公理化真理论
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摘要
传统的实质真理论存在不能令人满意之处,塔斯基等人的语义真理论需要使用更强的元语言。实质真理论和语义真理论都试图给真下定义,因此可被归为定义的真理论。人们开始认为真这个概念本身比其他定义的概念(如符合论中的事实等)更为清楚。而且许多哲学家希望能在一个语言内部给出该语言的真理论,塔斯基的不可定义性定理则宣告了定义方法的失败,因为定义的真理论不可避免地面临着无穷倒退的困境。直到上世纪80年代末和90年代初,才由弗里德曼、希尔德、费弗曼和坎蒂尼等人正式提出了公理化真理论,从而开辟了一个全新的研究领域,并得到了一些新的结论和定理。
与以前的定义真理论不同,公理化真理论不再给真下定义,而把真看作一个原始谓词,并且用一组公理和推理规则来规定它。公理化真理论可以给出自身语言中真谓词的意义,克服了无穷倒退的缺陷,而且可以对真的性质系统地推理。现有的公理化真理论绝大部分是以一阶算术PA系统为基础理论,并且可以分为类型真理论和无类型真理论两大类。类型真理论是指系统的公理仅仅允许证明不包含相同真谓词的句子的真,类型真理论主要有去引号真理论TB、组合真理论TC和阶层真理论RT<α等。无类型真理论则是指系统的公理允许证明包含相同真谓词的句子的真,无类型真理论主要有Friedman-Sheard理论、Kripke-Feferman理论、部分逻辑的Kripke-Feferman理论、无类型去引号真理论PUTB和确定真理论DT等。
本文系统地研究了公理化真理论及其在哲学上的应用,概括来说,主要工作包括如下四个方面：
第一,梳理了国外各个主流的公理化真理论,对每个公理化真理论分别详细阐述了其建立的背景、形式表述、导出定理及性质、语义模型、证明论强度等内容。
第二,现有的公理化真理论存在不足,PUTB系统没有关于真的一般原则,KF系统则过强而证明出了说谎者语句。所以基于这两点合理的改进需求,本文提出了两个新的公理化真理论TKF系统和LKF系统。证明了这两个系统是等价的,给出了LKF系统的模型。并在前人工作的基础上,得出了TKF和LKF系统的证明论强度为RT<ε0,真理论上的强度为弱于KF且强于PUTB。这两个新系统完全满足了开始的两点需求,具有较好的性质,这是本文主要的技术工作。
第三,探讨了紧缩真理论的新进展。紧缩论通常认为真是非实质的概念,并且采用了公理化方法而不是语义方法,紧缩论可以看作是公理化真理论的哲学解释。当代的紧缩论主要有两个核心主旨：1.真作为去引号策略用于概括：2.真应该对基础理论是保守的。为了解决真对数学不保守的困境,霍斯顿提出了推理的紧缩论,并以PKF系统作为其形式表达。本文认为推理的紧缩论由于放弃了保守性准则,存在许多问题,并从四个方面提出了质疑。最后在接受保守性准则的前提下,本文在哲学上提出了一种弱化的紧缩真理论的新进路,保留了真是浅层的概念这一特点。
第四,论证了公理化真理论对说谎者悖论语句的解决,通过考察悖论语句是否在各个系统中被证明来比较和评价不同的公理化真理论。在前人工作的基础上,独立给出了绝大多数悖论语句的详细证明过程。尤其对于一些较为复杂的悖论语句,以前的文献并没有涉及,本文则专门从悖论语句的视角,分析了这些较复杂的悖论语句,并系统地解决了它们的形式证明问题。
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.
与以前的定义真理论不同,公理化真理论不再给真下定义,而把真看作一个原始谓词,并且用一组公理和推理规则来规定它。公理化真理论可以给出自身语言中真谓词的意义,克服了无穷倒退的缺陷,而且可以对真的性质系统地推理。现有的公理化真理论绝大部分是以一阶算术PA系统为基础理论,并且可以分为类型真理论和无类型真理论两大类。类型真理论是指系统的公理仅仅允许证明不包含相同真谓词的句子的真,类型真理论主要有去引号真理论TB、组合真理论TC和阶层真理论RT<α等。无类型真理论则是指系统的公理允许证明包含相同真谓词的句子的真,无类型真理论主要有Friedman-Sheard理论、Kripke-Feferman理论、部分逻辑的Kripke-Feferman理论、无类型去引号真理论PUTB和确定真理论DT等。
本文系统地研究了公理化真理论及其在哲学上的应用,概括来说,主要工作包括如下四个方面：
第一,梳理了国外各个主流的公理化真理论,对每个公理化真理论分别详细阐述了其建立的背景、形式表述、导出定理及性质、语义模型、证明论强度等内容。
第二,现有的公理化真理论存在不足,PUTB系统没有关于真的一般原则,KF系统则过强而证明出了说谎者语句。所以基于这两点合理的改进需求,本文提出了两个新的公理化真理论TKF系统和LKF系统。证明了这两个系统是等价的,给出了LKF系统的模型。并在前人工作的基础上,得出了TKF和LKF系统的证明论强度为RT<ε0,真理论上的强度为弱于KF且强于PUTB。这两个新系统完全满足了开始的两点需求,具有较好的性质,这是本文主要的技术工作。
第三,探讨了紧缩真理论的新进展。紧缩论通常认为真是非实质的概念,并且采用了公理化方法而不是语义方法,紧缩论可以看作是公理化真理论的哲学解释。当代的紧缩论主要有两个核心主旨：1.真作为去引号策略用于概括：2.真应该对基础理论是保守的。为了解决真对数学不保守的困境,霍斯顿提出了推理的紧缩论,并以PKF系统作为其形式表达。本文认为推理的紧缩论由于放弃了保守性准则,存在许多问题,并从四个方面提出了质疑。最后在接受保守性准则的前提下,本文在哲学上提出了一种弱化的紧缩真理论的新进路,保留了真是浅层的概念这一特点。
第四,论证了公理化真理论对说谎者悖论语句的解决,通过考察悖论语句是否在各个系统中被证明来比较和评价不同的公理化真理论。在前人工作的基础上,独立给出了绝大多数悖论语句的详细证明过程。尤其对于一些较为复杂的悖论语句,以前的文献并没有涉及,本文则专门从悖论语句的视角,分析了这些较复杂的悖论语句,并系统地解决了它们的形式证明问题。
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.
