The Liar Paradox

The Crocodile Paradox

A croc catches a child by the Nile. The conversation follows:

--Please do not eat my child.
--OK. If you can predict what I will do I will free your child but if you are wrong, I will devour it.
--Ok croc! I predict you will eat my child.
--OK. If I do free your child, then you will have guessed wrong. So I have to eat it.
--No croc! If you eat it then I will have predicted correctly. So you have to let it go.

Source: H.M. Hubey, The Diagonal Infinity: Problems of Multiple Scales, World Scientific, Singapore, 1998; ISBN 9810230818

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The Liar Paradox
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Semantic paradox, known in antiquity, focus of much recent work. Jack says 'I am now speaking falsely', referring to the words he is then uttering. If Jack speaks truly when he says he is speaking falsely, he is speaking falsely. If he is speaking falsely when this is what he says is going on, he is speaking truly. So what he says is true if, and only if, it is false; which seems absurd. One response claims that Jack says nothing true and nothing false. But a variant makes trouble: Jill says 'I am now not speaking truly'. If Jill is not speaking truly when this is what she says she is up to, she is speaking truly. If she is speaking truly, then she must be doing what she says, that is, not speakingly truly. So, it seems, what she says is true if, and only if, it is not true.

The Oxford Companion to Philosophy, © Oxford University Press 1995

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The barber paradox

The barber in a certain village is *a man who shaves all and only those men in the village who do not shave themselves*. Is he a man who shaves himself? If he is then he isn't, and if he isn't then he is. It follows that he is a man who both does and does not shave himself. This contradiction shows that the apparently innocent italicized description can apply to no one.

Formally, the paradox resembles Russell's paradox of the class of all classes which are not members of themselves. The latter though is not so easy to dispose of, since it is generated by an assumption - that every predicate determines a class - which cannot simply be abandoned.

The Oxford Companion to Philosophy, © Oxford University Press 1995

Bibliography: T.S. Champlin, Reflexive Paradoxes, Routledge, London, 1988; ISBN 0415000831.

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Russell's paradox

Central paradox in the theory of classes. Most classes are not members of themselves, but some are; for example, the class of non-men, being itself not a man, is a member of itself. Let R be the class of all classes which are not members of themselves. If it exists, it is a member of itself if and only if it is not a member of itself: a contradiction. So it does not exist. This is paradoxical, because it conflicts with the seemingly inescapable view that any coherent condition determines a class. (Even a contradictory condition, like being round and square, determines a class: the class with no members.) Standard responses, like Russell's theory of types, aim to find some limitation on what classes there are which is (a) intuitively satisfactory, (b) excludes R, and (c) includes all classes needed by mathematicians.