Axiomatic Theories of Truth
Axiomatic Theories of Truth
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真理的公理理论
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发表时间：
2012
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通讯作者：
S. Michael
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文献类型：
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作者：
S. Michael
说一个普通语言的句子是真的，似乎是在断言句子与世界之间的某种关系，但这种关系的确切性质是一个相当有争议的话题--事实上，人们甚至不广泛同意句子是真理承载者的正确选择。面对这些巨大的定义障碍，人们可能会寻找一种方法，转而使用具有形式语义的形式语言。然而，在正式场合，人们很容易想象塔斯基在两个互补的里程碑式的贡献中解决了所有有趣的问题。首先，塔斯基提供了一种完全令人满意的方法，用适当的元语言为给定语言的句子定义真理，这种方法有效地将真理的概念从模糊的知识基础转移到数学精确的阳光下。第二，通过哥德尔式的对说谎者悖论的解释，他证明了一种语言的真理永远不能在语言本身中得到充分的定义。多年来，这些“已解决”的问题在两个主要方面受到了重新审视。一种方法，可以被描述为广义语义，改变了语言的底层逻辑或模型概念，或者两者兼而有之。一种不太广为人知的方法是公理化的;它试图分析创建一个公理化理论的后果，该理论捕捉到元语言中的（显然完全理解的）语言真理概念或语言本身中的（显然不一致的）语言真理概念的某些合适片段。在过去的四分之一世纪，这种公理化的方法已经变成了一个强大的领域，现在是哈尔巴赫的书的主题。作为背景的快速草图，让L是一个一阶语言，包括一些命名句子的机制（哥德尔编号是最知名的方法，但不是唯一的，也不一定是最自然的），并用一个新的一元谓词T来增加语言以创建语言LT。一个问题立刻出现了：我们谈论的是原始语言L的句子还是扩充语言LT的句子？这两种解释都是可行的;它们分别产生了类型化和无类型的真理系统。为了建立一个公理系统，人们选择这些选项之一，然后选择公理和可能的额外推理规则，以捕获真理概念应该如何运作的正常理解的一些元素。大多数公理系统的构建都由两个主要的直觉指导：（i）断言一个句子为真在某种意义上等同于断言句子本身（反引号），（ii）复合句的真值取决于其组成部分的真值（组合性）。
Saying that a sentence of ordinary language is true appears to assert some relation between the sentence and the world, but the exact nature of this relationship is a topic of considerable dispute – indeed it is not even widely agreed that sentences are the correct choice for truthbearers. Facing these massive definitional roadblocks right out of the gate, one might look for a way around, and turn instead to a formal language with a formal semantics. In a formal setting, though, there is a temptation to imagine that all interesting questions were settled by Tarski, in a pair of complementary landmark contributions. First, Tarski provided a wholly satisfactory method to define truth for sentences of a given language in an appropriate metalanguage, an approach which effectively moved the concept of truth away from its murky intellectual underpinnings into the sunlight of mathematical precision. Second, via a Gödel-like rendering of the Liar paradox, he showed that truth for a language can never be adequately defined in the language itself. Over the years, these ‘settled’ questions have come under renewed scrutiny in two main ways. One approach, which can be characterized as broadly semantic, alters either the underlying logic or the notion of model for the language, or both. A less widely known approach is axiomatic; it seeks to analyze the consequences of creating an axiomatized theory which captures some suitable fragment of either the (apparently fully understood) notion of truth for a language in a metalanguage or the (apparently inconsistent) notion of truth for a language within the language itself. In the past quarter century, this axiomatic approach has turned into a robust field, and now is the subject of Halbach’s book. As a quick sketch of the setting, let L be a first-order language, including some mechanism for naming sentences (Gödel numbering being the best known method, but not the only one, nor necessarily the most natural), and augment the language with a new unary predicate T to create the language LT. In the augmented language, the intended interpretation of T(x) is ‘x is a true sentence’. Immediately a question arises: are we talking about sentences of the original language L or of the augmented language LT? Both interpretations are viable; they give rise to systems of typed and type-free truth, respectively. To build an axiomatic system, one chooses one of these options, and then selects axioms and possibly additional rules of inference to capture some elements of a normal understanding of how the concept of truth should operate. The construction of most axiomatic systems are guided by two main intuitions: (i) asserting that a sentence is true is in some sense equivalent to asserting the sentence itself (disquotation), and (ii) the truth value of a compound sentence depends on the truth values of its components (compositionality).