A Study On Russell’s Theory Of Types

At the end of the 19 th century,mathematics,based on Cantor’s set theory,entered a period of rapid development.But with the discovery of set theory paradoxes,especially Russell’s paradox.They caused an uproar in the mathematics academic word,directly led to the third mathematical crisis.At that time,Russell was working on the relationship between logic and mathematics,that is,logicism.In order to realize logicism and solve the paradox of set theory,Russell constructed the theory of types.Russell type theory is one of the earliest solution to the paradox of set theory.It solves the paradox by distinguishing the logical types of meaningful classes or propositions.Different types of logic cannot be confused,otherwise,it is meaningless.Proposition,meaning and type are the basic concepts of type theory.Due to the difference of publication time and resolving method,type theory is divided into simple type theory and branch type theory.The solution of simple type theory is class stratification,that is,class hierarchy is determined to avoid paradox.On the basis of the principle of solving the contradiction of the vicious circle,the branch type theory solves the paradox by means of propositional order and propositional functional order.The former is to determine the order and logical type of propositions by means of propositional variables so as to avoid logical contradictions.The latter is to determine the logical type of proposition by the order of propositional functions on the basis of the order of proposition,so as to solve the paradox of Set theory.The branch type theory can analyze all property propositions and relation propositions,which makes up for the limited research object of simple type theory.Russell type theory not only solves the paradox of set theory,but also has an important influence on the development of logic and philosophy.In logic,the type theory plays an active role in maintaining traditional binary logic,solving semantic paradox and developing logicism.In philosophy,it promotes the rise and development of analytical philosophy and linguistic philosophy.It also has limitations,mainly embodied in the controversial "vicious circle principle" and "reducibility axiom".