Barber's Paradox

[Christopher]

There is a village where the barber shaves all those and only those who do not shave themselves. Who shaves the barber?

-- I have read some responses that suggest it is one of "Russell's paradoxes" that is ultimately unanswerable, and I have read some interesting responses that actually provide an answer that makes sense. What are your thoughts?

[Judith]

There is a village where the barber shaves all those and only those who do not shave themselves. Who shaves the barber?

As stated, it doesn't seem that difficult. He has an exclusive business, i.e., no competition. Some people shave themselves and don't use his services. And he shaves himself as well.

Judith

[Brant Gaede]

Where is this village?

What did he shave?

Why is a paradox a problem?

--Brant

there is a village where bats eat the fruit--who eats the bats? There is a village where everybody gets laid, where is this village? There is a village where no one starves--who starves the village? There is a village that get hit by an asteroid, who hits the asteroid? This may be me being stupid, but I tried to rise to the occasion.

--Brant

[Dragonfly]

There is a village where the barber shaves all those and only those who do not shave themselves. Who shaves the barber?

As stated, it doesn't seem that difficult. He has an exclusive business, i.e., no competition. Some people shave themselves and don't use his services. And he shaves himself as well.

Note that he shave only those who don't shave themselves, so if someone in the village shaves himself, the barber does not shave him. But that implies that the barber does not shave himself because he shaves himself, and that is a paradox that cannot be resolved.

[BaalChatzaf]

The Barber's Paradox is a non-mathematical way of stating Russell's Paradox.

See: http://en.wikipedia.org/wiki/Russell%27s_paradox

Russell showed that in Frege's theory of sets there is an uresolvable contradict:

Let S = the set of all sets which are not elements of them selves. Such sets exist. For example, the set of all books is not a book so that set is not an element of itself. On the other hand the set of all abstractions is an abstraction so that it is an element of itself.

Now is S and element of S or not? If it is, then by membership definition it is not an element of itself. On the other hand if the set S is not an element of itself, then by membership definition it is an element of itself. Hence S is an element of itself if and only if it is not an element of itself. That is a flat out contradiction.

The way set theory is "saved" from this paradox is to restrict the notion of set so that this paradox cannot arise. This is accomplished through the theory of types (proposed by Russell). Also Zermelo in his axiom system avoids the paradox. Both approaches restrict the generality of the set concept do that it does not apply to itself in a simple fashion.

The form of the paradox is exploited by Goedel in his famous Incompleteness Theorems. Again it is used to show that the Halting Problem for Turing Machines is not solvable using a Turing Machine.

Ba'al Chatzaf

[tjohnson]

Also Korzybski uses Russell's Theory of Types in his orders of abstraction theory. A proposition about previous propositions is a proposition (abstraction) of a higher order. This restricts universal statements so they cannot apply to themselves, only statements of lower order. For example, the statement "All my statements are false" is modified to "All my previous statements are false", which allows this one to be true.

[merjet]

It is a paradox because it is contradictory, if one thinks of only people that live in the hypothetical village. But the question is answerable -- somebody who who lives in another village shaves the barber.

[Dragonfly]

It is a paradox because it is contradictory, if one thinks of only people that live in the hypothetical village. But the question is answerable -- somebody who who lives in another village shaves the barber.

That doesn't solve the paradox: if someone from another village shaves the barber, the barber doesn't shave himself. But it was stated that the barber shaves all the people in the village who don't shave themselves; one of these people is the barber himself, so he shaves himself because he doesn't shave himself - the paradox is still there.

[Philip Coates]

Subject: Verbal Paradoxes

> he shaves himself because he doesn't shave himself - the paradox is still there.

I wouldn't use the term paradox. It often implies to the unwary that there are contradictions in reality. And there are no contradictions in reality.

Many of the alleged "paradoxes" they used to waste our time with in my (long-despised) freshman "Philosophy 101" college course rest on trying to deduce a conclusion from **a statement that is itself contradictory**.

This is one of those cases, since the first statement -itself- is not possible for the very reason that has been pointed out: it implies both that the barber does and does not shave himself. [Not all of the freshman philosophy paradoxes are of this form. Zeno's paradox is not. [One type, like the village barber, is more a 'verbal' paradox. The other is more a 'factual' paradox - not meaning that it is real - Aristotle refuted it, just that it is not just a word game but is about physics or an action taken in reality.]

There is no conundrum here - simply a statement at the very start that can't possibly be true. Once you see that, there is no actual head-scratching puzzle, case is closed, move on.

You can construct an infinite number of similarly self-contradictory statements:

1) "I am thinking of a thought no one has thought of before". What you mean is no one else. It's simply a sloppy statement or the grammatically permitted "elliptical statement" in which a qualification is omitted but implied. A case in which English 101 (and understanding grammar) trumps Philosophy 101.

2) "I always tell the truth, but this statement is a lie."

