THE STRENGTH OF TRUTH-THEORIES
THE STRENGTH OF TRUTH-THEORIES
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真理理论的力量
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发表时间：
2013
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通讯作者：
Richard G. Heck
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作者：
Richard G. Heck
塔斯基的经典论文《形式化语言中的真理概念》很好地代表了20世纪30年代的逻辑状态：它既是关于一个人能做什么，也是关于一个人不能做什么。在消极（或“限制”）方面，我们有塔斯基关于真理的不可定义性的著名定理。在积极的（或“建构的”）方面，我们有塔斯基的证明，对于大范围的理论T，有可能以这样一种方式给T添加一个真理理论，即所得到的理论不仅是一致的（如果T是），而且是富有成效的：在它里面，我们可以证明那些元数学结果，真理的概念当时已经被用于这些结果。特别地，如果我们给皮亚诺算术PA增加一个真值理论，也就是说，如果我们增加像“一个合取为真当且仅当它的两个合取为真”这样的公理，等等，那么我们将能够通过下面的论证证明PA是相容的：公理都为真;推理规则保持真值;因此PA的每个定理都为真;但有些句子，如'0 = 1'，是假的;所以有些句子不是PA的定理;所以PA是一致的。因为PA加上一个真值理论证明PA是相容的，所以从哥德尔的第二不完全性定理得出，前者比后者更强。因此，我们很想用这一事实来解释塔斯基在《真理的语义概念》中的著名主张，即元语言必须比对象语言“本质上更丰富”（塔斯基，1944，第354页）。然而，正如我们将看到的，这将把关于表达能力的问题与关于逻辑力量的问题混淆起来。对于集合论语言来说，有可能在元理论中形式化一个实质上足够的真理理论，该元理论尽可能弱，因为它是先验可能的：一个可以在罗宾逊算术中解释的理论。如果是这样，那么
Tarski’s classic paper “The Concept of Truth in Formalized Languages” is nicely representative of the state of logic in the 1930s: It is as much about what one cannot do as it is about what one can do. On the negative (or ‘limitative’) side, we have Tarski’s celebrated theorem on the indefinability of truth. On the positive (or ‘constructive’) side, we have Tarski’s demonstration that, for a large range of theories T , it is possible to add a theory of truth to T in such a way that the resulting theory is not only consistent (if T is) but also fruitful: Within it, we can prove the sorts of meta-mathematical results for which the notion of truth was then already being used. In particular, if we add a theory of truth to Peano arithmetic, PA—if, that is, we add axioms like “A conjunction is true iff both its conjuncts are true”, and so forth—then we will be able to prove that PA is consistent by the following sort of argument: The axioms are all true; the rules of inference preserve truth; hence every theorem of PA is true; but some sentences, such as ‘0 = 1’, are false; so some sentences are not theorems of PA; so PA is consistent. Since PA plus a truth-theory proves that PA is consistent, it follows from Gödel’s second incompleteness theorem that the former is stronger than the latter. It is tempting, therefore, to want to use this fact to interpret Tarski’s famous claim in “The Semantic Conception of Truth” that the metalanguage must be ‘essentially richer’ than the object langauge (Tarski, 1944, p. 354). As we shall see, however, that would be to confuse a question about expressive power with a question about logical strength. It is possible to formalize a materially adequate theory of truth for the language of set-theory in a meta-theory that is as weak as it is a priori possible for it to be: one interpretable in Robinson arithmetic. If so, then