模态系统T的Herbrand定理

On Herbrand Theorem for Modal System T

模态Herbrand定理研究是模态逻辑领域的重要开放问题。不含Barcan公式的模态系统T的Herbrand定理的获得必须完成两个方向的证明工作:首先是证明相对简单的从右到左方向;而对于相对复杂的从左到右方向,则可以通过在加标公式表列系统K的基础上增加特殊规则获得加标公式表列系统T,证明T的可靠性和完全性,并在此基础上完成从左到右方向的证明。同样的方法还可以用来证明模态系统D、K4、S4的Herbrand定理。

Modal Herbrand Theorem is an important open question in the field of modal logic.A Herbrand theorem for modal system T without Barcan formula must be obtained through proving in two directions,i.e.the right-to-left direction,which is relatively easy;and the left-to-right direction,which is relatively complicated and can be done by adding some special rules to the signed formulas tableau system K to obtain a signed formulas tableau system T and verify T＇s soundness and completeness.On the basis of this stage the left-to-right proving can be completed.This procedure also applies to the proving of Herbrand＇s theorems for modal systems D,K4,and S4.