The Barber’s Paradox

There’s this painter who is a woman. She has a strange condition placed upon her by the powers that be: she must paint on those and only those who do not paint on themselves. And so, if one lady doesn’t paint on herself, then the painter has to paint on her. And if another guy paints on himself, then the painter cannot paint on him. But what about the painter, does the painter paint on herself?

Let’s venture a guess: let’s assume the painter paints on herself. However, in this case, the painter cannot paint on herself, since the painter paints on only those who don’t paint on themselves. But then maybe that’s the answer we are looking for: the painter does not paint on herself. But in that case, the painter must paint on herself, because she paints on all who do not paint on themselves.

In this paradox, we assume the initial conditions and reason that the painter is the sort of thing that either paints on herself or does not paint on herself. Considering each disjunct in turn leads to the contradiction. This is the Barber’s . . . oh wait . . . Painter’s Paradox!

My Painter’s Paradox simply replicates the form of the Barber’s Paradox because the Barber’s Paradox has a general logical form that can be replicated with any two-place action (an action between a subject and an object) that can also be self-reflexive (an action by the subject upon the subject). The standard version of this paradox is of a barber who shaves those and only those who don’t shave themselves.

The logical form of the Barber’s Paradox is:

∃x ∀y (Pxy→~Pyy) & (~Pyy → Pxy)

Here, instead of the shaving relation, Pxy will be the two-place action that can also be self-reflective of painting, i.e. x paints on y. In English, the claim says: There is something, x, such that if x paints on anything y, then y does not paint on y, and if anything y does not paint on y, then x paints on y. We can derive a contradiction by giving an interpretation where x = y, and assuming the perfectly sensible law of excluded middle (LEM) that the painter either paints on herself or does not paint on herself:

  1. ∃x ∀y (Pxy→~Pyy) & (~Pyy → Pxy). The starting claim.
  2. ∀y(Pay→~Pyy) & (~Pyy → Pay). Let’s assign variable x in 1 to a particular thing, a; thus, x = a.
  3. (Paa→~Paa) & (~Paa → Paa). Let’s assign variable y in 2 to be that very same a, since y has anything at all as its scope; thus, y = a.
  4. Paa v ~Paa. Here is the LEM that a, which would be the painter, either paints or does not paint herself. Since a is the sort of thing that can either paint or not paint on herself, then this is a valid form of the law of excluded middle, which is universally true.
  5. Paa. Assume the first disjunct of 4.  
  6. Paa→~Paa. Conjunction Elimination from 3.
  7. ~Paa. Modus Ponens on 5 and 6.
  8. ~Paa. Assume the second disjunct of 4.
  9. Derive ~Paa. This is the result of either disjunct of 4, shown in 7 and 8 (Disjunction Elimination).
  10. Paa. Assume the first disjunct of 4.
  11. ~Paa. Assume the second disjunct of 4.
  12. ~Paa→ Paa. Conjunction Elimination from 3.
  13. Paa. Modus Ponens on 11 and 12.
  14. Derive Paa. This is the result of either disjunct of 4, shown in 10 and 13 (Disjunction Elimination).
  15. Paa & ~Paa. Conjunction Instantiation of 14 and 9.

Line 15 has the contradictory result that a (the painter) paints on herself and does not paint on herself. This result comes about when we offer an interpretation of 1 such that x is identical with y and then assume the law of excluded middle that Paa v ~Paa. The LEM is clearly sensible. That is, it is clear what it is for a painter to paint on herself and it is equally clear what it is for a painter to not paint on herself. Thus, the LEM is not the part of the claim that contains the contradiction, but rather the contradiction is contained in the initial conditions.

It is clear that the initial conditions on the painter embed the contradiction that results if we reword the initial conditions such that it is explicit that the painter herself can be considered under the scope of her own painting. Let’s reword the Barber’s/Painter’s Paradox to accurately reflect this possibility.

The Barber’s/Painter’s Paradox, where it’s explicit that the one who paints may be the one painted on:

The painter paints on all those and only those who do not paint on themselves, and, since this includes the painter herself, if the painter does not paint on herself, then the painter paints on herself, and if the painter paints on herself, then the painter does not paint on herself.

That the initial conditions themselves are contradictory is much more apparent here, the latter half claiming the equivalent of (Where P is a proposition–i.e. a claim that is either true or false) ~P → P & P→ ~P. The contradiction (P & ~P) results by merely assuming the law of excluded middle (P v ~P) which is hardly a step for being clearly meaningful in this case, and thus universally true.

Look at the boldface conditions on our painter once more. Would even a child accept such conditions, or think they’re without contradiction? Clearly no one would accept it!

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