论文标题
单调模态逻辑的算术完整定理
Arithmetical completeness theorems for monotonic modal logics
论文作者
论文摘要
我们研究了Provibaly Predicates $ \ Mathrm {pr} _t（x）$满足以下条件的模态逻辑方面： $ \ MATHBF {M} $：如果$ t \vdashφ\toψ$，则$ t \ vdash \ Mathrm {pr} _t（\ulcornerφ\ urcorner）\ to \ to \ mathrm {pr} 我们证明了单调模态逻辑$ \ Mathsf {Mn} $，$ \ Mathsf {Mn4} $，$ \ Mathsf {Mnp} $，$ \ Mathsf {mnp4} $，以及$ \ nathsf {Mathsf {Mathsf {Mathsf {Mathssf {Mathssf {Mathsf {Mathsf {Mathssf） $ \ mathbf {m} $。也就是说，我们证明，对于其中的每种逻辑$ l $，都存在$σ_1$可供应性谓词$ \ mathrm {pr} _t（x）$满足$ \ mathbf {m Mathbf {m} $，从而使$ \ mathrm {pr} _t（x）$ l $ l $ l $ $ l $ $ l $。特别是，模态公式$ \ MATHRM {p} $：$ \ neg \ box \ bot $和$ \ mathrm {d} $：$ \ neg（\ box a \ box a \ land \ box \ box \ neg a）$与非正态逻辑相比，对应于两个不同的正式$ \ neg $ \ neg \ n = 1 prm \ prm} \ urcorner）$和$ \ neg \ big（\ mathrm {pr} _t（\ulcornerφ\ urcorner）\ land \ mathrm {pr} _t（\ ulcorner \ neg或neg或φ\ urcorner）\ neg或neg或我们的结果从模态逻辑方面将这些形式化分开。
We investigate modal logical aspects of provability predicates $\mathrm{Pr}_T(x)$ satisfying the following condition: $\mathbf{M}$: If $T \vdash φ\to ψ$, then $T \vdash \mathrm{Pr}_T(\ulcorner φ\urcorner) \to \mathrm{Pr}_T(\ulcorner ψ\urcorner)$. We prove the arithmetical completeness theorems for monotonic modal logics $\mathsf{MN}$, $\mathsf{MN4}$, $\mathsf{MNP}$, $\mathsf{MNP4}$, and $\mathsf{MND}$ with respect to provability predicates satisfying the condition $\mathbf{M}$. That is, we prove that for each logic $L$ of them, there exists a $Σ_1$ provability predicate $\mathrm{Pr}_T(x)$ satisfying $\mathbf{M}$ such that the provability logic of $\mathrm{Pr}_T(x)$ is exactly $L$. In particular, the modal formulas $\mathrm{P}$: $\neg \Box \bot$ and $\mathrm{D}$: $\neg (\Box A \land \Box \neg A)$ are not equivalent over non-normal modal logic and correspond to two different formalizations $\neg \mathrm{Pr}_T(\ulcorner 0=1 \urcorner)$ and $\neg \big(\mathrm{Pr}_T(\ulcorner φ\urcorner) \land \mathrm{Pr}_T(\ulcorner \neg φ\urcorner) \bigr)$ of consistency statements, respectively. Our results separate these formalizations in terms of modal logic.
