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