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Barber Paradox (Russell's Paradox)


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#21 BoilingOil

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Posted 26 September 2007 - 08:17 PM

Barber Paradox (Russell's Paradox) - Back to the Paradoxes

Analogue paradox to the paradox of liar formulated English logician, philosopher and mathematician Bertrand Russell.
There was a barber in a village, who promised to shave everybody, who does not shave himself (or herself).
Can the barber shave himself and keep the mentioned promise?

Edited (better wording?):
In a village, the barber shaves everyone who does not shave himself/herself, but no one else.
Who shaves the barber?



The simple solution is lying in the fact that the barber never promised to shave ONLY people who do not shave themselves. He could shave anyone, including himself.
This way there would not be any paradox at all.

The edited version does not mean exactly the same as the original, after all, since the original didn't say he was excluding anyone who shaved themselves. In fact, the edited version IS a paradox.. or at least, so it seems to me.


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#22 heins57

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Posted 30 September 2007 - 08:32 AM

No one shaves the Barber. He does not shave himself. Or rather, personaly the barber does not shave, But he will shave anyone who wants to be shaved granted they live in the village.
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#23 Bach.nics

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Posted 23 October 2007 - 02:33 AM

The original version, "There was a barber in a village, who promised to shave everybody, who does not shave himself" translates in pure logic (if trigger, then result) to:

(1-A) If someone does not shave himself, then the barber will shave him.

The contrapositive is:

(1-B) If the barber does not shave someone, then that person must have shaved himself."

The triggers here are:
(1-A) someone does not shave himself
(1-B) the barber does not shave someone

When the barber shaves himself, neither of these triggers is activated, and thus neither of the results is required. This resolves the paradox. There is no contradiction; the barber CAN keep his promise.

Now, if you consider rooki1ja's edited wording up front, "but no one else," or Boiling Oil's comment on "ONLY," we get to the heart of the matter. Now there is a contradiction because "but no one else" creates a second conditional statement:

(2-A) If someone does shave himself, then the barber does not shave him.

and its contrapositive,

(2-B) If the barber does shave someone, then that person does not shave himself.

NOW when the barber shaves himself, he triggers both 2-A and 2-B which then requires the results of both 2-A and 2-B. Naturally, both results are violated and we have a contradiction. The barber CANNOT keep his promise. This is a paradox that does not resolve.

I'm saying the same thing as Boiling Oil, except in the first scenario he says "no paradox" using the word paradox to mean specifically a paradox with no resolution. Dictionary.com allows in separate definitions that a paradox may or may not have a resolution, so we can say scenario 1 is a paradox that resolves, and scenario 2 is a paradox that does not resolve.

Cheers!
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#24 peseta

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Posted 24 October 2007 - 08:24 PM

I think the original didn't add 'herself', leading to much embarassment at the gaffe, today understood to be sexism. I wonder how many paradoxes result from linguistic errors of sorts. All?
So here, if I shave all those who don't shave 'themselves', it first depends on what that means. How many shavers shave their brows? If not shaving one's brows means not shaving 'oneself,' but 'shaving' refers (as it did in Russell's paradox) to beards, the paradox is gone with a woman barber.
There's also the previous point about a PROMISE to shave. Are we to expect the barber to be a Sweeney Todd, shaving willy-nilly anyone who passes by?
Avoid such barbers, or risk a close shave.
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#25 biostrategem

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Posted 14 November 2007 - 12:02 AM

Firstly, I love Russell's work. However, his paradox (again, not really a paradox) has a rather simple solution.

===============================
Barber Paradox (Russell's Paradox)

Analogue paradox to the 'liar paradox' formulated by English logician, philosopher and mathematician Bertrand Russell. There was a barber in a village who promised to shave everybody that does not shave himself (or herself). Can the barber shave himself and keep the mentioned promise?
===============================

We have ten givens to consider in this challenge:

1) There exists bodies.
2) There exists a village; A village is a set of nonzero bodies.
3) There exists a kind of body called a Barber.
4) There exists an operation called Shaving.
5) Barbers may Shave bodies.
6) All barbers must be bodies.
7) All bodies must live in the village.
Some bodies may be barbers.
9) Some bodies may Shave themselves.
10) Barbers may shave bodies other than his own. (We assume female Barbers are called Barberellas and outside the scope of this challenge.)

The barber promises to shave everyone in the village who does not shave themselves.

This may be easier to understand if we restate the givens, substituting the word "Like" for the operation "Shave", Shape for Body, and Triangle for Barber. For kicks, let's finish up with substituting Box for Village.

1) There exists Shapes.
2) There exists a Box; A Box is a set of nonzero Shapes.
3) There exists a kind of Shape called a Triangle.
4) There exists an operation called Liking.
5) Triangles may Like (and therefore not Like) Shapes.
6) All Triangles must be Shapes.
7) All Shapes must live in the Box
Some Shapes may be Triangles.
9) Some Shapes may Like themselves.
10) a Triangle may Like Shapes other than his own.

Given: The triangle promises to like every shape in the Box that does not like itself.

In the Question we are given another truth:
Question: Can the triangle ("the" implies there is only one triangle in the box) like itself and like every shape in the box that does not like itself?

11) There is only one Triangle in the box.

Weeding through the extraneous data, the key to the solution lies in given 10. If a shape is a Triangle, it may like shapes other than his own. What given 10 does not say is that a shape may like shapes other than his own if and only if it is a triangle.

This is not a paradox, but rather a simple first-level logic exercise in the difference between if-then and if-and-only-if. Simply put, the people in the village can shave each other if the Barber cannot shave them. That would formulate the scenario that there is nobody in the village that needs a shave if the Barber shaves himself, therefore keeping the promise - The barber shaved everybody in the village that needed a shave after he shaved himself, and the number of needed villagers was zero.


Eric Mumford
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#26 callumny

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Posted 15 November 2007 - 03:15 AM

Gender is irrelevant. Some women do have beards, and besides, the question doesn't specify facial hair. But that's beside the point, really: the barber promised to shave each person who does not shave him or her self... not just those who actually need a shave.
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#27 callumny

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Posted 15 November 2007 - 03:39 AM

A lot of people seem to have missed the point. Say there is a guy in the village named Ted. One of the following must be true:
Ted shaves Ted, and the barber does not shave Ted.
Ted does not shave Ted, and the barber shaves Ted.

If Ted is the barber, you can replace "the barber" with "Ted":
Ted shaves Ted, and Ted does not shave Ted.
Ted does not shave Ted, and the barber shaves Ted.

This is not a trick question. There is no validity to any statement like, "Ted does not have a beard, so Ted does not count."

The barber cannot keep his promise, from a logical standpoint. From an ethical standpoint, on the other hand, most people would realise he was not including himself in "everyone", so he certainly could keep his promise.
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#28 Maddhatter77057

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Posted 19 November 2007 - 07:44 PM

The barber is a women. She doesn't shave
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#29 biostrategem

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Posted 19 November 2007 - 07:55 PM

What women are you dating that don't shave?
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#30 urkspleen

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Posted 21 November 2007 - 03:49 AM

In this particular paradox, there are many small factors that need to be addressed first:

-when will the Barber die
-how many people are there in the village
-what item will he shave with, and what happens if it breaks
-how much hair do people have
-just head hair or also facial hair, body hair, etc.
-will the cooperation of the people be a factor
????????????????????????????????????????????????????????????????????????????????????

In a real world, all of these little factors contribute to the overall outcome. All of these need to be addressed before a realistic answer can be given
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