塔斯基的真理定义解析 | 语义构想与形式构建_塔尔斯基的形式语言的真理论-优快云博客

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Tarski’s Truth Definitions

塔斯基的真理定义

First published Sat Nov 10, 2001; substantive revision Wed Sep 21, 2022

In 1933 the Polish logician Alfred Tarski published a paper in which he discussed the criteria that a definition of ‘true sentence’ should meet, and gave examples of several such definitions for particular formal languages. In 1956 he and his colleague Robert Vaught published a revision of one of the 1933 truth definitions, to serve as a truth definition for model-theoretic languages. This entry will simply review the definitions and make no attempt to explore the implications of Tarski’s work for semantics (natural language or programming languages) or for the philosophical study of truth. (For those implications, see the entries on truth and Alfred Tarski.)

1933 年，波兰逻辑学家阿尔弗雷德・塔斯基（Alfred Tarski）发表了一篇论文，在文中他讨论了 “真语句” 定义应满足的标准，并给出了针对特定形式语言的几个此类定义的示例。1956 年，他和同事罗伯特・沃特（Robert Vaught）发表了对 1933 年其中一个真理定义的修订版，用作模型论语言的真理定义。本文将仅回顾这些定义，不探讨塔斯基的工作对语义学（自然语言或编程语言）或对真理的哲学研究的影响。

1. The 1933 programme and the semantic conception

1933 年的计划与语义学概念

In the late 1920s Alfred Tarski embarked on a project to give rigorous definitions for notions useful in scientific methodology. In 1933 he published (in Polish) his analysis of the notion of a true sentence. This long paper undertook two tasks: first to say what should count as a satisfactory definition of ‘true sentence’ for a given formal language, and second to show that there do exist satisfactory definitions of ‘true sentence’ for a range of formal languages. We begin with the first task; Section 2 will consider the second.

20 世纪 20 年代后期，阿尔弗雷德・塔斯基着手开展一个项目，旨在为科学方法论中有用的概念给出严格定义。1933 年，他用波兰语发表了对 “真语句” 概念的分析。这篇长篇论文承担了两项任务：第一，说明对于给定的形式语言，什么应被视为 “真语句” 的令人满意的定义；第二，表明对于一系列形式语言，确实存在 “真语句” 的令人满意的定义。我们从第一项任务开始；第 2 节将讨论第二项任务。

We say that a language is fully interpreted if all its sentences have meanings that make them either true or false. All the languages that Tarski considered in the 1933 paper were fully interpreted, with one exception described in Section 2.2 below. This was the main difference between the 1933 definition and the later model-theoretic definition of 1956, which we shall examine in Section 3.

如果一种语言的所有句子都具有使其要么为真要么为假的意义，我们就说这种语言是完全解释的。塔斯基在 1933 年的论文中考虑的所有语言都是完全解释的，只有下文 2.2 节中描述的一个例外。这是 1933 年的定义与 1956 年后来的模型论定义之间的主要区别，我们将在第 3 节中探讨后者。

Tarski described several conditions that a satisfactory definition of truth should meet.

塔斯基描述了一个令人满意的真理定义应满足的几个条件。

1.1 Object language and metalanguage

1.1 对象语言和元语言

If the language under discussion (the object language) is $L$ , then the definition should be given in another language known as the metalanguage, call it $M$ . The metalanguage should contain a copy of the object language (so that anything one can say in $L$ can be said in $M$ too), and $M$ should also be able to talk about the sentences of $L$ and their syntax. Finally Tarski allowed $M$ to contain notions from set theory, and a 1-ary predicate symbol True with the intended reading ‘is a true sentence of $L$ ’. The main purpose of the metalanguage was to formalise what was being said about the object language, and so Tarski also required that the metalanguage should carry with it a set of axioms expressing everything that one needs to assume for purposes of defining and justifying the truth definition. The truth definition itself was to be a definition of True in terms of the other expressions of the metalanguage. So the definition was to be in terms of syntax, set theory and the notions expressible in $L$ , but not semantic notions like ‘denote’ or ‘mean’ (unless the object language happened to contain these notions).

如果正在讨论的语言（对象语言）是 $L$ ，那么定义应该在另一种称为元语言的语言中给出，将其称为 $M$ 。元语言应该包含对象语言的一个副本（这样在 $L$ 中能说的任何内容在 $M$ 中也能说），并且 $M$ 还应该能够谈论 $L$ 的句子及其句法。最后，塔斯基允许 $M$ 包含集合论中的概念，以及一个一元谓词符号 “True”，其预期含义是 “是 $L$ 的一个真语句”。元语言的主要目的是将关于对象语言所说的内容形式化，因此塔斯基还要求元语言应带有一组公理，这些公理表达了为定义和证明真理定义所需假设的一切内容。真理定义本身将是根据元语言的其他表达式对 “True” 的定义。所以这个定义将依据句法、集合论以及在 $L$ 中可表达的概念，而不是像 “指示” 或 “意指” 这样的语义概念（除非对象语言碰巧包含这些概念）。

Tarski assumed, in the manner of his time, that the object language $L$ and the metalanguage $M$ would be languages of some kind of higher order logic. Today it is more usual to take some kind of informal set theory as one’s metalanguage; this would affect a few details of Tarski’s paper but not its main thrust. Also today it is usual to define syntax in set-theoretic terms, so that for example a string of letters becomes a sequence. In fact one must use a set-theoretic syntax if one wants to work with an object language that has uncountably many symbols, as model theorists have done freely for over half a century now.

