AI 中文总结
该研究通过为模态公式构建内部克里普克语义，将模态集合论与布尔值模型相连，计算模态有效性并分析特殊代数行为，最终证明在特定语义下正规逻辑\(\KTB\)是相关模型中有效模态公式集。
AI 中文摘要
我们通过为原子命题为集合论语句的模态公式发展一种“内部”克里普克语义，将模态集合论与布尔值模型联系起来。给定一个完备布尔代数\(B\)，我们将其元素视为布尔值全域\(V^{(B)}\)内“关于真的局部视角”，并使用由共一致性（等价地，布尔兼容性）定义的\(B\)上的可达关系\(R\)来解释模态算子：\(aRb\)当且仅当\(a\wedge b\neq 0\)。我们计算了几个一般的和依赖于代数的模态有效性，并分析了完备原子布尔代数的特殊行为。最后，在非零部分\(B^+=B\setminus\{0\}\)上采用基于翻译的语义，我们证明了一个可靠性和完全性定理：正规逻辑\(\KTB\)恰好是在所有带参数的翻译共一致性模型中有效的模态公式集。
英文摘要
We connect modal set theory with Boolean-valued models by developing an \emph{internal} Kripke semantics for modal formulas whose atomic propositions are set-theoretic sentences. Given a complete Boolean algebra $B$, we view its elements as ``local perspectives on truth'' inside the Boolean-valued universe $V^{(B)}$ and interpret the modal operators using an accessibility relation $R$ on $B$ defined by \emph{co-consistency} (equivalently, Boolean compatibility): $aRb$ iff $a\wedge b\neq 0$. Our central conceptual point is that, for set-theoretic sentences $p$, the internal modality $\Diamond p$ holds at $b$ iff there is an ultrafilter $U$ of $B$ containing $b$ such that the classical quotient $V^{(B)}/U$ satisfies $p$. We compute several general and algebra-dependent modal validities, and analyze the special behavior of complete atomic Boolean algebras. Finally, adopting a translation-based semantics on the nonzero part $B^+=B\setminus\{0\}$, we prove a soundness-and-completeness theorem: the normal logic $\KTB$ is exactly the set of modal formulas valid in all translated co-consistency models with parameters.