引文
① Horsten L. The Tarskian Turn:Deflationism and Axiomatic Truth. Cambridge:MIT Press,2011, pp.11-14.
① Halbach V. Axiomatic Theories of Truth. New York:Cambridge University Press,2011, pp.3-4.
② Feferman S. Axiomatizing Truth:How and Why. Pillars of Truth Conference,2011.
① Horsten L. The Tarskian Turn:Deflationism and Axiomatic Truth. Cambridge:MIT Press,2011, p.65.
① Halbach V. Axiomatic Theories of Truth. New York:Cambridge University Press,2011.
② Horsten L. The Tarskian Turn:Deflationism and Axiomatic Truth. Cambridge:MIT Press,2011.
③ Fujimoto K. Relative Truth Definability of Axiomatic Truth Theories. Bulletin of Symbolic Logic,2010,16(3): pp.305-344.
④ Tarski A. The Concept of Truth in Formalized Languages. In:Logic, Semantics, Metamathematics. Indianapolis: Hackett,1983,2nd ed, pp.152-278.
① Davidson D. Truth and meaning. In:Davidson D, eds. Inquiries into Truth and Interpretation. Oxford:Oxford University Press,1984, pp.17-36.
① Halbach V. Conservative Theories of Classical Truth. Studia Logica,1999,62(3):pp.353-370.
① Feferman S. Reflecting on Incompleteness.Journal of Symbolic Logic,1991,56(1):pp.1-49.
① Halbach V, Horsten L. Axiomatizing Kripke's Theory of Truth. Journal of Symbolic Logic,2006,71(2): pp.677-712.
② Feferman S. Axioms for Determinateness and Truth. Review of Symbolic Logic,2008,1(2):pp.204-217.
① Friedman H, Sheard M. An Axiomatic Approach to Self-Referential Truth. Annals of Pure and Applied Logic, 1987,33(1):pp.1-21.
② 这里的FS系统得到哈尔巴赫的简化。
③ Gupta A. Truth and Paradox. Journal of Philosophical Logic,1982,11:pp.1-60.
① Halbach V. A System of Complete and Consistent Truth. Notre Dame Journal of Formal Logic,1994,35(3): pp.311-327.
① Kripke S. Outline of a Theory of Truth. The Journal of Philosophy,1975,72(19):pp.690-716.
② Feferman S. Reflecting on Incompleteness. Journal of Symbolic Logic,1991,56(1):pp.1-49.
① 为了详细阐述KF系统的模型,这里的KF暂不包含CONS公理。
② Cantini A. Notes on Formal Theories of Truth. Mathematical Logic Quarterly,1989,35(2):pp.97-130.
① 参见Gentzen G The Collected Papers of Gerhard Gentzen. Amsterdam:Horth-Holland,1969.
② Feferman S. Reflecting on Incompleteness. Journal of Symbolic Logic,1991,56(1):pp.1-49.
③ Reinhardt W. Some Remarks on Extending and Interpreting Theories with a Partial Predicate for Truth. Journal of Philosophical Logic,1986,15(2):pp.219-251.
① Halbach V, Horsten L. Axiomatizing Kripke's Theory of Truth. Journal of Symbolic Logic,2006,71(2): pp.677-712.
① Cantini A. Notes on Formal Theories of Truth. Mathematical Logic Quarterly,1989,35(2):pp.97-130.
② Cantini A. Notes on Formal Theories of Truth. Mathematical Logic Quarterly,1989,35(2):pp.97-130.
① McGee V. Maximal Consistent Sets of Instances of Tarski's Schema (T). Journal of Philosophical Logic,1992, 21(3):pp.235-241.
② McGee V. Maximal Consistent Sets of Instances of Tarski's Schema (T). Journal of Philosophical Logic,1992, 21(3):pp.235-241.
① Cieslinski C. Deflationism, Conservativeness and Maximality. Journal of Philosophical Logic,2007,36(6): pp.695-705.
② Cieslinski C. Deflationism, Conservativeness and Maximality. Journal of Philosophical Logic,2007,36(6): pp.695-705.
① Halbach V. Reducing Compositional to Disquotational Truth. Review of Symbolic Logic,2009,2(4): pp.786-798.
① Feferman S. Axioms for Determinateness and Truth. Review of Symbolic Logic,2008,1(2):pp.204-217.
