Axiomatic Theory Of Truth

Traditional substantial theories of truth are not satisfactory, and Tarski et al.’s semantic theories of truth need use more powerful metalanguage. Substantial theories of truth and semantic theories of truth both try to define truth, so they can be classified as definitional theories of truth. Some people begin to believe that the concept of truth is much clearer than the definiens (such as facts in correspondence theory). Many philosophers hope to give a truth theory for a language in this language itself, and Tarski’s undefinability theorem declares the failure of definitional approaches, as definitional theories of truth have to face the regress problem. In the late1980s and early1990s, Friedman, Sheard, Feferman, Cantini and others formally proposed axiomatic theories of truth, which opens up a new research field, and some new theorems and conclusions are obtained.In contrary to definitional theories of truth, axiomatic theories of truth don’t presuppose that truth can be defined but treat truth as a primitive predicate which is governed by certain axioms and rules. Axiomatic theories of truth can give the meaning of the truth predicate of their own languages. Therefore they overcome the regress problem and can reason systematically about the properties of truth. The existing axiomatic theories of truth mostly use first-order Peano arithmetic as the base theory, and they can be divided into typed theories of truth and type-free theories of truth. In typed theories of truth axioms only allow one to prove the truth of sentences not containing the same truth predicate, so typed theories of truth consist of disquotational theory TB, compositional theory TC, hierarchical theory RT<α, etc. In type-free theories of truth axioms allow one to prove the truth of sentences involving the same truth predicate, so type-free theories of truth contain Friedman-Sheard theory, Kripke-Feferman theory, Partial Kripke-Feferman theory, type-free disquotational theory PUTB, determinate theory DT, ect.This dissertation systematically studies axiomatic theories of truth and their application in philosophy. In summary, the main work in the dissertation includes the following four aspects:Firstly, it reviews various mainstream axiomatic theories of truth. For each axiomatic theory of truth, we discuss in detail the background of establishment, formal expression, theorems, derived properties, semantic model and proof-theoretic strength.Secondly, there are deficiencies in existing axiomatic theories of truth, as system PUTB can not prove generalizations about truth and system KF is so strong that it proves unexpectedly liar sentences. So based on these two reasonable improvement requirements this dissertation puts forward two new axiomatic theories of truth TKF system and LKF system. We prove that the two systems are equivalent and give the models of LKF system. On the basis of previous work, we also provide that the proof-theoretic strength of TKF and LKF is equivalent to system RT<ε0and that the truth-theoretic strength of TKF and LKF is weaker than system KF but stronger than system PUTB. The two new systems completely meet above two requirements and have good properties.Thirdly, this dissertation investigates the new progress of deflationary theories of truth. Deflationism believes that truth is an insubstantial concept; deflationism adopts axomatic approaches instead of semantic approaches; deflationism can be seen as philosophical interpretation of axiomatic theories of truth. Contemporary deflationism has two main doctrines:1. Truth is a device of disquotaion that serves expressing generalization.2. Truth theories should be conservative over their base theories. For resolving the trouble that truth is not conservative over mathematics, Horsten proposes inferential deflationism which can be reflected by system PKF. We believe that inferential deflationism has many problems as it abandons the conservativity criterion. In addition, we question inferential deflationism in four respects. On the premise of accepting the conservativity criterion, we finally put forward a new kind of weakened deflationism which reserves that truth is a light notion.Fourthly, this dissertation argues how axiomatic theories of truth treat the liar sentences and appreciates various axiomatic theories of truth by inspecting whether the liar sentences are proved in formal systems. Based on former work, we independently give the detailed proof of most liar sentences. Previous literature does not involve some more complex liar sentences, while specifically from perspective of liar sentences we analyze these more complex sentences and resolve systematically the matter of their formal proofs.