We prove that the provability logic of all provability predicates is exactly
Fitting, Marek, and Truszczy\'nski'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 derivabiity conditions, all
Rosser's provability predicates, and all Rosser's provability predicates
satisfying D3, respectively.Comment: 34 page