（奎因《短文集：一本不拘泥于分类学标准的哲学词典》，译者：翟玉章）
奎因哲学词条：函数
【单个的词的类型是其标记的类。词串的类型是其组成成分的类型的有限序列。某个词在某个表达式中的某次出现，是与从这个表达式开头至这个词的表达式片断一一对应的，后者可以用作前者的定义。】
Type versus Token
类型与标记
ES 1ST DER GEIST DER SICH DEN KORPER BAUT: such is the nine-word inscription on a Harvard museum. The count is nine because we count der both times;we are counting concrete physical objects, nine in a row. When on the other hand statistics are compiled regarding students' vocabularies, a firm line is drawn at repetitions; no cheating. Such are two contrasting senses in which we use the word word. A word in the second sense is not a physical object, not a dribble （点滴）of ink or an incision in granite, but an abstract object. In this second sense of the word word it is not two words derthatturn up in the inscription, but one word derthat gets inscribed twice. Words in the first sense have come to be called tokens; words in the second sense are called types.
ES IST DER GEIST DER SICH DEN KORPER BAUT[1]：这是哈佛的一个博物馆上的九词铭文。词的数目是9，是因为DER被数了两次；这个铭文，作为物理对象，是由9个连续的词所组成的。另一方面，当对学生词汇量进行统计时，则不会允许重复计数，否则就是作弊了。以上展示了“词”这个词的两种截然不同的含义。第二种意义上的词不是一个物理对象：既不是一个墨迹，也不是一个刻痕，而是一个抽象对象。根据“词”的第二种含义，铭文中出现的不是两个词DER，而是被铭刻了两次的同一个词DER。第一种意义上的词被称为标记（token）；第二种意义上的词则称为类型（type）。
A still further distinction can be drawn if we consult grammatical refinements. The word der figures as an article in its first occurrence and as a relative pronoun in its second. On this score it might be reckoned as two words, not one, even as types. My present purposes are better served, however, by abstracting from grammatical and semantical distinctions and addressing just the sheer forms of words.
如果我们考虑语法上的细微差别，还可以做出一个更进一步的区分。DER第一次出现时用作冠词，第二次出现时则用作关系代词。根据这个理由，甚至在类型的意义上，它都可以被视为两个词，而不应被视为一个词。然而，为了更好地实现我当前的目的，我选择忽略语法和语义上的区别，只关注语词的纯形式。
Thetype, one is prepared to say, is the class of all its tokens. The type derhas two of its members on the museum at Harvard and a host of others in books throughout Germany. This account fits derand other words well enough, but it breaks down when we press it to strings of words. What about two little lines of pentameter that are fated never to get thought up? Taken as classes of their tokens, each of the lines is identically the empty class; so there is but one. This we find unacceptable. We do not want to say that every possible line of pentameter, save one, is destined someday to be uttered or written.
人们会说，类型是其所有标记的类。我们在哈佛大学的那个博物馆里可以找到作为类型的DER的两个元素，在整个德国的书籍里则可以找到它的更多元素。这种说法对于DER这个词和其他词没有什么问题。但一旦应用到词串时，就会出现问题。设想两行永远不会被想到的五步格诗。如果把一个类型看成其标记的类，那么这两行诗的类型都将是空类，从而是同一个类型。但这是我们所无法接受的。我们不想说：每一个可能的诗行，除了和空类对应的那一个之外，都终有一天会被说出来或写出来。
When sharp formulations are offered for concepts that had been vague, they sometimes result in bizarre rulings along the edges, bizarre but harmless.Computer engineers call them don't-cares.Our present case, however, is quite the contrary: decidedly a care. Fruitful work in the mathematical theory of proof,stemming from FORMALISM and issuing in GODEL'STHEOREM andcomputer theory, hinges on the existence and distinctness of strings of signsof all finite lengths.
当人们为原本模糊的概念做清晰化处理时，有时会在边缘处产生怪异的结果；虽然怪异，但却是无害的。计算机工程师把这些边缘称为无所谓的情况（don't-cares）。我们现在的情况与此正好相反，两个不同的诗行绝对是有所谓的情况。发端于形式主义的数学证明论，之所以能导致哥德尔定理和计算机理论这些富有成效的成果，正是依赖于“每一个有限长度的符号串都存在而且具有惟一性”这个假设。
The postulate can be put thus: If a and b are different strings, then the string consisting of a followedby c differs from b followed by c.If types were themere classes of their tokens, this would be false. For, if the strings a and b have actually been written but are destined never to get written with cappended, then the two strings with c appended would both be the empty class,if construed as the classes of their tokens, and would thus be identical, contrary to the postulate.
