Discussion
In modern logic theory this scenario is not a paradox. The law of implication reconciles what Uncle Joe claims are incompatible hypotheticals. This law states that "if X then Y" is logically identical to "X is false or Y is true" (¬X ∨ Y). For example, given the statement "if you press the button then the light comes on," it must be true at any given moment that either you have not pressed the button, or the light is on.
In short, what obtains is not that ¬C yields a contradiction, only that it necessitates A, because ¬A is what actually yields the contradiction.
In this scenario, that means Carr doesn't have to be in, but that if he isn't in, Allen has to be in.
Read more about this topic: Barbershop Paradox
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