概率逻辑系统是与集合代数同态的布尔代数

A probabilistic logic system as a Boolean algebra homomorphic with set algebra

联结词的本质是命题的运算,只有对所有命题都适用的真值函数才能用于定义联结词.概率逻辑中由于命题的内涵相关性,任何[0,1]上的函数都不能完全适用于任意命题的运算,概率逻辑的联结词不能定义成真值函数.各种算子可以作为一种计算方法使用和研究,但不能代表一个逻辑系统研究系统的性质.概率逻辑系统是概率空间的逻辑表示,是与概率空间中的事件域(集合代数)同态的布尔代数.用事件域上的集合函数精确定义各种联结词,与经典二值逻辑相容,与事实相符,能够在经典逻辑框架内实现概率命题演算.

Connectives are essentially operations on propositions,and only the true value functions applicable to all propositions can be used to define connectives.In probabilistic logic,any function on is not completely applicable for the operation on all propositions,and the connectives of probabilistic propositional logic cannot be defined as a true value function because of propositional relativity in connotation.Every operator may be discussed and employed as a method of calculation,but not as a logic system.A probabilistic propositional logic system is the logical description of a probabilistic space,and is a Boolean algebra homomorphic with set algebra that is the event domain in the probabilistic space.All connectives which are compatible with those in classical two-valued logic and which accord with fact can be defined exactly by set functions on event domains.The classical formal system of propositional calculus is completely applicable to probabilistic propositional calculus.