萨奎斯特逻辑的格

The Lattice of Sahlqvist Logics

以萨奎斯特公式为额外公理添加到极小正规逻辑K上得到的逻辑都是完全的。这样得到的逻辑被称为萨奎斯特逻辑。所有的萨奎斯特逻辑组成了一个格。这个格中有可数无穷长的链以及可数无穷长的反链,格中的每个逻辑相对于格的不完全度是1。另外,萨奎斯特逻辑格的子格E具有规整的结构。

Every Sahlqvist formula is an elementary complete one. If we take Sahlqvist formulae as addition axioms to the minimal normal modal logic K, we get a strong elementary logic, namely Sahlqvist logic. In this paper we discuss properties of the class of Sahlqvist logics S. It is closed under intersection and another set operation that can be looked upon as logical union, so that S forms a lattice. Let E denote the class of logics K finmn. It is easy to determine that E is a subclass of S. We can deduce that every K finmn is a canonical logic, following from the Sahlqvist theorem. We can also get the same result by studying properties of the canonical frame of K finmn. E is an interesting subclass of S. It is a sublattice of S. It has a regular geometric form, in which there are infinite length chains, but every antichain in E is finite. Besides, the well known tabular logics are closely related to E, for every tabular logic is an extension of some K finmm. From the result that E has infinite length chains, we can easily know that S also has infinite length antichain. We can prove that in the class {K ◇i□p→□◇^ip │ i 〈 ω}, every logic is not contained in others, so that this class is an infinite length antichain of S, which is different from the result of E. The incomplete degree is an important concept when we study logical lattices. Usually, it is not easy to study the incomplete degree problems. However, we can easily conclude that the incomplete degree of Sahlqvist logic respect to the lattice S is 1.