A Sahlqvist theorem for distributive modal logic
A Sahlqvist theorem for distributive modal logic
复制标题
分配模态逻辑的 Sahlqvist 定理
DOI：
10.1016/j.apal.2004.04.007
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发表时间：
2005
期刊：
影响因子：
--
通讯作者：
Y. Venema
中科院分区：
文献类型：
--
作者：
M. Gehrke;Hideo Nagahashi;Y. Venema
在本文中，我们考虑分配模态逻辑，在这种设置中，我们可以在由连接、析取、真和假给出的经典命题逻辑片段中添加模态，例如经典模态类型以及弱否定形式。对于这些逻辑，我们既以分布模态代数的形式定义代数语义，又以有序Kripke结构的形式定义关系语义。本文的主要贡献在于将Sahlqvist公理的概念推广到我们的广义集合，并证明了由Sahlqvist公理公理化的分布模态逻辑的对应性和正则性结果。我们对对应结果的证明依赖于对经典情况的还原，但是我们的正则性证明脱离了传统的形式，使用了正则扩展的新扩展代数理论。
In this paper we consider distributive modal logic, a setting in which we may add modalities, such as classical types of modalities as well as weak forms of negation, to the fragment of classical propositional logic given by conjunction, disjunction, true, and false. For these logics we define both algebraic semantics, in the form of distributive modal algebras, and relational semantics, in the form of ordered Kripke structures. The main contributions of this paper lie in extending the notion of Sahlqvist axioms to our generalized setting and proving both a correspondence and a canonicity result for distributive modal logics axiomatized by Sahlqvist axioms. Our proof of the correspondence result relies on a reduction to the classical case, but our canonicity proof departs from the traditional style and uses the newly extended algebraic theory of canonical extensions.