按照他那个时代的方式，塔斯基假设对象语言 $L$ 和元语言 $M$ 将是某种高阶逻辑语言。如今，更常见的是采用某种非形式化的集合论作为元语言；这会影响塔斯基论文中的一些细节，但不会影响其主旨。如今，通常也用集合论的术语来定义句法，例如，一串字母被视为一个序列。事实上，如果想要处理具有不可数多个符号的对象语言（正如模型论者在过去半个多世纪里自由地做的那样），就必须使用集合论的句法。

1.2 Formal correctness

1.2 形式正确性

The definition of True should be ‘formally correct’. This means that it should be a sentence of the form

“True” 的定义应该是 “形式上正确的”。这意味着它应该是如下形式的句子：

For all $x$ , True $(x)$ if and only if $\phi (x)$ ,

对于所有 $x$ ，True $(x)$ 当且仅当 $\phi (x)$ ，

where True never occurs in $\phi$ ; or failing this, that the definition should be provably equivalent to a sentence of this form. The equivalence must be provable using axioms of the metalanguage that don’t contain True. Definitions of the kind displayed above are usually called explicit, though Tarski in 1933 called them normal.

其中 “True” 不在 $\phi$ 中出现；或者，如果不是这种形式，该定义应该可证等价于这种形式的句子。这种等价性必须使用不包含 “True” 的元语言公理来证明。上面展示的这种定义通常被称为显式定义，尽管塔斯基在 1933 年将它们称为正规定义。

1.3 Material adequacy

1.3 实质适当性

The definition should be ‘materially adequate’ (trafny – a better translation would be ‘accurate’). This means that the objects satisfying $\phi$ should be exactly the objects that we would intuitively count as being true sentences of $L$ , and that this fact should be provable from the axioms of the metalanguage. At first sight this is a paradoxical requirement: if we can prove what Tarski asks for, just from the axioms of the metalanguage, then we must already have a materially adequate formalisation of ‘true sentence of $L$ ’ within the metalanguage, suggesting an infinite regress. In fact Tarski escapes the paradox by using (in general) infinitely many sentences of $M$ to express truth, namely all the sentences of the form

这个定义应该是 “实质适当的”（“trafny”，更准确的翻译应该是 “准确的”）。这意味着满足 $\phi$ 的对象应该恰好是我们直观上认为是 $L$ 的真语句的那些对象，并且这个事实应该可以从元语言的公理中证明。乍一看，这是一个自相矛盾的要求：如果我们仅从元语言的公理就能证明塔斯基所要求的内容，那么我们在元语言中肯定已经有了对 “ $L$ 的真语句” 的实质适当的形式化，这就暗示了一种无穷倒退。实际上，塔斯基通过（通常）使用 $M$ 中的无穷多个句子来表达真理从而避开了这个悖论，即所有如下形式的句子：

$\phi (s)$ if and only if $\psi$

$\phi (s)$ 当且仅当 $\psi$

whenever $s$ is the name of a sentence $S$ of $L$ and $\psi$ is the copy of $S$ in the metalanguage. So the technical problem is to find a single formula $\phi$ that allows us to deduce all these sentences from the axioms of $M$ ; this formula $\phi$ will serve to give the explicit definition of True.

每当 $s$ 是 $L$ 的一个句子 $S$ 的名称，并且 $\psi$ 是 $S$ 在元语言中的副本时。所以技术问题在于找到一个单一的公式 $\phi$ ，使得我们能够从 $M$ 的公理中推导出所有这些句子；这个公式 $\phi$ 将用于给出 “True” 的显式定义。

Tarski’s own name for this criterion of material adequacy was Convention $T$ . More generally his name for his approach to defining truth, using this criterion, was the semantic conception of truth.

塔斯基将这个实质适当性标准称为约定 $T$ 。更一般地说，他将使用这个标准来定义真理的方法称为真理的语义学概念。

As Tarski himself emphasised, Convention $T$ rapidly leads to the liar paradox if the language $L$ has enough resources to talk about its own semantics. (See the entry on the revision theory of truth.) Tarski’s own conclusion was that a truth definition for a language $L$ has to be given in a metalanguage which is essentially stronger than $L$ .

正如塔斯基自己所强调的，如果语言 $L$ 有足够的资源来谈论自身的语义，那么约定 $T$ 会迅速导致说谎者悖论。（见关于真理修正理论的条目。）塔斯基自己的结论是，对于一种语言 $L$ 的真理定义必须在一种本质上比 $L$ 更强的元语言中给出。

There is a consequence for the foundations of mathematics. First-order Zermelo-Fraenkel set theory is widely regarded as the standard of mathematical correctness, in the sense that a proof is correct if and only if it can be formalised as a formal proof in set theory. We would like to be able to give a truth definition for set theory; but by Tarski’s result this truth definition can’t be given in set theory itself. The usual solution is to give the truth definition informally in English. But there are a number of ways of giving limited formal truth definitions for set theory. For example Azriel Levy showed that for every natural number $n$ there is a $\Sigma_n$ formula that is satisfied by all and only the set-theoretic names of true $\Sigma_n$ sentences of set theory. The definition of $\Sigma_n$ is too technical to give here, but three points are worth making. First, every sentence of set theory is provably equivalent to a $\Sigma_n$ sentence for any large enough $n$ . Second, the class of $\Sigma_n$ formulas is closed under adding existential quantifiers at the beginning, but not under adding universal quantifiers. Third, the class is not closed under negation; this is how Levy escapes Tarski’s paradox. (See the entry on set theory.) Essentially the same devices allow Jaakko Hintikka to give an internal truth definition for his independence friendly logic; this logic shares the second and third properties of Levy’s classes of formulas.

