一点说明

悖论问题属于逻辑学范畴，尽管该词条不长，但涉及到罗素悖论和埃庇蒙德斯悖论。前者为逻辑悖论，后者为语义悖论。为使读者更易于阅读和理解，在该词条前附上罗素的理发师悖论和埃庇蒙德斯的说谎者悖论。供参阅。

1、理发者悖论

一个男理发师在招牌上写着：

告示：城里所有不自己刮脸的男人都由我给他们刮脸，我也只给这些人刮脸。

那理发师可以给自己刮脸么？如果他不给自己刮脸，他就属于“不给自己刮脸的人”，他就要给自己刮脸，而如果他给自己刮脸呢？他又属于“给自己刮脸的人”，他就不该给自己刮脸。

2、说谎者悖论

“我现在说的这句话是谎话”。

如果这句话是真的，根据其语义，就否定了这句是“谎话”，那么这句话就是假的；如果这句话是假的，就肯定了这句话是“谎话”，那么这句话就是真的。因此这句话无解，是自相矛盾的。

PARADOX, a proposition that seems logically sound but leads to a contradiction or an absurdity. If logic is to be credited, then no valid proof, from true premises, can result in a self-contradictory conclusion. But such proofs can appear to exist. Such false appearance may arise because the premises are not all true---though they seem to be true—or because the argument leading to the contradictory conclusion is not in fact valid though it seems to be so. The term paradox is sometimes applied to inferences in which a contradiction appears to be derived validly from true premises.

悖论，一种在逻辑上似乎合理，但导致矛盾或荒谬的命题。如果逻辑是可信的，而没有来自真实前提的有效证据，就会导致自相矛盾的结论。但这样的证据似乎是存在的。出现这样的假象可能是因为这种前提并不都是真的---尽管它们似乎是真的---或者是因为导致矛盾结论的论据虽然看上去像真的，但事实上是无效的。悖论一词有时应用于从真实前提中有效推导出的一种矛盾推论。

   A paradox is said to be solved if and only if the invalidity of the supposed proof of a contradiction, or the falsity of its premise, is exposed. Indeed, the occurrence of paradox may be of theoretical importance when its solution displays the falsity of a premise that was antecedently accepted.

    据说，当且仅当证明了矛盾的假定证据无效，或证明其前提错误时，悖论便被解除。实际上，当悖论的解决办法展示了先前被接受的前提错误时，它的出现可能具有理论价值。

   When Bertrand Russell communicated his paradox about classes to Gottlob Frege, the German mathematician, the latter is said to have responded, “Alas, arithmetic totters.” For the concept of class is fundamental to mathematics, and Russell’s paradox called for revision of basic traditional assumptions pertaining to the accepted notion of class.

    当伯特兰·罗素与德国数学家戈特洛布·弗雷格交流了他的悖论时，据说后者回应到“哎，数学的基础动摇了”。因为集合论的概念是数学的基础，而罗素悖论要求修改与公认的集合论概念有关的基本传统假设。

   Russell’s paradox may be stated informally as follows. Some classes are members of themselves. Others are not. The class of dogs is not a dog and so is not a member of itself. The class of things that are not dogs is not a dog and therefore is a member of itself. Again, the class of all classes, being a class, is a self-member. The class of integers is not an integer and therefore is not a self-member. Consider the class of all classes that are not self-members. If this class were a self-member, it would not be a member of itself. And if it were not a self-member, it would be a member of itself.

罗素悖论可通俗地陈述如下：某些集合是其自身的元素。其它集合不是。犬的集合不是犬，因此不是其自身的元素。非犬的事物的集合不是犬，因此是其自身的元素。另外，属于一个集合的所有集合都是自身元素。整数的集合不是整数，因此不是自身的元素。考虑到所有集合都是非自身元素的集合。如果这个集合是自身元素的集合，那么它就不会是其本身的元素。而如果它不是自身元素的集合，那么它就是其本身的元素。

   Russell’s solution to the paradox is controversial. According to him, things are of different logical types. And logical types compose a hierarchical structure. Individuals are of type 0, which is the lowest type in the hierarchy. Classes of individuals are of type1. More generally, classes whose members are of type n belong to type n+1. It is significant to say that x belongs to the class A, if and only if the type of A is consecutive to the type of x. According to Russell, sentences that violate this rule have neither truth nor falsity.

    罗素对悖论的解决方法是有争议的。根据他的观点，事物具有不同的逻辑类型。而逻辑类型构成了一种层级结构。个体为类型0，是等级体系中最低的类型。个体的集合为类型1。概括起来，那些元素为类型n的集合属于类型n+1。当且仅当类型A与类型x连续时，说x属于A集合便有了重大意义。按照罗素的观点，违反这种规则的命题既不为真，也不为假。

   Russell’s theory of logical types is more elaborate than this account suggests. But the view presented here is a simple consequence of his more complex version.

