P --> q) -~ (Lp-+ Lq), qEp-->Lp} if and only if its has the form C~--->. C~ ... --->. Cn~'p, where each Ci is equivalent in E to either :p->p or ((p->p)--->p)~p.
In their book Entailment, [1] in Anderson and Belnap propose to define necessity in the system E in terms of entailment, as follows: D1.
LA -~ (A~A)~A.
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