It's just the nature of language. One can construct statements into which is smuggled the idea that S is P and in the same statement that S is Non-P. The solution is to simply discard the statement.

Best not to kill any more neurons on a topic this non-productive (and non-puzzling). In college, I remember the grinning professor confusing the more gullible students with a whole host of alleged 'paradoxes' under the guise of making them think, instead of doing actual philosophy about real, life and death issues in ethics or politics or epistemology. Once I unsnarled them, my contempt for this kind of class and contemporary academic philosophy simply grew or, in this case, had its start.

. . . Then you've got 'Venus is the Morning Star', 'Venus is the Evening Star' . . . . . Oh, never mind.

------

I just looked up 'paradox' on wikipedia. It makes some of the above points, such as self-contradiction, so I'm hardly original. For example, the philosopher WVO Quine makes a similar distinction to the one I just had between two types of paradox - he calls them falsidical and veridical.

[merjet]

That doesn't solve the paradox: if someone from another village shaves the barber, the barber doesn't shave himself. But it was stated that the barber shaves all the people in the village who don't shave themselves; one of these people is the barber himself, so he shaves himself because he doesn't shave himself - the paradox is still there.

Then try this. The barber doesn't live in the village. The paradox as given doesn't clearly say he/she does.

Edit: This paradox is addressed on Wikipedia. http://en.wikipedia.org/wiki/Barber_paradox

Edited July 26, 2009 by Merlin Jetton

[Dragonfly]

Then try this. The barber doesn't live in the village. The paradox as given doesn't clearly say he/she does.

Hmmm... I think that is a bit contrived. Of course if you want to be exact you must give unambiguous definitions or use symbolic logic, but this version is a translation of the paradox in a single sentence in everyday language with its ambiguities, to make it easy to understand the essence of the paradox. It's like those approximate proofs of mathematical theorems for non-mathematicians, which give the essence of the proof without the rigor of a formal proof with all its details and axioms. You cannot say that you solve the paradox by claiming that the barber doesn't live in the village, that implies only that you'd better formulate the paradox more accurately to avoid such ambiguities. And then the paradox cannot be solved.

[merjet]

You cannot say that you solve the paradox by claiming that the barber doesn't live in the village, that implies only that you'd better formulate the paradox more accurately to avoid such ambiguities. And then the paradox cannot be solved.

Avoiding all the ambiguities would make it contradictory and a paradox that fits only meaning 3 here.

[Brant Gaede]

All in all I think it best to keep shaving myself.

--Brant

[Selene]

Does the barber only shave the men in the village?

Adam

[BaalChatzaf]

Forget about barbers. Pay attention to the paradox, which is genuine. When Russell let Frege know about this, Frege was sorrowful. Several years of his life went up in smoke.

Ba'al Chatzaf

[Christopher]

General Semanticist, can you explain more about Korzybski's use of abstraction theory as it regards this paradox? I'm trying to understand it better.

Regarding Goedels' Incompleteness Theorem, here it is, let's try to figure this out:

1. Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true,[1] but not provable in the theory (Kleene 1967, p. 250).

2. For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent.

Relation to the liar paradox

The liar paradox is the sentence "This sentence is false." An analysis of the liar sentence shows that it cannot be true (for then, as it asserts, it is false), nor can it be false (for then, it is true). A Gödel sentence G for a theory T makes a similar assertion to the liar sentence, but with truth replaced by provability: G says "G is not provable in the theory T." The analysis of the truth and provability of G is a formalized version of the analysis of the truth of the liar sentence.

It is not possible to replace "not provable" with "false" in a Gödel sentence because the predicate "Q is the Gödel number of a false formula" cannot be represented as a formula of arithmetic. This result, known as Tarski's undefinability theorem, was discovered independently by Gödel (when he was working on the proof of the incompleteness theorem) and by Alfred Tarski.

I think in both cases 1 & 2 above, Goedel is asserting that systems are built upon premises which: A) cannot be self-proving nor can prove anti-premises (although such statements might be capable of being made); and B) if all truths including the premises or anti-premises are proved, then outside logic was necessary to make those proofs stable, and therefore the system itself is non-closed or inconsistent.

Relating to liar's paradox, it sounds like "this sentence is false" means that one cannot prove the following sentence is true given the theory that the sentence is true. In other words, since the sentence directly relates to the premises of the system, there are no set of tools capable of proving such a statement within the system built upon the premises. (?)... perhaps. It's hard to be sure of one's own logic when addressing such problems.

Chris

[tjohnson]

General Semanticist, can you explain more about Korzybski's use of abstraction theory as it regards this paradox? I'm trying to understand it better.

From here.

This solution to Russell's paradox is motivated in large part by the so-called vicious circle principle, a principle which, in effect, states that no propositional function can be defined prior to specifying the function's scope of application.