这对数学基础有影响。一阶策梅洛 - 弗兰克尔集合论被广泛认为是数学正确性的标准，从某种意义上说，一个证明是正确的，当且仅当它可以形式化为集合论中的形式证明。我们希望能够为集合论给出一个真理定义；但根据塔斯基的结果，这个真理定义不能在集合论本身中给出。通常的解决方案是用英语非正式地给出真理定义。但是有多种方法可以为集合论给出有限的形式真理定义。例如，阿兹列尔・列维（Azriel Levy）表明，对于每个自然数 $n$ ，都存在一个 $\Sigma_n$ 公式，它被且仅被集合论中真 $\Sigma_n$ 句子的集合论名称所满足。 $\Sigma_n$ 的定义在这里过于技术化而无法给出，但有三点值得一提。第一，对于足够大的 $n$ ，集合论的每个句子都可证等价于一个 $\Sigma_n$ 句子。第二， $\Sigma_n$ 公式类在开头添加存在量词时是封闭的，但在添加全称量词时不是。第三，这个类在否定下不是封闭的；这就是列维避开塔斯基悖论的方式。（见关于集合论的条目。）本质上相同的方法使雅各・辛提卡（Jaakko Hintikka）能够为他的独立友好逻辑给出一个内部真理定义；这种逻辑具有列维公式类的第二和第三个属性。

2. Some kinds of truth definition on the 1933 pattern

一些符合 1933 年模式的真理定义

In his 1933 paper Tarski went on to show that many fully interpreted formal languages do have a truth definition that satisfies his conditions. He gave four examples in that paper. One was a trivial definition for a finite language; it simply listed the finitely many true sentences. One was a definition by quantifier elimination; see Section 2.2 below. The remaining two, for different classes of language, were examples of what people today think of as the standard Tarski truth definition; they are forerunners of the 1956 model-theoretic definition.

在 1933 年的论文中，塔斯基接着表明，许多完全解释的形式语言确实有满足他条件的真理定义。他在论文中给出了四个例子。一个是针对有限语言的平凡定义，它只是列出了有限数量的真语句。一个是通过量词消去法给出的定义；见下文 2.2 节。剩下的两个，针对不同类别的语言，是如今人们认为的标准塔斯基真理定义的示例；它们是 1956 年模型论定义的前身。

2.1 The standard truth definitions

2.1 标准真理定义

The two standard truth definitions are at first glance not definitions of truth at all, but definitions of a more complicated relation involving assignments $a$ of objects to variables:

这两个标准真理定义乍一看根本不是真理的定义，而是涉及将对象赋值给变量的更复杂关系的定义：

$a$ satisfies the formula $F$

$a$ 满足公式 $F$

(where the symbol ‘ $F$ ’ is a placeholder for a name of a particular formula of the object language). In fact satisfaction reduces to truth in this sense: $a$ satisfies the formula $F$ if and only if taking each free variable in $F$ as a name of the object assigned to it by $a$ makes the formula $F$ into a true sentence. So it follows that our intuitions about when a sentence is true can guide our intuitions about when an assignment satisfies a formula. But none of this can enter into the formal definition of truth, because ‘taking a variable as a name of an object’ is a semantic notion, and Tarski’s truth definition has to be built only on notions from syntax and set theory (together with those in the object language); recall Section 1.1. In fact Tarski’s reduction goes in the other direction: if the formula $F$ has no free variables, then to say that $F$ is true is to say that every assignment satisfies it.

（其中符号 “ $F$ ” 是对象语言中特定公式名称的占位符）。事实上，在这个意义上，满足关系可归结为真理： $a$ 满足公式 $F$ 当且仅当将 $F$ 中的每个自由变量视为由 $a$ 分配给它的对象的名称时，使公式 $F$ 成为一个真语句。因此，我们关于一个句子何时为真的直觉可以指导我们关于一个赋值何时满足一个公式的直觉。但这些都不能进入真理的形式定义，因为 “将一个变量视为一个对象的名称” 是一个语义概念，而塔斯基的真理定义必须仅基于句法、集合论（以及对象语言中的那些概念）来构建；回想 1.1 节。实际上，塔斯基的推导方向相反：如果公式 $F$ 没有自由变量，那么说 $F$ 为真就是说每个赋值都满足它。

The reason why Tarski defines satisfaction directly, and then deduces a definition of truth, is that satisfaction obeys recursive conditions in the following sense: if $F$ is a compound formula, then to know which assignments satisfy $F$ , it’s enough to know which assignments satisfy the immediate constituents of $F$ . Here are two typical examples:

塔斯基之所以直接定义满足关系，进而推导出真理的定义，是因为满足关系在以下意义上遵循递归条件：如果 $F$ 是一个复合公式，那么要知道哪些赋值满足 $F$ ，只需知道哪些赋值满足 $F$ 的直接子公式即可。下面是两个典型例子：

The assignment $a$ satisfies the formula ‘ $F$ and $G$ ’ if and only if $a$ satisfies $F$ and $a$ satisfies $G$ .

赋值 $a$ 满足公式 “ $F$ 且 $G$ ”，当且仅当 $a$ 满足 $F$ 并且 $a$ 满足 $G$ 。

The assignment $a$ satisfies the formula ‘For all $x$ , $G$ ’ if and only if for every individual $i$ , if $b$ is the assignment that assigns $i$ to the variable $x$ and is otherwise exactly like $a$ , then $b$ satisfies $G$ .

赋值 $a$ 满足公式 “对所有 $x$ ， $G$ ”，当且仅当对于每一个体 $i$ ，如果 $b$ 是将 $i$ 赋给变量 $x$ 且在其他方面与 $a$ 完全相同的赋值，那么 $b$ 满足 $G$ 。

We have to use a different approach for atomic formulas. But for these, at least assuming for simplicity that $L$ has no function symbols, we can use the metalanguage copies $\#(R)$ of the predicate symbols $R$ of the object language. Thus:

对于原子公式，我们必须采用不同的方法。但至少为简单起见，假设 $L$ 没有函数符号，我们可以使用对象语言中谓词符号 $R$ 在元语言中的副本 $\#(R)$ 。如下：

The assignment $a$ satisfies the formula $R (x, y)$ if and only if $\#(R)(a (x), a (y))$ .

赋值 $a$ 满足公式 $R (x, y)$ ，当且仅当 $\#(R)(a (x), a (y))$ 成立。

(Warning: the expression $\#$ is in the metametalanguage, not in the metalanguage $M$ . We may or may not be able to find a formula of $M$ that expresses $\#$ for predicate symbols; it depends on exactly what the language $L$ is.)