    罗素的逻辑类型理论比这种陈述暗示的要更为复杂。但这里展示的观点是其更为复杂版本的一种简单结果。

   The paradox of classes is solved by the consideration that membership in the class of classes, which are not members of themselves, is tantamount to not being a self-member. According to Russell, however, to say of something that it is, or is not, a self-member is to employ a nonsensical EXPRESSION that in effect violates the significance condition imposed by the theory of types.

集合悖论得以解决，是通过考虑了在集合类的集合元素中那些不是它们自身元素的情况。然而，罗素认为，要说某事是，或不是一个自身元素，是使用了一种荒谬的表达式，实际上违反了由类型理论强加的意义条件。

   A characteristic of the theory of logical types is that one may not speak of the universal class but only of the universal class of a given type. Many think that this prohibition precludes the possibility of that generality which is the traditional objective of logical theory.

    逻辑类型理论的一个特性就是人们无法论及普遍的集合，只能论及一种给定类型的普遍集合。许多人认为，这种禁止妨碍了一般性的可能性，这是逻辑理论的传统目标。

   Russell’s paradox is one of infinitely many paradoxes that can be constructed within a system of mathematical logic such as his Principia Mathematica. These are called logical paradoxes.

    罗素悖论是一种可以在数学逻辑系统中建立的无穷多悖论中的一个悖论，比如他的《数学原理》。这些都被称为逻辑悖论。

   In addition, a variety of paradoxes—some of which were known in antiquity---utilize such concepts as truth, falsity, meaning, and reference. These paradoxes cannot be formulated within logic but occur in a language in which the semantic features of words and sentences are discussed. These are the semantic paradoxes. An example of a semantic paradox is that propounded by Epimentides. Epimenides asserted, during a certain interval of time, t, that what he asserted during that period of time was false. And he asserted nothing else during t. It follows that what he asserted at t is true if and only if it is false. The very existence of this paradox seems to show not only that the premises are untrue but that they could not be true. Yet it is difficult to believe that this is so. The problem seems to arise not merely because certain words were uttered but because of some underlying conceptual confusion.

    此外，各种悖论---其中一些在古代已为人所知---利用如真、假、含义以及参照等。这些悖论无法在逻辑中表述，但会出现在讨论词汇和句子语义特征的语言中。这些都是语义悖论。语义悖论的一个例子是由埃庇蒙德斯提出的。埃庇蒙德斯断言，在一定的时间间隔内，t，他在那段时期间内断言的东西为假。并且他断言在t期间没有其它的东西。因此，他在t时所断言的东西为真，当且仅当它为假。这种悖论的存在似乎不仅表明那个前提是不真实的，而且它们不可能为真。然而，很难让人相信这是真的。提出这个问题不只是因为使用的某些词汇，而是因为一些潜在的概念混淆。

   A number of ways are available to solve this and other similar paradoxes, though none of these ways seem to be in thorough accord with common sense. One way is to construct a hierarchy of sentences reminiscent of the hierarchy of classes. At the zeroth level are those significant sentences that do not ascribe truth or falsity to any sentences. At level 1 are the sentences ascribing truth or falsity to sentences of level 0. And, in general sentences of level n ascribe truth or falsity to sentences of the level immediately preceding n.

    有许多方法可用于解决这种和类似的其它悖论，尽管这些方法似乎与常识并不完全一致。有一种方法是构建一种语句的层级结构，对集合层级结构引发联想。在0级是那些不属于任何语句真或假的重要语句。而在1级是那些将真或假归属于0级的语句。另外，在n级的一般语句中会将真或假归属于紧接着n级前的语句。

   A solution to the Epimennides paradox is that a significant ascription of truth value assigns truth or falsity not to itself but only to sentences of the preceding level. Because the Epimenides statement violates this rule, it is not significant and has no truth value. But if the statement of Epimenides is construed to mean that all statements of level n made by Epimentides are false, the paradox would not be forthcoming, since his statement would be of level n+1.

    埃庇蒙德斯悖论的解决方法是一种不把真或假归属于自身，仅归属于上层语句真值的重要归属。因为，埃庇蒙德斯的陈述违反了这一规则，它并不重要，而且无真值。但如果解释埃庇蒙德斯的陈述意味着由埃庇蒙德斯提出的所有n级陈述都为假，那么悖论就不会出现，因为他的陈述应该属于n+1级。

                                                ARTHUR SMULLYAN

                                         Author of “Fundamentals of Logic”

                                                 亚瑟·斯穆里安

                                             “逻辑学基础”的作者

                                              2022年10月14日译

（译者注：该词条位列《大美国百科全书》1985年版，第21卷，第399页至400页）