① Fujimoto K. Relative Truth Definability of Axiomatic Truth Theories. Bulletin of Symbolic Logic,2010,16(3): p.319.
① Fujimoto K. Relative Truth Definability of Axiomatic Truth Theories. Bulletin of Symbolic Logic,2010,16(3): p.319.
② Fujimoto K. Relative Truth Definability of Axiomatic Truth Theories. Bulletin of Symbolic Logic,2010,16(3): pp.305-344.
① 上述详细证明过程参见Halbach V. Axiomatic Theories of Truth. New York:Cambridge University Press, 2011, pp.218-224.
① 上述详细证明过程参见Halbach V. Axiomatic Theories of Truth. New York:Cambridge University Press, 2011, pp.280-282.
① 参见Halbach V. Axiomatic Theories of Truth. New York:Cambridge University Press,2011, pp.283-284.
① Horsten L. The Tarskian Turn:Deflationism and Axiomatic Truth. Cambridge:MIT Press,2011, p.20.
② Halbach V. Axiomatic Theories of Truth. New York:Cambridge University Press,2011, p.306.
① Field H. Critical Notice:Paul Horwich's "Truth". Philosophy of Science,1992,59:p.322.
② Halbach V, Horsten L. Principles of Truth. Hanselhohenhausen,2002.
① Horsten L. The Semantical Paradoxes, the Neutrality of Truth, and the Neutrality of the Minimalist Theory of Truth, in:Cortois P, eds. The Many Problems of Realism. Tilburg University Press,1995, pp.173-187.
② Shapiro S. Proof and Truth:Through Thick and Thin. The Journal of Philosophy,1998,95(10):pp.497-498.
① Halbach V. How Innocent is Deflationism?. Synthese,2001,126(1):pp.167-194.
① Field H. Deflating the Conservativeness Argument. The Journal of Philosophy,1999,96(10):pp.533-540.
② 参见Horsten L. The Tarskian Turn:Deflationism and Axiomatic Truth. Cambridge:MIT Press,2011, p.83.
① Halbach V. How Innocent is Deflationism?. Synthese,2001,126(1):pp.187-188.
② Horsten L. The Tarskian Turn:Deflationism and Axiomatic Truth. Cambridge:MIT Press,2011, pp.92-93.
① 霍斯顿这里引用的PKF系统使用了自然演绎的推理规则,与第三章论述的矢列演算略有不同,但两者是等价的。
② Horsten L. The Tarskian Turn:Deflationism and Axiomatic Truth. Cambridge:MIT Press,2011, pp.143-145.
① Horsten L. The Tarskian Turn:Deflationism and Axiomatic Truth. Cambridge:MIT Press,2011, pp.146-147.
① Gupta A. A Critique of Deflationism. In:Blackburn S, Simmons K, eds. Truth. Oxford University Press,1999, pp.282-307.
② 本节相关内容将发表在,刘大为,李娜.真理论的转向：由定义到公理化.哲学研究,2013(5)。
① Horsten L. The Tarskian Turn:Deflationism and Axiomatic Truth. Cambridge:MIT Press,2011, p.151.
② Horsten L. The Tarskian Turn:Deflationism and Axiomatic Truth. Cambridge:MIT Press,2011, pp.147-148.
③ Horsten L. The Tarskian Turn:Deflationism and Axiomatic Truth. Cambridge:MIT Press,2011, p.143.
① Halbach V, Horsten L. Axiomatizing Kripke's Theory of Truth. Journal of Symbolic Logic,2006,71(2):p.682.
① Horsten L. The Tarskian Turn:Deflationism and Axiomatic Truth. Cambridge:MIT Press,2011, p.112.
① Halbach V. Axiomatic Theories of Truth. New York:Cambridge University Press,2011, p.160.
① Halbach V, Horsten L. Axiomatizing Kripke's Theory of Truth. Journal of Symbolic Logic,2006,71(2):p.682.
② Halbach V. Axiomatic Theories of Truth. Stanford Encyclopedia of Philosophy,2009, URL: http://stanford.library.usyd.edu.au/entries/truth-axiomatic.
① Reinhardt W. Some Remarks on Extending and Interpreting Theories with a Partial Predicate for Truth. Journal of Philosophical Logic,1986,15(2):pp.219-251.
① Halbach V. Axiomatic Theories of Truth. New York:Cambridge University Press,2011, pp.241-242.
① Feferman S. Axioms for Determinateness and Truth. Review of Symbolic Logic,2008,1(2):p.209.
① 该定理的证明过程由笔者补充给出。
② Feferman S. Axioms for Determinateness and Truth. Review of Symbolic Logic,2008, 1(2):p.213.
① Feferman S. Axioms for Determinateness and Truth. Review of Symbolic Logic,2008,1 (2):p.213.
① Halbach V. Axiomatic Theories of Truth. New York:Cambridge University Press,2011, pp.278-279.
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[17]Soames S. Understanding truth. New York:Oxford University Press,1999.
[18]Takeuti G. Proof Theory. Amsterdam:North Holland,1987,2nd ed.
[1]李娜,刘大为.公理化真理论研究述评.哲学动态,2012(8)：91-95.
[2]刘大为,李娜.真理论的转向：由定义到公理化.哲学研究,2013(5).
[3]刘大为.公理化真理论与悖论语句.待发.