（注意：符号 $\#$ 属于元元语言，而非元语言 $M$ 。我们或许能够或许无法找到 $M$ 中的某个公式来表达谓词符号的 $\#$ 关系，这完全取决于语言 $L$ 的具体情况。）

Subject to the mild reservation in the next paragraph, Tarski’s definition of satisfaction is compositional, meaning that the class of assignments which satisfy a compound formula $F$ is determined solely by (1) the syntactic rule used to construct $F$ from its immediate constituents and (2) the classes of assignments that satisfy these immediate constituents. (This is sometimes phrased loosely as: satisfaction is defined recursively. But this formulation misses the central point, that (1) and (2) don’t contain any syntactic information about the immediate constituents.) Compositionality explains why Tarski switched from truth to satisfaction. You can’t define whether ‘For all $x$ , $G$ ’ is true in terms of whether $G$ is true, because in general $G$ has a free variable $x$ and so it isn’t either true or false.

在有下一段所述的轻微保留条件的情况下，塔斯基对满足关系的定义具有组合性，这意味着满足复合公式 $F$ 的赋值类完全由以下两点决定：（1）从其直接成分构建 $F$ 所使用的句法规则；（2）满足这些直接成分的赋值类。（这有时被不严谨地表述为：满足关系是递归定义的。但这种表述忽略了关键一点，即（1）和（2）并不包含关于直接成分的任何句法信息。）组合性解释了塔斯基为什么从定义真理转向定义满足关系。你无法依据 $G$ 的真假来定义 “对所有 $x$ ， $G$ ” 的真假，因为一般来说 $G$ 含有自由变量 $x$ ，所以它本身无所谓真假。

The reservation is that Tarski’s definition of satisfaction in the 1933 paper doesn’t in fact mention the class of assignments that satisfy a formula $F$ . Instead, as we saw, he defines the relation ‘ $a$ satisfies $F$ ’, which determines what that class is. This is probably the main reason why some people (including Tarski himself in conversation, as reported by Barbara Partee) have preferred not to describe the 1933 definition as compositional. But the class format, which is compositional on any reckoning, does appear in an early variant of the truth definition in Tarski’s paper of 1931 on definable sets of real numbers. Tarski had a good reason for preferring the format ‘ $a$ satisfies $F$ ’ in his 1933 paper, namely that it allowed him to reduce the set - theoretic requirements of the truth definition. In sections 4 and 5 of the 1933 paper he spelled out these requirements carefully.

保留条件是，塔斯基在 1933 年论文中对满足关系的定义实际上并未提及满足公式 $F$ 的赋值类。相反，正如我们所见，他定义了 “ $a$ 满足 $F$ ” 这一关系，该关系确定了这个赋值类。这可能就是为什么有些人（包括据芭芭拉・帕蒂所述，塔斯基本人在谈话中）不倾向于将 1933 年的定义描述为具有组合性的主要原因。但是，无论从何种角度来看都具有组合性的赋值类形式，确实出现在塔斯基 1931 年关于实数可定义集的论文中真理定义的早期版本里。塔斯基在 1933 年的论文中选择 “ $a$ 满足 $F$ ” 这种形式是有充分理由的，即它使他能够降低真理定义对集合论的要求。在 1933 年论文的第 4 和第 5 节中，他详细阐述了这些要求。

The name ‘compositional (ity)’ first appears in papers of Putnam in 1960 (published 1975) and Katz and Fodor in 1963 on natural language semantics. In talking about compositionality, we have moved to thinking of Tarski’s definition as a semantics, i.e. a way of assigning ‘meanings’ to formulas. (Here we take the meaning of a sentence to be its truth value.) Compositionality means essentially that the meanings assigned to formulas give at least enough information to determine the truth values of sentences containing them. One can ask conversely whether Tarski’s semantics provides only as much information as we need about each formula, in order to reach the truth values of sentences. If the answer is yes, we say that the semantics is fully abstract (for truth). One can show fairly easily, for any of the standard languages of logic, that Tarski’s definition of satisfaction is in fact fully abstract.

“组合性（compositional (ity)）” 这个术语最早出现在 1960 年普特南的论文（1975 年发表）以及 1963 年卡茨和福多关于自然语言语义学的论文中。当我们讨论组合性时，我们已经将塔斯基的定义视为一种语义学，即一种为公式赋予 “意义” 的方式。（这里我们将句子的意义看作其真值。）组合性本质上意味着，赋予公式的意义至少提供了足够的信息，以确定包含这些公式的句子的真值。反之，人们可能会问，塔斯基的语义学是否只为我们提供了确定句子真值所需的关于每个公式的信息。如果答案是肯定的，我们就说这种语义学（对于真值而言）是完全抽象的。对于任何一种标准逻辑语言，人们都可以相当容易地证明，塔斯基对满足关系的定义实际上是完全抽象的。

As it stands, Tarski’s definition of satisfaction is not an explicit definition, because satisfaction for one formula is defined in terms of satisfaction for other formulas. So to show that it is formally correct, we need a way of converting it to an explicit definition. One way to do this is as follows, using either higher order logic or set theory. Suppose we write $S$ for a binary relation between assignments and formulas. We say that $S$ is a satisfaction relation if for every formula $G$ , $S$ meets the conditions put for satisfaction of $G$ by Tarski’s definition. For example, if $G$ is ‘ $G_1$ and $G_2$ ’, $S$ should satisfy the following condition for every assignment $a$ :

就目前而言，塔斯基对满足关系的定义不是一个显式定义，因为对一个公式的满足是依据对其他公式的满足来定义的。所以，为了表明它在形式上是正确的，我们需要一种方法将其转换为显式定义。一种方法如下，可使用高阶逻辑或集合论。假设我们用 $S$ 表示赋值与公式之间的二元关系。我们称 $S$ 为满足关系，如果对于每个公式 $G$ ， $S$ 都满足塔斯基定义中对 $G$ 的满足条件。例如，如果 $G$ 是 “ $G_1$ 且 $G_2$ ”，那么对于每个赋值 $a$ ， $S$ 应该满足以下条件：

$S (a, G)$ if and only if $S (a, G_1)$ and $S (a, G_2)$ .

