The provability logic of all provability predicates

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We prove that the provability logic of all provability predicates is exactly Fitting, Marek, and Truszczyński’s pure logic of necessitation N⁠. Moreover, we introduce three extensions N4⁠, NR and NR4 of N and investigate the arithmetical semantics of these logics. In fact, we prove that N4⁠, NR and NR4 are the provability logics of all provability predicates satisfying the third condition D3 of the derivability conditions, all Rosser provability predicates and all Rosser provability predicates satisfying D3⁠, respectively.

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