Arithmetical completeness theorems for monotonic modal logics

We investigate modal logical aspects of provability predicates PrT(x) satisfying the following condition: M: If T⊢φ→ψ, then T⊢PrT(⌜φ⌝)→PrT(⌜ψ⌝). prove arithmetical completeness theorems for monotonic logics MN, MN4, MNP, MNP4, and MND with respect to condition M. That is, we that each logic L them, there exists a Σ1 predicate M such is exactly L. In particular, formulas P: ¬□⊥ D: ¬(□A∧□¬A) are not equivalent over non-normal correspond two different formalizations ¬PrT(⌜0=1⌝) ¬(PrT(⌜φ⌝)∧PrT(⌜¬φ⌝)) consistency statements, respectively. Our results separate these in terms logic.