$S (a, G)$ 当且仅当 $S (a, G_1)$ 且 $S (a, G_2)$ 。

We can define ‘satisfaction relation’ formally, using the recursive clauses and the conditions for atomic formulas in Tarski’s recursive definition. Now we prove, by induction on the complexity of formulas, that there is exactly one satisfaction relation $S$ . (There are some technical subtleties, but it can be done.) Finally we define

我们可以利用塔斯基递归定义中的递归子句和原子公式条件，形式地定义 “满足关系”。现在，我们通过对公式复杂度进行归纳证明，存在唯一的满足关系 $S$ 。（这其中存在一些技术细节，但这是可行的。）最后我们定义：

$a$ satisfies $F$ if and only if: there is a satisfaction relation $S$ such that $S (a, F)$ .

$a$ 满足 $F$ 当且仅当：存在一个满足关系 $S$ ，使得 $S (a, F)$ 。

It is then a technical exercise to show that this definition of satisfaction is materially adequate. Actually one must first write out the counterpart of Convention $T$ for satisfaction of formulas, but I leave this to the reader.

然后，证明这个满足关系的定义在实质上是适当的是一个技术问题。实际上，我们必须首先写出与公式满足相关的约定 $T$ 的对应内容，但这部分留给读者完成。

2.2 The truth definition by quantifier elimination

2.2 通过量词消去法给出的真理定义

The remaining truth definition in Tarski’s 1933 paper – the third as they appear in the paper – is really a bundle of related truth definitions, all for the same object language $L$ but in different interpretations. The quantifiers of $L$ are assumed to range over a particular class, call it $A$ ; in fact they are second order quantifiers, so that really they range over the collection of subclasses of $A$ . The class $A$ is not named explicitly in the object language, and thus one can give separate truth definitions for different values of $A$ , as Tarski proceeds to do. So for this section of the paper, Tarski allows one and the same sentence to be given different interpretations; this is the exception to the general claim that his object language sentences are fully interpreted. But Tarski stays on the straight and narrow: he talks about ‘truth’ only in the special case where $A$ is the class of all individuals. For other values of $A$ , he speaks not of ‘truth’ but of ‘correctness in the domain $A$ ’.

塔斯基 1933 年论文中剩下的真理定义（在论文中排第三）实际上是一组相关的真理定义，它们都针对同一对象语言 $L$ ，但解释不同。假设 $L$ 的量词作用于一个特定的类，称其为 $A$ ；实际上它们是二阶量词，因此它们实际上作用于 $A$ 的子类的集合。类 $A$ 在对象语言中没有被明确命名，因此人们可以像塔斯基那样，为 $A$ 的不同取值分别给出真理定义。所以在论文的这部分内容中，塔斯基允许对同一个句子进行不同的解释；这是对他所说的对象语言句子都是完全解释这一普遍主张的一个例外。但塔斯基严格遵循标准：他仅在 $A$ 是所有个体的类这种特殊情况下才谈论 “真理” 。对于 $A$ 的其他取值，他不使用 “真理” 一词，而是说 “在域 $A$ 中的正确性” 。

These truth or correctness definitions don’t fall out of a definition of satisfaction. In fact they go by a much less direct route, which Tarski describes as a ‘purely accidental’ possibility that relies on the ‘specific peculiarities’ of the particular object language. It may be helpful to give a few more of the technical details than Tarski does, in a more familiar notation than Tarski’s, in order to show what is involved. Tarski refers his readers to a paper of Thoralf Skolem in 1919 for the technicalities.

这些真理或正确性定义并非源自满足关系的定义。事实上，它们采用了一条迂回得多的路径，塔斯基将其描述为一种 “纯粹偶然” 的可能性，依赖于特定对象语言的 “特殊特性” 。用比塔斯基更易懂的符号，比他更详细地给出一些技术细节，可能有助于说明其中的原理。塔斯基让读者参考托拉尔夫・斯科伦 1919 年的一篇论文以了解相关技术细节。

One can think of the language $L$ as the first - order language with predicate symbols $\subseteq$ and $=$ . The language is interpreted as talking about the subclasses of the class $A$ . In this language we can define:

我们可以将语言 $L$ 看作是具有谓词符号 $\subseteq$ 和 $=$ 的一阶语言。这种语言被解释为谈论类 $A$ 的子类。在这种语言中，我们可以定义：

‘ $x$ is the empty set’ (viz. $x\subseteq$ every class).
“ $x$ 是空集”（即 $x$ 包含于每一个类）。

‘ $x$ is an atom’ (viz. $x$ is not empty, but every subclass of $x$ not equal to $x$ is empty).
“ $x$ 是一个原子”（即 $x$ 非空，但 $x$ 的每个不等于 $x$ 的子类都是空的）。

‘ $x$ has exactly $k$ members’ (where $k$ is a finite number; viz. there are exactly $k$ distinct atoms $\subseteq x$ ).
“ $x$ 恰好有 $k$ 个成员”（其中 $k$ 是一个有限数；即恰好有 $k$ 个不同的原子包含于 $x$ ）。

‘There are exactly $k$ elements in $A$ ’ (viz. there is a class with exactly $k$ members, but there is no class with exactly $k + 1$ members).
“ $A$ 中恰好有 $k$ 个元素”（即存在一个恰好有 $k$ 个成员的类，但不存在恰好有 $k + 1$ 个成员的类）。

Now we aim to prove:
现在我们旨在证明：

Lemma. Every formula $F$ of $L$ is equivalent to (i.e. is satisfied by exactly the same assignments as) some boolean combination of sentences of the form ‘There are exactly $k$ elements in $A$ ’ and formulas of the form ‘There are exactly $k$ elements that are in $v_1$ , not in $v_2$ , not in $v_3$ and in $v_4$ ’ (or any other combination of this type, using only variables free in $F$ ).

