强可能性与弱必然性的逻辑

本文考察了强可能性和弱必然性这两种真势模态的逻辑性质。称一个命题是强可能的,当且仅当它在某个可及的但非现实的可能世界中成立;称一个命题是弱必然的,当且仅当它在所有可及的但非现实的可能世界中都成立。强可能性与弱必然性互为对偶。在表达力上,强可能性算子不同于可能性算子。尽管如此,刻画强可能性和弱必然性的逻辑(简称为强可弱必逻辑)仍然是正规模态逻辑,而且它的极小逻辑在形态上类似于极小正规模态逻辑K。本文主要在形式技术上对强可弱必逻辑作了初步研究:比较了它和模态逻辑以及一阶逻辑在语言表达力上的区别,给出了它的极小和扩张的公理刻画以及必然和弱必然的双模态逻辑的公理刻画。

Logics for Strong Possibility and Weak Necessarity

In this paper, we investigate the logical properties of two alethic modalities: strong possibility and weak necessarity. A proposition is said to be strongly possible if and only if it holds in all accessible but non-actual possible worlds. Weak necessarity is the dual of strong possibility. The expressivity of strong possibility operator is different from possibility operator. Nevertheless, the logics for strong possibility and weak necessarity(LSW for short) are still normal modal logics, and the minimal system for LSW is similar to minimal normal modal logic system K. In this paper, we made a preliminary work on the formal properties of LSW: we compared its expressivity with modal logic and first-order logic, axiomatize LSW over various classes of frames, and axiomatize the bi-modal logics for necessarity and weak necessarity.