Actually the Barber of Seville is an orthodox Jew and just trims his own beard.

But I think that the best answer will be that the Barber of Seville is female.

Now I am going to look at the answer

The Barber of Seville shaves only the men of Seville who don't shave themselves.

Does the Barber of Seville shave himself?

There's the paradox...

In this context, to claim that there is a paradox is to show how a contradiction is drawn. But there is no contradiction drawn from "The Barber of Seville shaves only the men of Seville who don't shave themselves". Three possibilities are each separately consistent with "The Barber of Seville shaves only the men of Seville who don't shave themselves": It is consistent that the barber is not shaven, and it is also consistent that the barber is shaved by someone who is not a man of Seville, and it is also consistent that the barber is shaved by someone who is a man of Seville.

(1) "The Barber of Seville shaves all the men of Seville apart from himself." [original post]

(2) "The Barber of Seville shaves only the men of Seville who don't shave themselves." [latest post]

Those are two different propositions.

Let 'Sx' stand for 'x is a man of Seville'.

Let 'Hxy' stand for 'x shaves y'.

(1) would be taken to mean "the barber is a man of Seville and if x is a man of Seville who is not the barber, then the barber shaves x", and probably tacit is "the barber does not shave himself".

Sb & Ax((Sx & ~x=b) -> Hbx) & ~Hbb

(2) would be taken to mean "the barber is a man of Seville and if the barber shaves x, then x is a man of Seville who does not shave himself."

Sb & (Hbx -> (Sx & ~Hxx))

But neither (1) nor (2) are paradoxical.

With (1) it is consistent that the barber is not shaven: ~Ex Hxb, and it is also consistent with (1) that the barber is shaved by someone who is not a man of Seville: Ex(~Sx & Hxb), and it is also consistent with (1) that the barber is shaved by someone who is a man of Seville: Ex(Sx & Hxb).

With (2) it is consistent that the barber is not shaven: ~Ex Hxb, and it is also consistent with (2) that the barber is shaved by someone who is not a man of Seville: Ex(~Sx & Hxb), and it is also consistent with (2) that the barber is shaved by someone who is a man of Seville: Ex(Sx & Hxb).

/

The actual contradictory formulation is (and we don't even need to mention maleness, a location such as Seville, or being a barber):

There is someone who shaves all and only those who do not shave themselves:

EbAx(Hbx <-> ~Hxx)

Put another way:

There is someone b such that for all x, if b shaves x then x does not shave x and if x does not shave x then b shaves x:

EbAx((Hbx -> ~Hxx) & (~Hxx -> Hbx))

We could add that men of Seville are shaved only by men of Seville and that every man of Seville is shaved, and still we would not have a paradox in either case (1a) or (2a):

(1a) The barber is a man of Seville, and the barber shaves all the men of Seville who are not the barber, and the barber does not shave himself, and every man of Seville is shaved by a man of Seville.

Sb &
Ax((Sx & ~x=b) -> Hbx) &
~Hbb &
Ax(Sx -> Ey(Sy & Hyx))

It is consistent that the barber is shaved by a man of Seville:

Ey(Sy & Hyb)

(2a) The barber is a man of Seville, and the barber shaves only the men of Seville who do not shave themselves, and the barber does not shave himself, and every man of Seville is shaved by a man of Seville.

Sb &
Hbx -> (Sx & ~Hxx) &
~Hbb &
Ax(Sx -> Ey(Sy & Hyx))

It is consistent that the barber is shaved by a man of Seville:

Ey(Sy & Hyb)

Consistency proof for both (1a) and (2a)

Let the universe be {barber, Charlie}

Let the men of Seville be {barber, Charlie}

Let the barber shave Charlie and Charlie shave the barber.

Symbolized:

U = {b c}
S = {b c}
H = {<b c> <c b>}

it is also consistent that the barber is shaved by someone who is a man of Seville. — TonesInDeepFreeze

Then that person becomes 'the' Barber of Seville, so that's not consistent.

'the barber of Seville' is a definite description. There are different ways of handling definite descriptions, including, at least, both the Fregean and Russellian. To bring that complication into the example would require being exact in how we do it.

In any case, one can shave a person without being a barber. So, still, there is no paradox.

On the other hand, you could add a premise: There is one and only one man in Seville who does any shaving.

E!x(Sx & Ey Hxy)

But that premise is not in your statements.

