The Fixed Point Method And An Investigation Of Dynamic Truth Theory

What is truth?The question has existed for a long time.However,a systemat-ic and rigorous investigation had not existed until the emergence of the model theory.Alfred Tarski offered a semantics for language and characterized Truth with models.Since then,model characterization of Truth became the most important approach of investigating into Truth.Model characterization means constructing a model and in which there exists a consistent interpretation.Besides consistency,model characteri-zation should satisfy our intuition of Truth.Alfred Tarski was the first logician to characterize the truth semantically with rigor.He suggested a criterion of characterizing Truth and proposed a theory that the concept of truth with respect to a language does not belong to that language but to a richer language.Most researchers accept the first but criticize the latter.The root of criticisms is that his theory is static.It does not characterize Truth with respect to a language dynamically,but characterize Truth outside that language.Saul Kripke recommends that we should find a model through the fixed point method,and after him,many theories by model characterization have come to exist.The method of Kripke is totally different with Tarski' s.The fixed point method constructs a sequence of models and this sequence find a characterization of truth dy-namically.In the sequence Kripke choose the(least)fixed point to characterize Truth,and the characterization is both static and dynamical.The minimal fixed point of Krip-ke has some nice properties and can be justified philosophically by the intuition of" groundness",but the minimal fixed point theory is deficient from both internal and external perspectives.Many logicians and philosophers try their best to do justice to the minimal fixed point theory,but their efforts are unsuccessful.The root of failure lies in the paracompleteness of the theory.For this reason,the minimal fixed point theory is also called naive paracomplete theory.Because of the paracompleteness of the fixed point theory,Gupta proposes the revision method by changing the fixed point theory.The revision method is similar to the fixed point theory.The revision method also constructs a sequence of models to characterize Truth.The essence of the revision method is the same as the fixed point method.The core notion of the revision method is the revision rule.The philosophical justification of the revision rule seems easy,but actually there are many difficulties.Besides,because of the uncertainty of revision rule in limit rule,there exists various revision theories.By summarizing and analyzing three nice revision theories,Hartry Field concludes that in the revision theory of truth the rules of truth do not preserve truth.Besides,the law of exclude middle holds in revision theory,but there exists a sentence ? such that ??(?)? hods but the? and(?)? does not hold,to some extent this is counter intuition.By improving the revision method and combing methods of fixed point and re-vision,Hartry Field proposes the theory with nice conditionals,which combines the advantages of the fixed point theory and revision theory.Field constructs a sequence of models,and in the sequence there exists an acceptable model which is similar to the fixed point model.The T-schema holds in the acceptable model and Truth is transpar-ent.Many intuitively true sentences which are not true in the minimal fixed point model are true in the acceptable model.Besides,by introducing the determinate operator,it overcomes the paracompleteness of the minimal fixed point theory.However,Field's truth theory is essentially paracomplete and his theory is named 'advanced paracom-plete theory'.One of the main problems of model characterization of Truth is the Revenge of the Liar,the Revenge problem concerns the limits of model characterization of Truth.By analyzing the actual language and the model language we can see that the model language cannot reflect the actual language perfectly.The above truth theory is formed in classical set theory,there also exists truth theories in non-classical set theory,such as the situation theory of John Barwise.Although there exists huge difference in philo-sophical interpretation between situation theory and classical theory.I assume that in formal construction situation theory is similar to the above truth theory.The critic of situation theory to the fixed point theory is a critic of the paracompleteness of the minimal fixed point theory.The difference lies in its metalanguage,situation uses the Azel set theory which enables the construction of self-referential proposition rather than the Godel-code.In sum,the situation theory reconstructs the fixed point theory with non-well founded set theory.The truth theories of Kripke,Gupta and Field are the most important truth theories of the recent 40 years,to some extent,the situation theory is related to the fixed point theory.Therefore,if the revision method is considered as a variant of the fixed point method and the philosophical justification of each theory is ignored,the claim that the method of fixed point dominates the model characterization of truth may not be an overstatement.The major task of this essay is analyzing the development of truth theory in the last 40 years.By a deep analysis of the Kripke's minimal fixed point,Gupta's revision theory and Field's truth theory,this essay thoroughly compares and contrasts these truth theories' methods,and consequently this essay reveals that the model character-ization of Truth is a dynamical process of constructing model sequences.In a model sequence,as the sequence is put forward,the characterization of Truth becomes better and eventually arrives at a stable state.The process of finding a model is dynamical and stable,and by the 'dynamical' finding,the process arrives at a stable state.