[4]Apostoli P. The analytic conception of truth and the foundations of arithmetic. Journal of Symbolic Logic,2000,65(1):33-102.
[5]Barrio E. Theories of Truth without Standard Models and Yablo's Sequences. Studia Logica, 2010,96(3):375-391.
[6]Bays T. Beth's Theorem and Deflationism. Mind,2009,118(472):1061-1073.
[7]Beall J C, Glanzberg M. Where the Paths Meet:Remarks on Truth and Paradox. Truth and Its Deformities,2008,32:169-198.
[8]Burgess H P. Friedman and the axiomatization of Kripke's theory of truth. Conference Paper, 2009.
[9]Cantini A. Notes on Formal Theories of Truth. Mathematical Logic Quarterly,1989,35(2): 97-130.
[10]Cantini A. A theory of formal truth arithmetically equivalent to ID1. Journal of Symbolic Logic,1990,55(1):244-259.
[11]Cantini A. Extending the first-order theory of combinators with self-referential truth. Journal of Symbolic Logic,1993,58(2):477-513.
[12]Cieslinski C. Deflationism, Conservativeness and Maximality. Journal of Philosophical Logic,2007,36(6):695-705.
[13]Cieslinski C. Truth, Conservativeness, and Provability. Mind,2010,119(474):409-422.
[14]Cieslinski C. T-equivalences for positive sentences. Review of Symbolic Logic,2011,4(2): 319-325.
[18]Davidson D. The Folly of Trying to Define Truth. The Journal of Philosophy,1996,93(6): 263-278.
[15]Feferman S. Toward Useful Type-Free Theories.1. Journal of Symbolic Logic,1984,49(1): 75-111.
[16]Feferman S. Hilbert's Program Relativized:Proof-Theoretical and Foundational Reductions. The Journal of Symbolic Logic,1988,53(2):364-384.
[17]Feferman S. Reflecting on incompleteness. Journal of Symbolic Logic,1991,56(1):1-49.
[18]Feferman S. Axioms for determinateness and truth. Review of Symbolic Logic,2008,1(2): 204-217.
[19]Feferman S. Axiomatizing truth:how and why. Pillars of Truth Conference,2011.
[20]Field H. Critical Notice:Paul Horwich's "Truth". Philosophy of Science,1992,59:321-330.
[21]Field H. Deflationist Views of Meaning and Content. Mind,1994,103(411):249-285.
[22]Field H. Deflating the Conservativeness Argument. The Journal of Philosophy,1999,96(10): 533-540.
[23]Field H. A revenge-immune solution to the semantic paradoxes. Journal of Philosophical Logic,2003,32(2):139-177.
[24]Fischer M. Minimal truth and interpretability. Review of Symbolic Logic,2009,2(4): 799-815.
[25]Friedman H, Sheard M. An axiomatic approach to self-referential truth. Annals of Pure and Applied Logic,1987,33(1):1-21.
[26]Friedman H, Sheard M. The disjunction and existence properties for axiomatic systems of truth. Annals of Pure and Applied Logic,1988,40(1):1-10.
[27]Fujimoto K. Relative truth definability of axiomatic truth theories. Bulletin of Symbolic Logic,2010,16(3):305-344.
[28]Fujimoto K. Autonomous progression and transfinite iteration of self-applicable truth. Journal of Symbolic Logic,2011,76(3):914-945.
[29]Glanzberg M. Discussion-Truth, disquotation, and expression-On McGinn's theory of truth. Philosophical Studies,2004,118(3):413-423.
[30]Gupta A. Truth and Paradox. Journal of Philosophical Logic,1982,11:pp.1-60.
[31]Halbach V. A System of Complete and Consistent Truth. Notre Dame Journal of Formal Logic,1994,35(3):311-327.
[32]Halbach V. Conservative Theories of Classical Truth. Studia Logica,1999,62(3):353-370.
[33]Halbach V. Disquotationalism and infinite conjunctions. Mind,1999,108(429):1-22.
[34]Halbach V. How Innocent Is Deflationism? Synthese,2001,126(1):167-194.
[35]Halbach V. Reducing Compositional To Disquotational Truth. Review of Symbolic Logic, 2009,2(4):786-798.
[36]Halbach V. Axiomatic Theories of Truth. Stanford Encyclopedia of Philosophy,2009, URL: http://stanford.library.usyd.edu.au/entries/truth-axiomatic.
[37]Halbach V, Horsten L. Axiomatizing Kripke's theory of truth. Journal of Symbolic Logic, 2006,71(2):677-712.
[38]Heck R G. Truth and Disquotation. Synthese,2004,142(3):317-352.
[39]Horsten L. Levity. Mind,2009,118(471):555-581.
[40]Horsten L. Truth and paradox. Network PPT,2011:
[41]Hyttinen T, Sandu G. Deflationism and Arithmetical Truth. Dialectica,2004,58(3):413-426.
[42]Ketland J. Deflationism and Tarski's Paradise. Mind,1999,108(429):69-94.
[43]Ketland J. Beth's Theorem and Deflationism-Reply to Bays. Mind,2009,118(472): 1075-1079.
[44]Ketland J. Truth, Conservativeness, and Provability:Reply to Cieslinski. Mind,2010, 119(474):423-436.
[45]Kreisel G, Levy A. Reflection Principles and their Use for Establishing the Complexity of Axiomatic Systems. Mathematical Logic Quarterly,1968,14(7-12):97-142.