Why don't you just look up the barber paradox to see that it involves:

one who shaves all and only those who do not shave themselves

You have to have both - all of those and only those.

If you say, "there is someone who shaves all and only those who do not shave themselves" then you do have a contradiction.

To reiterate, the paradox doesn't even need any mention of maleness, barbers, towns or even humans. It is best seen in its starkest form:

Something shaves all and only those that do not shave themselves. Contradiction.

For set theory (replacing the 2-place relation 'shaves' with the 2-place relation 'is a member of'):

There is an x that such that for all y, y is a member of x if and only if y is not a member of y. Contradiction.

And most generally, in logic alone, for any 2-place relation R:

There is an x such that for all y, x bears the relation R to y if and only y does not bear the relation R to y. Contradiction.

In symbols:

ExAy(Rxy <-> ~Ryy). Contradiction.

In any case, one can shave a person without being a barber. So, still, there is no paradox. — TonesInDeepFreeze

Yes, but can you shave a barber without being to a certain extent a barber yourself? What I mean to say is this, that if a barber has someone else shave him, that person would require a degree of professionality, otherwise the role would cease to exist. This is also what the paradox is alluding to when you include the social setting of Seville. If you have a barber who has an amateur shave him, then he is diminishing his profession. I think there is a logical way to approach this paradox, but what is more interesting for me is to discover whether one can advance one's understanding of how society works.

Such utterly incidental questions as to the meaning of 'barber' can't seriously be considered part of the subject of paradox.

Anyway, looking in several dictionaries, I find that merely shaving someone does not constitute being a barber.

Again, if you add the premise that there is only one man in Seville who does any shaving (viz. the barber), and that the barber does not shave himself, but that every man in Seville is shaved is by some man in Seville, then of course that is a contradiction. But so what?

On the other hand, if, as you now suggest, you're interested in looking at the subject of tonsorial practices in various societies, then you don't need Russell's paradox for that.

/

Yes, there is a logical way to look at the paradox. The most obvious is to observe the theorem of logic:

~ExAy(Rxy <-> ~Ryy)

which in set theory yields:

~ExAy(xey <-> ~yey)

"there is no set of all the sets that are not members of themselves"

which in set theory yields:

~ExAy yex

"there is no set of all sets"

The perspective I am advancing is the meaning of the paradox. This is interpretation. There are different approaches as has been shown here. So there's the logical aspect, the interpretive aspect and others perhaps. The Barber paradox is well-known and also alludes to Russell's paradox. I am not going to confine myself to purely logic here.

Take the Millet seed paradox. One seed falling on the ground makes no sound. Take a fistful, a thousand and they make a noise. How could something that makes no noise individually make noise collectively. Now you could approach this paradox from different angles (e.g. the collective over the individual). That's what I understand to be, in principle, interesting about them.

A paradox is like a puzzle without a definitive answer.

I'll leave it at that.

The barber paradox is also related to the Epimenides paradox ("damn Cretins") and the liar paradox.

"You shall be careful with self-reference."

A function cannot be its own argument, because the functional sign already contains the prototype of its own argument and it cannot contain itself.

If, for example, we suppose that the function F(fx) could be its own argument, then there would be a proposition “F(F(fx))”, and in this the outer function F and the inner function F must have different meanings; for the inner has the form ϕ(fx), the outer the form ψ(ϕ(fx)). Common to both functions is only the letter “F”, which by itself signifies nothing.

This is at once clear, if instead of “F(F(u))” we write “(∃ϕ) : F(ϕu) . ϕu = Fu”.

Herewith Russell’s paradox vanishes. — Wittgenstein

I'm guessing these sorts of things show up elsewhere as well.

Of course there are many kinds of paradoxes ranging formal to informal. But you referred to Russell's barber paradox, which is an informal illustration standing for concerns in logic and mathematics, so I wrote in that context.

In any case, hopefully now you see that your presentations of Russell's barber paradox were incorrect and not paradoxes since they have easy non-perplexing answers.

In any case, hopefully now you see that your presentations of Russell's barber paradox were incorrect — TonesInDeepFreeze

No, it does appear as if my original expositions of the paradox were erroneous.

That said, it is difficult to find a definitive version on the web.

The Barber of Seville remains elusive.

It was explained exactly why your versions are not paradoxes.

Clear versions are available on the Internet. Moreover, I stated clear versions in this thread. Here again, in greatest generality:

Someone shaves all and only those who do not shave themselves.