引理。 $L$ 的每个公式 $F$ 都等价于（即被完全相同的赋值所满足）某些形式为 “ $A$ 中恰好有 $k$ 个元素” 的句子和形式为 “恰好有 $k$ 个元素在 $v_1$ 中，不在 $v_2$ 中，不在 $v_3$ 中且在 $v_4$ 中”（或任何其他这种类型的组合，仅使用 $F$ 中的自由变量）的公式的布尔组合。

The proof is by induction on the complexity of formulas. For atomic formulas it is easy. For boolean combinations of formulas it is easy, since a boolean combination of boolean combinations is again a boolean combination. For formulas beginning with $\forall$ , we take the negation. This leaves just one case that involves any work, namely the case of a formula beginning with an existential quantifier. By induction hypothesis we can replace the part after the quantifier by a boolean combination of formulas of the kinds stated. So a typical case might be:

证明通过对公式复杂度进行归纳。对于原子公式，证明很容易。对于公式的布尔组合，证明也很容易，因为布尔组合的布尔组合仍然是布尔组合。对于以 $\forall$ 开头的公式，我们取其否定形式。这样就只剩下一种需要费些功夫的情况，即以存在量词开头的公式。根据归纳假设，我们可以用量词后面所述类型的公式的布尔组合来替换量词后面的部分。所以一个典型的例子可能是：

$\exists z$ (there are exactly two elements that are in $z$ and $x$ and not in $y$ ).

$\exists z$ （恰好有两个元素既在 $z$ 中又在 $x$ 中且不在 $y$ 中）。

This holds if and only if there are at least two elements that are in $x$ and not in $y$ . We can write this in turn as: The number of elements in $x$ and not in $y$ is not 0 and is not 1; which is a boolean combination of allowed formulas. The general proof is very similar but more complicated.

这个式子成立当且仅当在 $x$ 中且不在 $y$ 中的元素至少有两个。我们可以依次将其写为：在 $x$ 中且不在 $y$ 中的元素数量不是 0 且不是 1；这是允许的公式的布尔组合。一般的证明非常相似但更为复杂。

When the lemma has been proved, we look at what it says about a sentence. Since the sentence has no free variables, the lemma tells us that it is equivalent to a boolean combination of statements saying that $A$ has a given finite number of elements. So if we know how many elements $A$ has, we can immediately calculate whether the sentence is ‘correct in the domain $A$ ’.

引理得证后，我们来看看它对句子的说明。由于句子没有自由变量，引理告诉我们，它等价于关于 $A$ 具有特定有限数量元素的陈述的布尔组合。所以，如果我们知道 $A$ 中有多少元素，就能立即算出该句子在 “域 $A$ 中是否正确”。

One more step and we are home. As we prove the lemma, we should gather up any facts that can be stated in $L$ , are true in every domain, and are needed for proving the lemma. For example we shall almost certainly need the sentence saying that $\subseteq$ is transitive. Write $T$ for the set of all these sentences. (In Tarski’s presentation $T$ vanishes, since he is using higher order logic and the required statements about classes become theorems of logic.) Thus we reach, for example:

再进一步我们就完成了。在证明引理的过程中，我们应该收集所有能用 $L$ 表述、在每个域中都为真且证明引理所需的事实。例如，我们几乎肯定需要用到表述 $\subseteq$ 具有传递性的句子。将所有这些句子构成的集合记为 $T$ 。（在塔斯基的论述中 $T$ 并未出现，因为他使用的是高阶逻辑，关于类的所需陈述成为了逻辑定理。）这样我们就能得出，例如：

Theorem. If the domain $A$ is infinite, then a sentence $S$ of the language $L$ is correct in $A$ if and only if $S$ is deducible from $T$ and the sentences saying that the number of elements of $A$ is not any finite number.

定理。如果域 $A$ 是无限的，那么语言 $L$ 中的句子 $S$ 在 $A$ 中正确，当且仅当 $S$ 可从 $T$ 以及表述 $A$ 的元素数量不是任何有限数的句子中推导出来。

The class of all individuals is infinite (Tarski asserts), so the theorem applies when $A$ is this class. And in this case Tarski has no inhibitions about saying not just ‘correct in $A$ ’ but ‘true’; so we have our truth definition.

所有个体构成的类是无限的（塔斯基如此断言），所以当 $A$ 是这个类时，该定理适用。在这种情况下，塔斯基毫无顾忌地说句子不只是 “在 $A$ 中正确”，而是 “为真”；这样我们就得到了真理定义。

The method we have described revolves almost entirely around removing existential quantifiers from the beginnings of formulas; so it is known as the method of quantifier elimination. It is not as far as you might think from the two standard definitions. In all cases Tarski assigns to each formula, by induction on the complexity of formulas, a description of the class of assignments that satisfy the formula. In the two previous truth definitions this class is described directly; in the quantifier elimination case it is described in terms of a boolean combination of formulas of a simple kind.

我们所描述的方法几乎完全围绕从公式开头消除存在量词展开，因此它被称为量词消去法。它与前面两种标准定义的差异并不像你想象的那么大。在所有情况下，塔斯基都通过对公式复杂度进行归纳，为每个公式赋予对满足该公式的赋值类的描述。在前两种真理定义中，这个赋值类是直接描述的；而在量词消去法的情况中，它是用一类简单公式的布尔组合来描述的。

At around the same time as he was writing the 1933 paper, Tarski gave a truth definition by quantifier elimination for the first - order language of the field of real numbers. In his 1931 paper it appears only as an interesting way of characterising the set of relations definable by formulas. Later he gave a fuller account, emphasising that his method provided not just a truth definition but an algorithm for determining which sentences about the real numbers are true and which are false.