[46]Kremer M. Kripke and the logic of truth. Journal of Philosophical Logic,1988,17(3): 225-278.
[47]Kripke S. Outline of a Theory of Truth. The Journal of Philosophy,1975,72(19):690-716.
[48]Leigh G E, Rathjen M. An ordinal analysis for theories of self-referential truth. Archive for Mathematical Logic,2010,49(2):213-247.
[49]Leitgeb H. Theories of Truth Which Have No Standard Models. Studia Logica,2001,68(1): 69-87.
[50]Leitgeb H. What Theories of Truth Should be Like (but Cannot be). Philosophy Compass, 2007,2(2):276-290.
[51]McDonald B E. On meaningfulness and truth. Journal of Philosophical Logic,2000,29(5): 433-482.
[52]McGee V. How truthlike can a predicate be? A negative result. Journal of Philosophical Logic,1985,14(4):399-410.
[53]McGee V. Maximal Consistent Sets of Instances of Tarski's Schema (T). Journal of Philosophical Logic,1992,21(3):235-241.
[54]Reinhardt W N. Remarks on significance and meaningful applicability. Mathematical logic and formal systems. Lecture Notes in Pure and Applied Mathematics,1985:
[55]Reinhardt W N. Some Remarks on Extending and Interpreting Theories with a Partial Predicate for Truth. Journal of Philosophical Logic,1986,15(2):219-251.
[56]Shapiro S. Proof and Truth:Through Thick and Thin. The Journal of Philosophy,1998, 95(10):493-521.
[57]Sheard M. A Guide to Truth Predicates in the Modern Era. Journal of Symbolic Logic,1994, 59(3):1032-1054.
[58]Sheard M. Weak and Strong Theories of Truth. Studia Logica,2001,68(1):89-101.
[59]Tarski A. The Semantic Conception of Truth:and the Foundations of Semantics. Philosophy and Phenomenological Research,1944,4(3):341-376.
[60]Tarski A. The Concept of Truth in Formalized Languages. In:Logic, Semantics, Metamathematics. Indianapolis:Hackett,1983,2nd ed,152-278.
[61]Tennant N. Deflationism and the Godel Phenomena. Mind,2002,111(443):551-582.
[62]Tennant N. Deflationism and the Godel Phenomena:Reply to Cieslinski. Mind,2005, 114(453):75-88.
[63]Yablo S. Grounding, Dependence, and Paradox. Journal of Philosophical Logic,1982,11: 117-137.
① Halbach V. Axiomatic Theories of Truth. New York:Cambridge University Press,2011, pp.3-4.
② Feferman S. Axiomatizing Truth:How and Why. Pillars of Truth Conference,2011.
① Horsten L. The Tarskian Turn:Deflationism and Axiomatic Truth. Cambridge:MIT Press,2011, p.65.
① Halbach V. Axiomatic Theories of Truth. New York:Cambridge University Press,2011.
② Horsten L. The Tarskian Turn:Deflationism and Axiomatic Truth. Cambridge:MIT Press,2011.
③ Fujimoto K. Relative Truth Definability of Axiomatic Truth Theories. Bulletin of Symbolic Logic,2010,16(3): pp.305-344.
④ Tarski A. The Concept of Truth in Formalized Languages. In:Logic, Semantics, Metamathematics. Indianapolis: Hackett,1983,2nd ed, pp.152-278.
① Davidson D. Truth and meaning. In:Davidson D, eds. Inquiries into Truth and Interpretation. Oxford:Oxford University Press,1984, pp.17-36.
① Halbach V. Conservative Theories of Classical Truth. Studia Logica,1999,62(3):pp.353-370.
① Feferman S. Reflecting on Incompleteness.Journal of Symbolic Logic,1991,56(1):pp.1-49.
① Halbach V, Horsten L. Axiomatizing Kripke's Theory of Truth. Journal of Symbolic Logic,2006,71(2): pp.677-712.
② Feferman S. Axioms for Determinateness and Truth. Review of Symbolic Logic,2008,1(2):pp.204-217.
① Friedman H, Sheard M. An Axiomatic Approach to Self-Referential Truth. Annals of Pure and Applied Logic, 1987,33(1):pp.1-21.
② 这里的FS系统得到哈尔巴赫的简化。
③ Gupta A. Truth and Paradox. Journal of Philosophical Logic,1982,11:pp.1-60.
① Halbach V. A System of Complete and Consistent Truth. Notre Dame Journal of Formal Logic,1994,35(3): pp.311-327.
① Kripke S. Outline of a Theory of Truth. The Journal of Philosophy,1975,72(19):pp.690-716.
② Feferman S. Reflecting on Incompleteness. Journal of Symbolic Logic,1991,56(1):pp.1-49.
① 为了详细阐述KF系统的模型,这里的KF暂不包含CONS公理。
② Cantini A. Notes on Formal Theories of Truth. Mathematical Logic Quarterly,1989,35(2):pp.97-130.
① 参见Gentzen G The Collected Papers of Gerhard Gentzen. Amsterdam:Horth-Holland,1969.
② Feferman S. Reflecting on Incompleteness. Journal of Symbolic Logic,1991,56(1):pp.1-49.
③ Reinhardt W. Some Remarks on Extending and Interpreting Theories with a Partial Predicate for Truth. Journal of Philosophical Logic,1986,15(2):pp.219-251.