In this case the propositional function is "There is a village where the barber shaves all those and only those who do not shave themselves". Now if you don't limit the scope of this, ie. you must not include the barber himself, it results in a contradiction. So it needs to be reworded to "There is a village where the barber shaves all those and only those, excluding the barber, who do not shave themselves." So really it is a case of illegitimate totalities, whether we are discussing objects or propositions, it seems we must be careful using the little word "all" (and other absolute words - see my signature ).

[Philip Coates]

> needs to be reworded to "There is a village where the barber shaves all those and only those, **excluding the barber**, who do not shave themselves." [general semanticist]

Very true.

In grammar, leaving out the above highlighted phrase is called 'ellipsis' (not the kind with the three dots...). A part of a sentence is left out which is implied and understood by the listener. The reason for ellipsis is cognitive economy. If writers and speakers spelled everything out or made every qualification explicit, communication would be impossibly wordy. Like any grammatical tool, it evolved because it is entirely rational and necessary. And not 'paradoxical' or puzzling for those (unlike some analytic philosophers) who learned grammar thoroughly in high school.

Four examples of ellipsis or sentence shortening:

1. "It is raining." --> i) here and now, not in Russia or in Ancient Greece; ii) significant moisture falling in drops; not mist and not snow.

2. "He is as tall as his sister" --> as tall as his sister is.

3. "In four innings, the starting pitcher pitched to everyone" --> on the opposing team.

4. "Mary, John, and Peter walked home" --> Mary walked home. John walked home. Peter walked home.

Once again, logically, this should end the angels on a head of a pin discussion of a 'puzzle' in the Barber's paradox (and quite a few others). Note that the parentheses in my last sentence contained, guess what?, an ellipsis. Whose meaning is crystal clear to any careful reader.

[Philip Coates]

> it seems we must be careful using the little word "all" (and other absolute words) [g.s.]

The word 'all' can properly be used elliptically in many, many cases to mean all within a certain limited context.

Or to put it in Oist terminology, contextually: "All the soldiers returned safely". [Not the ones who never went out on patrol; not the generals; not the support staff at base.] "All the land was soaked by the rain". . . .

[BaalChatzaf]

Forget the god-damned barber. Look at the paradox!

Ba'al Chatzaf

[Dragonfly]

Forget the god-damned barber. Look at the paradox!

Exactly! This isn't some semantic issue as some people seem to think.

[Philip Coates]

> Forget the god-damned barber. Look at the paradox! [baal] / Exactly! This isn't some semantic issue. [Dragonfly]

Yes it is.

Baal and Dragonfly, I carefully analyzed in posts #9, #19, #20 why there is no real problem. After three [ignored] posts, I can't imagine what else I could say to clarify this issue. If you won't take it from me, I also linked to an on-the-whole good article on Wikipedia which makes some similar points. I've discussed the grammatical concept often involved. [i forgot to add that the barber statement can be viewed as *ambiguity* rather than merely *ellipsis*, depending on intent of the speaker.]

I've made the metpahysical point (which I hope those who study Rand would agree with) that there are no contradictions (or paradoxes) in reality. Another ignored point.

Category error: You both don't see that this is an issue in English or in Linguistics. Not in Philosophy.

I also indicated the wider point that often philosophical conundrums (especially from modern philosophy) are simply based on semantic muddles, verbal omissions or ambiguities.

[Philip Coates]

Or imprecise definitions. Including those which are self-contradictory or otherwise invalid.

[Dragonfly]

The Barber paradox is just an illustration for the layman of Russell's paradox, and it has all the disadavantages of using imprecise layman's terms. Russel's paradox is not trivial and is not about semantics. See for example http://en.wikipedia.org/wiki/Russell's_paradox .

[Christopher]

I'm going to echo Philip's words: paradoxes cannot exist in reality. Although I find this one endlessly entertaining, I equally enjoy our discussion about how our use of logic allows for the statement of such a paradox.

Here's something more that will blow your mind concerning logic and paradoxes (the simplified version!):

We will first state a simplified version of Tarski's theorem, then state and prove in the next section the theorem Tarski actually proved in 1936. Let L be the language of first-order arithmetic, and let N be the standard structure for L. Thus (L, N) is the "interpreted first-order language of arithmetic." Let T denote the set of L-sentences true in N, and T* the set of code numbers of the sentences in T. The following theorem answers the question: Can T* be defined by a formula of first-order arithmetic?

Tarski's undefinability theorem: There is no L-formula True(x) which defines T*. That is, there is no L-formula True(x) such that for every L-formula x, True(x) ↔ x is true.

Informally, the theorem says that given some formal arithmetic, the concept of truth in that arithmetic is not definable using the expressive means that arithmetic affords. This implies a major limitation on the scope of "self-representation." It is possible to define a formula True(x) whose extension is T*, but only by drawing on a metalanguage whose expressive power goes beyond that of L, second-order arithmetic for example.

I forget what Rand called this... when swinging her arm and declaring in essence "here it is" for everything that exists but cannot be proved. And anyway, you know what they say: give a paradox some time and privacy, and eventually you'll have a family of little ducklings...