2016-01-05
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The Liar Paradox (and other beasties)
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Barber Paradox
[A short fictive dialogue between a detective (d) and a suspect known as Jack (j).]
d: your whiskers are well tended, and your lips and kin are exceptionally smooth. May I ask who your barber is? j: I'm afraid I'm my own barber. d: are you really? j: I confess that I am somewhat paranoid. I do not trust anybody with a knife in the vicinity of my throat. d: you must be very good with knives, then. j: I can shave myself, if that's what you mean.
Could Jack be the barber that everyone is talking about? The one that shaves all men in the village that do not shave themselves? I see no reason why not. Two reasons for that:
1) you can be your own barber, 2) you do not have to be a barber to shave yourself.
The Barber paradox only makes sense if we make another assumption: 3) If you shave yourself, you are not a barber.
When the barber shaves himself, he does what men throughout history have done: shave themselves. The paradox rests on the false assumption that when he is shaving himself he is still behaving as a barber. Being a barber is the job or function of the person we are referring to. But in the apparent paradox, the term gains an existence of its own and replaces the real person as it were. 'Barber' becomes its own reference.
Last but not least, 'the barber' in this apparent paradox behaves like a logical variable, and logicians are very attached to the rule that states that one may not change the meaning of a variable during an argumentation. A rule which of course is happily violated anytime a real discussion in natural language takes place. Quine's solution, that such a barber as depicted by the classic paradox cannot exist, is the result of a very poor analysis of natural language rhetorics. He takes the formulation of the paradox at its literal value and sees then no other way to save Logic (as a human faculty and as a discipline) than to magically declare the barber as a non-entity.
To sum up: "The barber" is not the person who does the shaving, but the name or description we give of him. Kripke made abundantly clear that a description is other than a name, and that a person, miraculously, could turn out to be someone entirely in some alternate time line. Whatever conception strikes your fancy, "the barber" remains a string of sounds to which we first must give a meaning, and not necessarily the same meaning each time.
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