如何从E□M◇O-3广义模态三段论推导出其他20个有效的广义模态三段论

How to Derive the Other 20 Valid Generalized Modal Syllogisms from the Generalized Modal Syllogism E□M◇O-3

首先根据集合论和可能世界语义学给出直言命题和模态直言命题的真值定义,证明广义模态三段论E□M◇O-3的有效性;然后,在广义量词理论和模态逻辑的基础上,充分利用广义量词的内否定与外否定、亚氏量词some与no的对称性、必然模态词□与可能模态词◇之间的对偶关系、后件弱化规则和反三段论推理规则等化归运算,根据E□M◇O-3的有效性,推导出了其他20个有效的广义模态三段论。这一创新性研究不仅为其他种类的三段论(如亚氏三段论、亚氏模态三段论和广义三段论)的可化归性提供了统一的数学研究范式,而且为人工智能的知识表示与知识推理提供了理论支撑。这一研究方法顺应了大数据时代“对自然语言信息的形式化转换的”需求。

Firstly,according to set theory and the possible worlds semantics,the validity of the generalized modal syllogism E□M◇O-3 is proved in line with the truth value definitions of categorical propositions and modal categorical propositions.Then,on the basis of generalized quantifier theory and modal logic,taking advantage of the inner and outer negation of a generalized quantifier,the symmetry of the two Aristotelian quantifiers‘some’and‘no’,the dual relationship between the necessary modality□and the possible modality◇,the subsequent weakening rule,the anti-syllogism inference rules,and other reduction operations,the paper derives the other 20 valid generalized modal syllogisms from the validity of the syllogism E□M◇O-3.This innovative research not only provides a unified mathematical research paradigm for reducible relationships between/among other kinds of syllogisms(such as Aristotelian syllogisms,Aristotelian modal syllogisms,and generalized syllogisms),but also provides theoretical support for knowledge representation and knowledge reasoning in artificial intelligence.This research method conforms to the demand of formal transformation for natural language information in the era of big data.