乌鸦悖论

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一只黑乌鸦

乌鸦悖论，也叫做亨佩尔的乌鸦或亨佩尔悖论，是二十世纪四十年代德国逻辑学家卡尔·古斯塔夫·亨佩尔（-{Carl Gustav Hempel}-）为了说明归纳法违反直觉而提出的一个悖论。

[编辑] 问题的综述

几千年以来，无数人观察了许多事务，比如地心引力法则，人们趋于相信其极可能是真理。这种类型的推理可以总结成“归纳法原理”：

如果实例 X 被观察到和论断 T 相符合，那么论断 T 正确的概率增加。

亨佩尔给出了归纳法原理的一个例子： “所有乌鸦都是黑色的”论断。我们可以出去观察成千上万只乌鸦，然后发现他们都是黑的。在每一次观察之后，我们对“所有乌鸦都是黑的”的信任度会逐渐提高。归纳法原理在这里看起来合理的。

非黑非乌鸦

现在问题出现了。“所有乌鸦都是黑的” 的论断在逻辑上和“所有不是黑的东西不是乌鸦”等价。如果我们观察到一只红苹果，它不是黑的，也不是乌鸦，那么这次观察必会增加我们对“所有不是黑的东西不是乌鸦”的信任度，因此更加确信“所有的乌鸦都是黑的”！这个问题被总结成 (摘自吉利特·伯吉斯的诗：(en:Gelett Burgess) ：

我从未见过紫牛（-{I never saw a purple cow}-）

如果我见到一头（-{But if I were to see one}-）

乌鸦皆黑的概率（-{Would the probability ravens are black}-）

将会更加可能么（-{Have a better chance to be one?}-）

[编辑] 可能的解决

解决它和直觉的冲突，哲学家们提出了一些方法。美国逻辑学家纳尔逊·古德曼（en:Nelson Goodman）建议对我们的推理添加一些限制，比如永远不要考虑支持论断“所有P满足Q”且同时也支持“没有P满足Q” 的实例。

其他一些哲学家质疑“等价原理”。也许红苹果能够增加我们对论断“所有不是黑的东西不是乌鸦”的信任度，而不增加我们对 “所有乌鸦都是黑色的”信任。这个提议受到质疑，因为你不能对等价的两个命题有不同的信任度，如果你知道他们都是真的或都是假的。

古德曼，以及其后的蒯因，使用术语 projectible predicate to describe those expressions, such as 乌鸦 and 黑色, which do allow inductive generalization; non-projectible predicates are by contrast those such as non-black and non-raven which apparently do not. (See also grue, another non-projectible predicate invented by Goodman.) Quine suggests that it is an empirical question which, if any, predicates are projectible; and notes that in an infinite domain of objects the complement of a projectible predicate ought always be non-projectible.

This would have the consequence that, although "All ravens are black" and "All non-black things are non-ravens" must be equally supported, they both derive all their support from black ravens and not from non-black non-ravens.

Some philosophers have argued that it's only our intuition that is flawed. Observing a red apple really does increase the probability that all ravens are black! After all, if someone gave you all the non-black things in the universe, and you noticed that there were no ravens in the collection, then you could indeed conclude that all ravens are black. The example only seems counterintuitive because the set of non-black-things is far, far larger than the set of ravens. Thus observing one more non-black-thing which is not a raven should make a tiny difference to our degree of belief in the proposition compared to the difference made by observing one more raven which is black.

There is an alternative to the "归纳法原理" described above.

设 X 表示论断T的一个实例, and I represent all of our background information.
Let $\Pr(\bullet | \circ)$ represent the probability of $\bullet$ given $\circ$ . Then,

$\Pr(T|XI) = \frac{\Pr(T|I) \cdot \Pr(X|TI)}{\Pr(X|I)}$

This principle is known as "Bayes' theorem". It is foundational to the mathematics of probability and statistics. When scientists publish analyses of experimental results and calculate that they are "statistically significant", they are implicitly using this principle. It could be argued that this principle is a better representation of how scientists actually reason than the original "principle of induction" described above.

Using this principle, the paradox does not arise. If you ask someone to select an apple at random and show it to you, then the probability of seeing a red apple is independent of the colors of ravens. The numerator will equal the denominator, the ratio will equal one, and the probability will remain unchanged. Seeing a red apple will not affect your belief about whether all ravens are black.

If you ask someone to select a non-black-thing at random, and they show you a red apple, then the numerator will exceed the denominator by an extremely small amount. Seeing the red apple will only slightly increase your belief that all ravens are black. You'll have to see almost every non-black-thing in the universe (and see they're all non-ravens) before your belief in "all ravens are black" increases appreciably. In both cases, the result agrees with intuition.

However, a hypothetical experiment can demonstrate that there is a problem with the above reasoning: Suppose all the ravens in the universe were magically placed inside a large box, and that you have not seen a raven yet. Now imagine going over every non-black thing in the universe (with the exception of looking inside the box) and verifying it is indeed not a raven. The problem is that after doing this you could still not make any statement regarding the color of a raven. The reason is that you could also have verified in the same manner that every non-red, non-green or non-blue thing in the universe is also not a raven (since the ravens are safely in the box). So at the end of the experiment you have seen almost every non-black thing in the universe and verified it is not a raven, but having done that did not contribute anything to knowing the color of one raven.

[编辑] 参见

贝叶斯推理

[编辑] 参考书目

Hempel, C. G. A Purely Syntactical Definition of Confirmation. J. Symb. Logic 8, 122-143, 1943.
Hempel, C. G. Studies in Logic and Confirmation. Mind 54, 1-26, 1945.
Hempel, C. G. Studies in Logic and Confirmation. II. Mind 54, 97-121, 1945.
Hempel, C. G. Studies in the Logic of Confirmation. In Marguerite H. Foster and Michael L. Martin, eds. Probability, Confirmation, and Simplicity. New York: Odyssey Press, 1966. Pp 145-183
Falletta, Nicholas. The Paradoxicon: a Collection of Contradictory Challenges, Problematical Puzzles, and Impossible Illustrations. 1983. Pp 126-131. ISBN 0385179324