Finite bases of admissible rules for the logic S5₂ C

Abstract

We study bimodal logic system S5₂ C having two modal operators □₀ and □₁, each of which satisfies the axioms of S5 and in addition, an axiom for commutability of modal operators: □₀□₁ p]≡□₁□₀ p. The main result of this paper establishes that the bimodal logic S5₂ C and all its extensions have finite bases for admissible inference rules. We also show that even though the logic S5₂ C is not locally finite, any proper extension of S5₂ C is already locally finite. Moreover, the universal theory of the free algebra of any S5₂ C-logic is decidable. It is shown also that any S5₂ C-logic λ with the adjoined inference rule

is structurally complete and that logic has the same set of theorems as the logic λ.

The research was supported by Grant Center of Novosibirsk State University and RFFI grant 96-01-00228