这个假设可以这样来表达：如果a和b是不同的字符串，那么，由a附加c组成的字符串，与b附加c组成的字符串，这两者也将是不同的字符串。如果类型仅仅是其标记的类，这个假设将是假的。如果字符串a和b实际上已经被写出，但注定不会附加c，那么这两个附加c的字符串的类型将都是空集，而这是与上面的假设不符的。
Classes are abstract objects (see CLASSES VERSUS PROPERTIES), but classes of tokens are not in general abstract enough for types. They do well enough for types of single words or signs,we saw, for we can assure the existence of tokens at that level, and thus avert emptiness of the classes. So far, so good; let us then construe the types of single signs as the classes of their tokens. Types of strings of signs thereupon call for a different logical tack: we can construe them as finite sequences of the types of thecomponent signs, taking 'sequence' not in its physical, spatial, or temporal sense but in its abstract mathematical sense, where failureof existence is no longer to be apprehended.
类是抽象对象（请参阅类与属性），但标记类，对于泛化的类型来说，抽象得还不够。我们看到，对于单个的词或符号的类型来说，它们已经足够抽象了，因为我们可以确保这个水平上的标记的存在，从而避免空集。到此为止，一切都好；让我们把单个符号的类型定义为其标记的类。但符号串的类型则需要一种不同的逻辑处理：我们可以把它们解释为组成符号的类型的有限序列（这里的序列是数学抽象意义上的序列，而不是物理意义上的或有时空涵义的序列）；采用这个定义后，我们就不用担心会出现不存在的问题了。
And just whatis that mathematical sense? A finite sequence is a FUNCTION,q. v., that assigns things to the positive integers up to some point. Where f is the function, the nth thing in the sequence is f ofn.
到底是怎样的数学意义呢？一个有限序列是一个函数，即一个将事物分配给一定范围内的正整数的函数。如果f是这样的函数，那么这个序列中的第n个事物就是f(n)。
Type and token nicely span the abstract and concrete, fromlexicon to inscription. It is seldom appreciated that occurrence is a third thing: not token, but something between. The word der has two occurrences in the sentence Es ist der Geist der sich den Körper baut; and I speak now of types, not tokens. Tokens occur in tokens, types in types.
从类型到标记，正好是从抽象到具体的跨越，从词汇到铭文的跨越。很少有人意识到，出现是第三种事物：既不是类型，也不是标记，而是两者之间的某种东西。DER这个词在语句Es ist der Geist der sich den Körper baut中出现了两次；我在这里谈到的是类型，而不是标记。类型只出现在类型中，标记只出现在标记中。
Just what sort of thing, then, is an occurrence? In 1940 I had to take an explicit stand on the matter, in writing about GODEL'S THEOREM[2].I found that I could construe an occurrence ofaword or sign in an expression as the initial segment of the expression up to and including that word or sign. Thus the first of the two occurrences of der in question is the type Es ist der,and the second is Es ist derGeistder. This version is artificial but adequate; whatever needs to be said by reference to occurrences of words or signs can be said by reference to these segments, thanks to a one-to-one correspondence.
那么，“出现”到底是怎样的事物呢？1940年，我在谈到哥德尔定理时，不得不对这个问题明确表态。我发现，我可以将一个词或一个符号在某个表达式中的某次出现解释为：从该表达式开头至这个词或符号的表达式片断。因此正在讨论的两个出现中的第一个是类型Es ist der，第二个是类型Es ist derGeistder。这个做法是人为的，但也是适当的；由于这两者之间是一一对应的，所以任何需要提到语词或符号的出现才能到的事情，通过提到这些表达式片断也能做到。
[1]英语：It is the spirit that builds itself the body. 汉语：精神本身构建了身体。——译者
[2]Mathematical Logic (1940; Harvard, 1951), last chapter.
《数理逻辑》（1940；哈佛，1951），最后一章。
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