The concept of implicative pseudo valuations on hoops is introduced and some related properties are investigated. As an application of properties of pseudo valuations, we find the relationship between a pseudo valuati...The concept of implicative pseudo valuations on hoops is introduced and some related properties are investigated. As an application of properties of pseudo valuations, we find the relationship between a pseudo valuation and an implicative pseudo valuation and obtain some characterizations of implicative pseudo valuations. In particular, we show that a pseudo valuation on regular hoops is implicative if and only if it satisfies φ(x■x') = 0.This result will provide a more general algebraic foundation for pseudo valuations theory on algebraic structures based on substructure logic.展开更多
Valuation spaces with respect to diverse implication operators are investigated in a unified way where the Lebesgue measure is a commonly used measure, and it is proved that all the logic formulas are measurable funct...Valuation spaces with respect to diverse implication operators are investigated in a unified way where the Lebesgue measure is a commonly used measure, and it is proved that all the logic formulas are measurable functions with respect to popularly used implication operations. The concept of t-(α-tautology) is introduced and rules of generalized modus ponens (MP) and hypothetic syllogism (HS) are established in the sense of semantics. The concept of truth degree of a logic formula is introduced and rules of integral MP and integral HS are proposed. Finally, a kind of pseudo-metric is introduced to the set consisting of all logic formulas by establishing a universal logical metric space, making it possible to develop a new type of approximate reasoning arise.展开更多
基金Supported by a grant of National Natural Science Foundation of China（11571281）
文摘The concept of implicative pseudo valuations on hoops is introduced and some related properties are investigated. As an application of properties of pseudo valuations, we find the relationship between a pseudo valuation and an implicative pseudo valuation and obtain some characterizations of implicative pseudo valuations. In particular, we show that a pseudo valuation on regular hoops is implicative if and only if it satisfies φ(x■x') = 0.This result will provide a more general algebraic foundation for pseudo valuations theory on algebraic structures based on substructure logic.
文摘Valuation spaces with respect to diverse implication operators are investigated in a unified way where the Lebesgue measure is a commonly used measure, and it is proved that all the logic formulas are measurable functions with respect to popularly used implication operations. The concept of t-(α-tautology) is introduced and rules of generalized modus ponens (MP) and hypothetic syllogism (HS) are established in the sense of semantics. The concept of truth degree of a logic formula is introduced and rules of integral MP and integral HS are proposed. Finally, a kind of pseudo-metric is introduced to the set consisting of all logic formulas by establishing a universal logical metric space, making it possible to develop a new type of approximate reasoning arise.
