The Barbershop paradox was proposed by Lewis Carroll in a three-page essay entitled "A Logical Paradox" which appeared in the July 1894 issue of Mind. The name comes from the "ornamental" short story that Carroll uses to illustrate the paradox (although it had appeared several times in more abstract terms in his writing and correspondence before the story was published). Carroll claimed that it illustrated "a very real difficulty in the Theory of Hypotheticals" in use at the time.
Read more about Barbershop Paradox: The Paradox, Simplification, Discussion
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