布尔模态的公理化(英文)

On Axiomatization of Boolean Modalities

布尔模态(即模态生成算子补、交、并)涉及到了完全性问题。并模态是模态可定义的,但是补和交都不是。这意味着证明包含这三个模态的逻辑的完全性不是一件简单直接的事情。Gargov和Passy使用复制方法从整体上处理这三个模态,但是,这个方法不适用于强完全性,也不能单独处理这三个模态。本文改进了这个方法,使得改进后的方法更具一般性,并且没有这两个不足之处。

The modality constructors complement, intersection and union, called boolean modalities, raise the issue of completeness. Union is modally definable but neither of the other two is. This means that it is nontrivial to prove the completeness of the logics having them. G. Gargov and S. Passy introduces a way, called copy method, to handle them as a whole. This method does not work for strong completeness and can not separately handle the three modalities either. We in this paper improve this method to a more general one without these disadvantages.