① Halbach V, Horsten L. Axiomatizing Kripke's Theory of Truth. Journal of Symbolic Logic,2006,71(2): pp.677-712.
① Cantini A. Notes on Formal Theories of Truth. Mathematical Logic Quarterly,1989,35(2):pp.97-130.
② Cantini A. Notes on Formal Theories of Truth. Mathematical Logic Quarterly,1989,35(2):pp.97-130.
① McGee V. Maximal Consistent Sets of Instances of Tarski's Schema (T). Journal of Philosophical Logic,1992, 21(3):pp.235-241.
② McGee V. Maximal Consistent Sets of Instances of Tarski's Schema (T). Journal of Philosophical Logic,1992, 21(3):pp.235-241.
① Cieslinski C. Deflationism, Conservativeness and Maximality. Journal of Philosophical Logic,2007,36(6): pp.695-705.
② Cieslinski C. Deflationism, Conservativeness and Maximality. Journal of Philosophical Logic,2007,36(6): pp.695-705.
① Halbach V. Reducing Compositional to Disquotational Truth. Review of Symbolic Logic,2009,2(4): pp.786-798.
① Feferman S. Axioms for Determinateness and Truth. Review of Symbolic Logic,2008,1(2):pp.204-217.
① Fujimoto K. Relative Truth Definability of Axiomatic Truth Theories. Bulletin of Symbolic Logic,2010,16(3): p.319.
① Fujimoto K. Relative Truth Definability of Axiomatic Truth Theories. Bulletin of Symbolic Logic,2010,16(3): p.319.
② Fujimoto K. Relative Truth Definability of Axiomatic Truth Theories. Bulletin of Symbolic Logic,2010,16(3): pp.305-344.
① 上述详细证明过程参见Halbach V. Axiomatic Theories of Truth. New York:Cambridge University Press, 2011, pp.218-224.
① 上述详细证明过程参见Halbach V. Axiomatic Theories of Truth. New York:Cambridge University Press, 2011, pp.280-282.
① 参见Halbach V. Axiomatic Theories of Truth. New York:Cambridge University Press,2011, pp.283-284.
① Horsten L. The Tarskian Turn:Deflationism and Axiomatic Truth. Cambridge:MIT Press,2011, p.20.
② Halbach V. Axiomatic Theories of Truth. New York:Cambridge University Press,2011, p.306.
① Field H. Critical Notice:Paul Horwich's "Truth". Philosophy of Science,1992,59:p.322.
② Halbach V, Horsten L. Principles of Truth. Hanselhohenhausen,2002.
① Horsten L. The Semantical Paradoxes, the Neutrality of Truth, and the Neutrality of the Minimalist Theory of Truth, in:Cortois P, eds. The Many Problems of Realism. Tilburg University Press,1995, pp.173-187.
② Shapiro S. Proof and Truth:Through Thick and Thin. The Journal of Philosophy,1998,95(10):pp.497-498.
① Halbach V. How Innocent is Deflationism?. Synthese,2001,126(1):pp.167-194.
① Field H. Deflating the Conservativeness Argument. The Journal of Philosophy,1999,96(10):pp.533-540.
② 参见Horsten L. The Tarskian Turn:Deflationism and Axiomatic Truth. Cambridge:MIT Press,2011, p.83.
① Halbach V. How Innocent is Deflationism?. Synthese,2001,126(1):pp.187-188.
② Horsten L. The Tarskian Turn:Deflationism and Axiomatic Truth. Cambridge:MIT Press,2011, pp.92-93.
① 霍斯顿这里引用的PKF系统使用了自然演绎的推理规则,与第三章论述的矢列演算略有不同,但两者是等价的。
② Horsten L. The Tarskian Turn:Deflationism and Axiomatic Truth. Cambridge:MIT Press,2011, pp.143-145.
① Horsten L. The Tarskian Turn:Deflationism and Axiomatic Truth. Cambridge:MIT Press,2011, pp.146-147.
① Gupta A. A Critique of Deflationism. In:Blackburn S, Simmons K, eds. Truth. Oxford University Press,1999, pp.282-307.
② 本节相关内容将发表在,刘大为,李娜.真理论的转向：由定义到公理化.哲学研究,2013(5)。
① Horsten L. The Tarskian Turn:Deflationism and Axiomatic Truth. Cambridge:MIT Press,2011, p.151.
② Horsten L. The Tarskian Turn:Deflationism and Axiomatic Truth. Cambridge:MIT Press,2011, pp.147-148.
③ Horsten L. The Tarskian Turn:Deflationism and Axiomatic Truth. Cambridge:MIT Press,2011, p.143.
① Halbach V, Horsten L. Axiomatizing Kripke's Theory of Truth. Journal of Symbolic Logic,2006,71(2):p.682.
① Horsten L. The Tarskian Turn:Deflationism and Axiomatic Truth. Cambridge:MIT Press,2011, p.112.
① Halbach V. Axiomatic Theories of Truth. New York:Cambridge University Press,2011, p.160.
① Halbach V, Horsten L. Axiomatizing Kripke's Theory of Truth. Journal of Symbolic Logic,2006,71(2):p.682.
② Halbach V. Axiomatic Theories of Truth. Stanford Encyclopedia of Philosophy,2009, URL: http://stanford.library.usyd.edu.au/entries/truth-axiomatic.