大约在撰写 1933 年那篇论文的同一时期，塔斯基为实数域的一阶语言给出了通过量词消去法得到的真理定义。在他 1931 年的论文中，这只是作为一种刻画可由公式定义的关系集的有趣方式出现。后来他给出了更全面的阐述，强调他的方法不仅提供了真理定义，还提供了一种算法，用于判定关于实数的句子哪些为真，哪些为假。

3. The 1956 definition and its offspring

3. **1956 年的定义及其发展 **

In 1933 Tarski assumed that the formal languages that he was dealing with had two kinds of symbol (apart from punctuation), namely constants and variables. The constants included logical constants, but also any other terms of fixed meaning. The variables had no independent meaning and were simply part of the apparatus of quantification.

1933 年，塔斯基假设他所处理的形式语言（除标点符号外）有两种符号，即常量和变量。常量包括逻辑常量，也包括任何其他具有固定含义的项。变量没有独立的含义，仅仅是量化机制的一部分。

Model theory by contrast works with three levels of symbol. There are the logical constants ( $=$ , $\neg$ , & for example), the variables (as before), and between these a middle group of symbols which have no fixed meaning but get a meaning through being applied to a particular structure. The symbols of this middle group include the nonlogical constants of the language, such as relation symbols, function symbols and constant individual symbols. They also include the quantifier symbols $\forall$ and $\exists$ , since we need to refer to the structure to see what set they range over. This type of three - level language corresponds to mathematical usage; for example we write the addition operation of an abelian group as +, and this symbol stands for different functions in different groups.

相比之下，模型论使用三个层次的符号。有逻辑常量（例如 $=$ 、 $\neg$ 、&）、变量（和之前一样），在它们之间还有一组中间符号，这些符号没有固定含义，但通过应用于特定结构而获得意义。这组中间符号包括语言的非逻辑常量，如关系符号、函数符号和个体常量符号。它们还包括量词符号 $\forall$ 和 $\exists$ ，因为我们需要参照结构来确定它们的作用域。这种三层符号的语言类型与数学用法相符；例如，我们将阿贝尔群的加法运算写作 + ，这个符号在不同的群中代表不同的函数。

So one has to work a little to apply the 1933 definition to model - theoretic languages. There are basically two approaches: (1) Take one structure $A$ at a time, and regard the nonlogical constants as constants, interpreted in $A$ . (2) Regard the nonlogical constants as variables, and use the 1933 definition to describe when a sentence is satisfied by an assignment of the ingredients of a structure $A$ to these variables. There are problems with both these approaches, as Tarski himself describes in several places. The chief problem with (1) is that in model theory we very frequently want to use the same language in connection with two or more different structures – for example when we are defining elementary embeddings between structures (see the entry on first - order model theory). The problem with (2) is more abstract: it is disruptive and bad practice to talk of formulas with free variables being ‘true’. (We saw in Section 2.2 how Tarski avoided talking about truth in connection with sentences that have varying interpretations.) What Tarski did in practice, from the appearance of his textbook in 1936 to the late 1940s, was to use a version of (2) and simply avoid talking about model - theoretic sentences being true in structures; instead he gave an indirect definition of what it is for a structure to be a ‘model of’ a sentence, and apologised that strictly this was an abuse of language. (Chapter VI of Tarski 1994 still contains relics of this old approach.)

因此，要将 1933 年的定义应用于模型论语言，还需要费些功夫。基本上有两种方法：（1）每次取一个结构 $A$ ，将非逻辑常量视为在 $A$ 中被解释的常量。（2）将非逻辑常量视为变量，使用 1933 年的定义来描述当用结构 $A$ 的组成部分对这些变量进行赋值时，句子何时被满足。正如塔斯基在多处描述的那样，这两种方法都存在问题。方法（1）的主要问题是，在模型论中，我们经常希望用同一语言处理两个或更多不同的结构 —— 例如在定义结构之间的基本嵌入时（见关于一阶模型论的条目）。方法（2）的问题更抽象：谈论含有自由变量的公式 “为真” 会造成混乱，属于不当做法。（我们在 2.2 节中看到塔斯基是如何避免谈论具有不同解释的句子的真假的。）从 1936 年他的教科书出版到 20 世纪 40 年代后期，塔斯基在实际操作中采用了方法（2）的一种变体，并且干脆避免谈论模型论句子在结构中为真；相反，他给出了结构是句子 “模型” 的间接定义，并为这种严格来说属于语言滥用的做法致歉。（塔斯基 1994 年著作的第六章中仍保留着这种旧方法的痕迹。）

By the late 1940s it had become clear that a direct model - theoretic truth definition was needed. Tarski and colleagues experimented with several ways of casting it. The version we use today is based on that published by Tarski and Robert Vaught in 1956. See the entry on classical logic for an exposition.

到 20 世纪 40 年代后期，显然需要一个直接的模型论真理定义。塔斯基和他的同事尝试了多种定义方式。我们如今使用的版本基于塔斯基和罗伯特・沃特 1956 年发表的内容。相关阐述可参见关于经典逻辑的条目。

The right way to think of the model - theoretic definition is that we have sentences whose truth value varies according to the situation where they are used. So the nonlogical constants are not variables; they are definite descriptions whose reference depends on the context. Likewise the quantifiers have this indexical feature, that the domain over which they range depends on the context of use. In this spirit one can add other kinds of indexing. For example a Kripke structure is an indexed family of structures, with a relation on the index set; these structures and their close relatives are fundamental for the semantics of modal, temporal and intuitionist logic.

理解模型论定义的正确方式是，我们的句子真值会根据使用情境而变化。所以非逻辑常量不是变量；它们是限定摹状词，其指称依赖于上下文。同样，量词也具有这种索引特征，即它们的作用域取决于使用上下文。本着这种精神，人们可以添加其他类型的索引。例如，克里普克结构是一个带索引的结构族，索引集上有一个关系；这些结构及其相关结构是模态逻辑、时态逻辑和直觉主义逻辑语义学的基础。

Already in the 1950s model theorists were interested in formal languages that include kinds of expression different from anything in Tarski’s 1933 paper. Extending the truth definition to infinitary logics was no problem at all. Nor was there any serious problem about most of the generalised quantifiers proposed at the time. For example there is a quantifier $Q x y$ with the intended meaning:

早在 20 世纪 50 年代，模型论者就对包含与塔斯基 1933 年论文中不同表达式的形式语言产生了兴趣。将真理定义扩展到无穷逻辑根本不成问题。对于当时提出的大多数广义量词，也没有什么严重问题。例如，有一个量词 $Q x y$ ，其预期含义是：

$Q x y F (x, y)$ if and only if there is an infinite set $X$ of elements such that for all $a$ and $b$ in $X$ , $F (a, b)$ .

