Assuming that the barber properly finishes his job, he's going to shave
or not shave himself... He probably does not shave all of his body
hair, or of any other man in the town, so in that sense he shaves
himself to a certain degree. If the question being asked was 'Does the
barber shave all men who do not shave themselves?', one could answer in
a particularly vague fuzzy way, which would be to divide the men of the
town who do not shave themselves minus the barber by the men of the town
who do not shave themselves (1/m), and call the rest of the ratio
ambigous. So, if the town had 20 men who do not shave themselves, the
barber would shave all men who do not shave themselves to a .95 degree,
with the other .05 being ambigous or paradoxical. However, this has
nothing to do with Russell's paradox, as the question is whether the
barber shaves himself. The point of Russell's Paradox was that within
formal logic there is no way out, vague or otherwise(vague being
Russell's term for what we now call fuzzy). Following the time that he
formulated this paradox, Russell attempted to formulate a system of
logic that would disallow paradoxes, but failed. Somewhat later Kurt
Godel proved that within any formal system of logic there would be at
least one statement that could not be proved to be true or not true, so
we are trapped with the paradoxical barber forever (at least, the
formalists are).
-Christian Whitaker
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