用伽罗瓦联络刻画中间时态逻辑
Characterizing intermediate tense logics in terms of Galois connections
摘要：
我们提出了一个统一的方式来定义每个逻辑${\sf L}$之间的直觉和经典逻辑，相应的中间最小时态逻辑${\sf LK_t}$的中间。这是通过建立两个中间逻辑副本与伽罗瓦连接${\sf LGC}$的融合，然后通过两个Fischer Servi公理将它们的运算符相互连接来完成的。由此产生的系统在这里称为${\sf L2GC{+}FS}$。在直觉逻辑${\sf Int}$和经典逻辑${\sf Cl}$的情况下，注意到${\sf IK_t}$在语法上等价于W. B. Ewald和${\sf Cl2GC{+}FS}$等价于经典最小时态逻辑${\sf K_t}$。这证明将${\sf L2GC{+}FS}$视为任何中间逻辑${\sf L}$的最小${\sf L}$时态逻辑${\sf LK_t}$是合理的。本文定义H2 GC + FS-代数为E.奥洛斯卡和我。雷维茨基对于每一个中间逻辑${\sf L}$，我们证明了${\sf L2GC{+}FS}$的代数完备性及其在${\sf L}$上的保守性。我们证明了${\sf IK}$关于定义在G.菲舍尔·塞尔维我们还证明了一个表示定理，说明每个H2 GC + FS-代数可以嵌入到它的标准${\sf IK}$ -框架的复代数。
英文摘要：
We propose a uniform way of defining for every logic ${\sf L}$ intermediate between intuitionistic and classical logics, the corresponding intermediate minimal tense logic ${\sf LK_t}$. This is done by building the fusion of two copies of intermediate logic with a Galois connection ${\sf LGC}$, and then interlinking their operators by two Fischer Servi axioms. The resulting system is called here ${\sf L2GC{+}FS}$. In the cases of intuitionistic logic ${\sf Int}$ and classical logic ${\sf Cl}$, it is noted that ${\sf Int2GC{+}FS}$ is syntactically equivalent to intuitionistic minimal tense logic ${\sf IK_t}$ by W. B. Ewald and ${\sf Cl2GC{+}FS}$ equals classical minimal tense logic ${\sf K_t}$. This justifies to consider ${\sf L2GC{+}FS}$ as minimal ${\sf L}$-tense logic ${\sf LK_t}$ for any intermediate logic ${\sf L}$. We define H2GC+FS-algebras as expansions of HK1-algebras, introduced by E. Orlowska and I. Rewitzky. For each intermediate logic ${\sf L}$, we show algebraic completeness of ${\sf L2GC{+}FS}$ and its conservativeness over ${\sf L}$. We prove relational completeness of ${\sf Int2GC{+}FS}$ with respect to the models defined on ${\sf IK}$-frames introduced by G. Fischer Servi. We also prove a representation theorem stating that every H2GC+FS-algebra can be embedded into the complex algebra of its canonical ${\sf IK}$-frame.
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