命题模态逻辑的模型构造和完备性证明

MODEL CONSTRUCTION AND COMPLETENESS PROOF OF PROPOSITIONAL MODAL LOGIC

模态逻辑是研究必然、可能及其相关概念的逻辑。模态公式的可满足性问题和证明系统的完备性问题是模态逻辑中的两个经典的问题。为了解决这两个问题,提出一个构造模态公式的canonical model的方法。通过这个方法,对于给定模态公式φ,如果φ是可满足的,可以得到φ的一个canonical model;如果φ是不可满足的,可以得到φ的证明。此外,还给出命题模态逻辑完备性的一个构造性证明方法。

Propositional modal logic is the logic studying the necessity,possibility and their correlated concepts. The satisfiability problem of model formula and the proof of system completeness problem are two classic problems in modal logic. To solve these two problems,we propose a method of canonical model to construct the modal formula. By this method,for a given model formula φ,if φ is satisfiable,a canonical model can be obtained for it; but if φ is not satisfiable,a proof of its negation  φ is to be obtained. In addition,in the paper we also give a constructive proof method of the completeness of the propositional modal logic.