① Reinhardt W. Some Remarks on Extending and Interpreting Theories with a Partial Predicate for Truth. Journal of Philosophical Logic,1986,15(2):pp.219-251.
① Halbach V. Axiomatic Theories of Truth. New York:Cambridge University Press,2011, pp.241-242.
① Feferman S. Axioms for Determinateness and Truth. Review of Symbolic Logic,2008,1(2):p.209.
① 该定理的证明过程由笔者补充给出。
② Feferman S. Axioms for Determinateness and Truth. Review of Symbolic Logic,2008, 1(2):p.213.
① Feferman S. Axioms for Determinateness and Truth. Review of Symbolic Logic,2008,1 (2):p.213.
① Halbach V. Axiomatic Theories of Truth. New York:Cambridge University Press,2011, pp.278-279.
[1]Beall J C, Armour-Garb B. Deflationism and Paradox. Oxford:Clarendon Press,2005.
[2]Blackburn S, Simmons K. Truth. Oxford:Oxford University Press,1999.
[3]Cantini A. Logical frameworks for truth and abstraction:an axiomatic study. Amsterdam: Elsevier Science B V,1996.
[4]Cortois P. The Many Problems of Realism. Tilburg University Press,1995.
[5]Davidson D. Inquiries into Truth and Interpretation. Oxford:Oxford University Press,1984.
[6]Field H. Saving truth from paradox. Oxford:Oxford University Press,2008.
[7]Gentzen G The Collected Papers of Gerhard Gentzen. Amsterdam:Horth-Holland,1969.
[8]Halbach V. Axiomatic Theories of Truth. New York:Cambridge University Press,2011.
[9]Halbach V, Horsten L. Principles of Truth. Hanselhohenhausen,2002.
[10]Hinman P. Recursion Theoretic Hierarchies. Berlin:Springer,2004.
[11]Horsten L. The Tarskian Turn:Deflationism and Axiomatic Truth. Cambridge:MIT Press, 2011.
[12]Horwich P. Truth. Oxford:Basil Blackwell,1990.
[13]Kunnr W. Conceptions of Truth. Oxford:Oxford University Press,2003.
[14]Maudlin T. Truth and paradox:solving the riddles. Oxford:Clarendon Press,2004.
[15]McGee V. Truth, Vagueness and Paradox:An Essay on the Logic of Truth. Indianapolis: Hackett,1991.
[16]Quine W V O. Philosophy of Logic. Cambridge:Harvard University Press,1970.
[17]Soames S. Understanding truth. New York:Oxford University Press,1999.
[18]Takeuti G. Proof Theory. Amsterdam:North Holland,1987,2nd ed.
[1]李娜,刘大为.公理化真理论研究述评.哲学动态,2012(8)：91-95.
[2]刘大为,李娜.真理论的转向：由定义到公理化.哲学研究,2013(5).
[3]刘大为.公理化真理论与悖论语句.待发.
[4]Apostoli P. The analytic conception of truth and the foundations of arithmetic. Journal of Symbolic Logic,2000,65(1):33-102.
[5]Barrio E. Theories of Truth without Standard Models and Yablo's Sequences. Studia Logica, 2010,96(3):375-391.
[6]Bays T. Beth's Theorem and Deflationism. Mind,2009,118(472):1061-1073.
[7]Beall J C, Glanzberg M. Where the Paths Meet:Remarks on Truth and Paradox. Truth and Its Deformities,2008,32:169-198.
[8]Burgess H P. Friedman and the axiomatization of Kripke's theory of truth. Conference Paper, 2009.
[9]Cantini A. Notes on Formal Theories of Truth. Mathematical Logic Quarterly,1989,35(2): 97-130.
[10]Cantini A. A theory of formal truth arithmetically equivalent to ID1. Journal of Symbolic Logic,1990,55(1):244-259.
[11]Cantini A. Extending the first-order theory of combinators with self-referential truth. Journal of Symbolic Logic,1993,58(2):477-513.
[12]Cieslinski C. Deflationism, Conservativeness and Maximality. Journal of Philosophical Logic,2007,36(6):695-705.
[13]Cieslinski C. Truth, Conservativeness, and Provability. Mind,2010,119(474):409-422.
[14]Cieslinski C. T-equivalences for positive sentences. Review of Symbolic Logic,2011,4(2): 319-325.
[18]Davidson D. The Folly of Trying to Define Truth. The Journal of Philosophy,1996,93(6): 263-278.
[15]Feferman S. Toward Useful Type-Free Theories.1. Journal of Symbolic Logic,1984,49(1): 75-111.
[16]Feferman S. Hilbert's Program Relativized:Proof-Theoretical and Foundational Reductions. The Journal of Symbolic Logic,1988,53(2):364-384.
[17]Feferman S. Reflecting on incompleteness. Journal of Symbolic Logic,1991,56(1):1-49.
[18]Feferman S. Axioms for determinateness and truth. Review of Symbolic Logic,2008,1(2): 204-217.
[19]Feferman S. Axiomatizing truth:how and why. Pillars of Truth Conference,2011.
[20]Field H. Critical Notice:Paul Horwich's "Truth". Philosophy of Science,1992,59:321-330.
[21]Field H. Deflationist Views of Meaning and Content. Mind,1994,103(411):249-285.
[22]Field H. Deflating the Conservativeness Argument. The Journal of Philosophy,1999,96(10): 533-540.
