Tarski's Definition Of Truth In The Calculus Of Classes

In this article,firstly,I will systemize the process in which Tarski defined the truth in the calculus of classes,and will study deeply some problems of the process.Secondly,I will prove some characters of Tarski's definition of truth ,which could indicate the perfection of this definition.Finaly,I will compare different methods to define the truth in the calculus of classes.Tarski thought that a satisfactory definition of truth should satisfy two conditions: (1)It is formally correct; (2)It is materially adequate.The correctness condition require that this definition should comply with common laws of definition.The adequacy condition require that all the equivalence of the form (T) should are logical consequence of this definition.Tarski found that defining the truth of a language under this language would make it semantically closed.Because the investigation of natural language had shown that we could construct successfully semantical antinomy in any semantically closed language,Tarski thought that the definition of truth in a fomalized language could be only constructed in its metalanguage.This is the famous thoughts of language-delamination.Tarski constructed strictly the calculus of classes and its metalanguage,then defined some important concepts of the calculus of classes,such as sentential function,free variable, sentence,operation,consequence.After analyzing the structure of sentences,Tarski found that, it was impossible to define the truth of the calculus of classes directly,but making use of the structure character of sentential functions could give us the approach to escape the dilemma:Firstly,we could defined the concept ofsatisfaction in sentential functions by means of recursive method;Then,we could regard sentences as a peculiar form of sentential functions;Finally,with the help of the concept of satisfaction,we could defined successfully the truth of the calculus of classes.Tarski's definition of truth of the calculus of classes is: x is true if and only if x is a sentence of the calculus of classes and every infinite sequence of classes satisfies x .This definition satisfy completely the correctness condition and adequacy condition,and the class of true sentences from it is a consistent and complete deductive system.Such characters make us believe that this definition is a perfect one.Nowadays,if we define the concept of truth in calculus of classes,we usually introduce the concept of interpretation. After defining interpretation,adopting not only the concept of sequence but also the concept of evaluation,we could get the perfect definition of truth. Although there is much difference between the two methods and the method by means of which Tarski defined the truth,the former are essentially mends of the latter.The language of calculus of classes is a sort of the language of first-order,and the language of first-order is the foundation of all fomalized languages,so Tarski contributed greatly to logical semantics.