$Q x y F (x, y)$ 当且仅当存在一个无限元素集 $X$ ，使得对于 $X$ 中的所有 $a$ 和 $b$ ， $F (a, b)$ 成立。

This definition itself shows at once how the required clause in the truth definition should go.

这个定义本身立刻表明了真理定义中所需的子句应该如何表述。

In 1961 Leon Henkin pointed out two sorts of model - theoretic language that didn’t immediately have a truth definition of Tarski’s kind. The first had infinite strings of quantifiers:

1961 年，利昂・亨金指出了两种无法直接用塔斯基式定义给出真理定义的模型论语言。第一种语言含有无限量词串：

$\forall v_1\exists v_2\forall v_3\exists v_4\ldots R (v_1, v_2, v_3, v_4, \ldots)$ .

The second had quantifiers that are not linearly ordered. For ease of writing I use Hintikka’s later notation for these:

第二种语言的量词不是线性排序的。为了书写方便，我使用辛提卡后来提出的符号来表示：

$\forall v_1\exists v_2\forall v_3 (\exists v_4/\forall v_1) R (v_1, v_2, v_3, v_4)$ .

Here the slash after $\exists v_4$ means that this quantifier is outside the scope of the earlier quantifier $\forall v_1$ (and also outside that of the earlier existential quantifier).

这里 $\exists v_4$ 后面的斜线表示这个量词不在前面的量词 $\forall v_1$ 的作用域内（也不在前面的存在量词的作用域内）。

Henkin pointed out that in both cases one could give a natural semantics in terms of Skolem functions. For example the second sentence can be paraphrased as

亨金指出，在这两种情况下，都可以用斯科伦函数给出一种自然语义。例如，第二个句子可以改写为：

$\exists f\exists g\forall v_1\forall v_3R (v_1, f (v_1), v_3, g (v_3))$ ,

which has a straightforward Tarski truth condition in second - order logic. Hintikka then observed that one can read the Skolem functions as winning strategies in a game, as in the entry on logic and games. In this way one can build up a compositional semantics, by assigning to each formula a game. A sentence is true if and only if the player Myself (in Hintikka’s nomenclature) has a winning strategy for the game assigned to the sentence. This game semantics agrees with Tarski’s on conventional first - order sentences. But it is far from fully abstract; probably one should think of it as an operational semantics, describing how a sentence is verified rather than whether it is true.

这个式子在二阶逻辑中有直接的塔斯基式真值条件。辛提卡随后指出，可以将斯科伦函数看作是逻辑与博弈条目中提到的博弈中的获胜策略。通过这种方式，给每个公式分配一个博弈，就可以构建一种组合语义。一个句子为真，当且仅当 “我”（按照辛提卡的术语）在分配给该句子的博弈中有获胜策略。这种博弈语义在传统一阶句子上与塔斯基语义一致。但它远非完全抽象；或许应该将其视为一种操作语义，描述的是句子如何被验证，而非句子是否为真。

The problem of giving a Tarski - style semantics for Henkin’s two languages turned out to be different in the two cases. With the first, the problem is that the syntax of the language is not well - founded: there is an infinite descending sequence of subformulas as one strips off the quantifiers one by one. Hence there is no hope of giving a definition of satisfaction by recursion on the complexity of formulas. The remedy is to note that the explicit form of Tarski’s truth definition in Section 2.1 above didn’t require a recursive definition; it needed only that the conditions on the satisfaction relation $S$ pin it down uniquely. For Henkin’s first style of language this is still true, though the reason is no longer the well - foundedness of the syntax.

为亨金提出的这两种语言给出塔斯基式语义的问题，在两种情况下有所不同。对于第一种语言，问题在于其语法不是良基的：当逐个去掉量词时，会出现一个无限下降的子公式序列。因此，无法通过对公式复杂度进行递归给出满足关系的定义。解决办法是注意到，上文 2.1 节中塔斯基真理定义的显式形式并不要求递归定义；它只要求满足关系 $S$ 的条件能唯一确定它。对于亨金的第一种语言，这一点仍然成立，尽管原因不再是语法的良基性。

For Henkin’s second style of language, at least in Hintikka’s notation (see the entry on independence friendly logic), the syntax is well - founded, but the displacement of the quantifier scopes means that the usual quantifier clauses in the definition of satisfaction no longer work. To get a compositional and fully abstract semantics, one has to ask not what assignments of variables satisfy a formula, but what sets of assignments satisfy the formula ‘uniformly’, where ‘uniformly’ means ‘independent of assignments to certain variables, as shown by the slashes on quantifiers inside the formula’. (Further details of revisions of Tarski’s truth definition along these lines are in the entry on dependence logic.) Henkin’s second example is of more than theoretical interest, because clashes between the semantic and the syntactic scope of quantifiers occur very often in natural languages.

对于亨金的第二种语言，至少在辛提卡的符号表示下（见关于独立友好逻辑的条目），其语法是良基的，但量词作用域的变动意味着，在满足关系的定义中，通常的量词子句不再适用。为了得到一种具有组合性且完全抽象的语义，人们不能再问哪些变量赋值满足一个公式，而要问哪些赋值集合 “一致地” 满足该公式，这里 “一致地” 意思是 “独立于对某些变量的赋值，就像公式中量词上的斜线所表明的那样” 。（沿着这些思路对塔斯基真理定义进行修订的更多细节，可在关于依赖逻辑的条目中找到。）亨金的第二个例子不仅具有理论意义，因为量词的语义作用域和句法作用域之间的冲突在自然语言中经常出现。

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