[23]Field H. A revenge-immune solution to the semantic paradoxes. Journal of Philosophical Logic,2003,32(2):139-177.
[24]Fischer M. Minimal truth and interpretability. Review of Symbolic Logic,2009,2(4): 799-815.
[25]Friedman H, Sheard M. An axiomatic approach to self-referential truth. Annals of Pure and Applied Logic,1987,33(1):1-21.
[26]Friedman H, Sheard M. The disjunction and existence properties for axiomatic systems of truth. Annals of Pure and Applied Logic,1988,40(1):1-10.
[27]Fujimoto K. Relative truth definability of axiomatic truth theories. Bulletin of Symbolic Logic,2010,16(3):305-344.
[28]Fujimoto K. Autonomous progression and transfinite iteration of self-applicable truth. Journal of Symbolic Logic,2011,76(3):914-945.
[29]Glanzberg M. Discussion-Truth, disquotation, and expression-On McGinn's theory of truth. Philosophical Studies,2004,118(3):413-423.
[30]Gupta A. Truth and Paradox. Journal of Philosophical Logic,1982,11:pp.1-60.
[31]Halbach V. A System of Complete and Consistent Truth. Notre Dame Journal of Formal Logic,1994,35(3):311-327.
[32]Halbach V. Conservative Theories of Classical Truth. Studia Logica,1999,62(3):353-370.
[33]Halbach V. Disquotationalism and infinite conjunctions. Mind,1999,108(429):1-22.
[34]Halbach V. How Innocent Is Deflationism? Synthese,2001,126(1):167-194.
[35]Halbach V. Reducing Compositional To Disquotational Truth. Review of Symbolic Logic, 2009,2(4):786-798.
[36]Halbach V. Axiomatic Theories of Truth. Stanford Encyclopedia of Philosophy,2009, URL: http://stanford.library.usyd.edu.au/entries/truth-axiomatic.
[37]Halbach V, Horsten L. Axiomatizing Kripke's theory of truth. Journal of Symbolic Logic, 2006,71(2):677-712.
[38]Heck R G. Truth and Disquotation. Synthese,2004,142(3):317-352.
[39]Horsten L. Levity. Mind,2009,118(471):555-581.
[40]Horsten L. Truth and paradox. Network PPT,2011:
[41]Hyttinen T, Sandu G. Deflationism and Arithmetical Truth. Dialectica,2004,58(3):413-426.
[42]Ketland J. Deflationism and Tarski's Paradise. Mind,1999,108(429):69-94.
[43]Ketland J. Beth's Theorem and Deflationism-Reply to Bays. Mind,2009,118(472): 1075-1079.
[44]Ketland J. Truth, Conservativeness, and Provability:Reply to Cieslinski. Mind,2010, 119(474):423-436.
[45]Kreisel G, Levy A. Reflection Principles and their Use for Establishing the Complexity of Axiomatic Systems. Mathematical Logic Quarterly,1968,14(7-12):97-142.
[46]Kremer M. Kripke and the logic of truth. Journal of Philosophical Logic,1988,17(3): 225-278.
[47]Kripke S. Outline of a Theory of Truth. The Journal of Philosophy,1975,72(19):690-716.
[48]Leigh G E, Rathjen M. An ordinal analysis for theories of self-referential truth. Archive for Mathematical Logic,2010,49(2):213-247.
[49]Leitgeb H. Theories of Truth Which Have No Standard Models. Studia Logica,2001,68(1): 69-87.
[50]Leitgeb H. What Theories of Truth Should be Like (but Cannot be). Philosophy Compass, 2007,2(2):276-290.
[51]McDonald B E. On meaningfulness and truth. Journal of Philosophical Logic,2000,29(5): 433-482.
[52]McGee V. How truthlike can a predicate be? A negative result. Journal of Philosophical Logic,1985,14(4):399-410.
[53]McGee V. Maximal Consistent Sets of Instances of Tarski's Schema (T). Journal of Philosophical Logic,1992,21(3):235-241.
[54]Reinhardt W N. Remarks on significance and meaningful applicability. Mathematical logic and formal systems. Lecture Notes in Pure and Applied Mathematics,1985:
[55]Reinhardt W N. Some Remarks on Extending and Interpreting Theories with a Partial Predicate for Truth. Journal of Philosophical Logic,1986,15(2):219-251.
[56]Shapiro S. Proof and Truth:Through Thick and Thin. The Journal of Philosophy,1998, 95(10):493-521.
[57]Sheard M. A Guide to Truth Predicates in the Modern Era. Journal of Symbolic Logic,1994, 59(3):1032-1054.
[58]Sheard M. Weak and Strong Theories of Truth. Studia Logica,2001,68(1):89-101.
[59]Tarski A. The Semantic Conception of Truth:and the Foundations of Semantics. Philosophy and Phenomenological Research,1944,4(3):341-376.
[60]Tarski A. The Concept of Truth in Formalized Languages. In:Logic, Semantics, Metamathematics. Indianapolis:Hackett,1983,2nd ed,152-278.
[61]Tennant N. Deflationism and the Godel Phenomena. Mind,2002,111(443):551-582.
[62]Tennant N. Deflationism and the Godel Phenomena:Reply to Cieslinski. Mind,2005, 114(453):75-88.
[63]Yablo S. Grounding, Dependence, and Paradox. Journal of Philosophical Logic,1982,11: 117-137.