Subject:  The Barber Paradox

We now have enough "machinery" to state the Barber Paradox more
precisely.

First, here is the syntax and semantics of a first-order language
for expressing it:

SYNTAX		SEMANTICS
---------------------------
b		the barber
S(x,y)		x shaves y

Note:  S(x,x) will therefore mean:  x shaves themself
and    S(b,x) will therefore mean:  the barber shaves x,
				i.e., x is shaved by the barber

Now, the problem says that the barber shaves "those people, and only
those people, who do not shave themselves".  Let the domain be: people.

Then we have two propositions to consider:

(1) The barber shaves those people who don't shave themselves.
(2) The barber shaves only those people who don't shave themselves.

Let's consider (1):  There's no quantifier in the English sentence, but,
typically, when we say something like that in English, we mean "all".
So, (1) becomes:

The barber shaves all those people who don't shave themselves; i.e.:
All those people who don't shave themselves are shaved by the barber.

In first-order logic (FOL), that becomes:

	Ax[-S(x,x) -> S(b,x)]

Now let's consider (2).  What does "only those" mean?  Consider a
simpler sentence:

Only smart students take 191.

That means that:

	If you take 191, then you're smart.

So, (2) means:

If you are shaved by the barber, then you don't shave yourself.

In FOL, it becomes:

	Ax[S(b,x) -> -S(x,x)]

Conjoining these, we get:

	Ax[S(x,x) <-> -S(b,x)]

But the barber is a person, so the predicate "S" applies to the barber:

	S(b,b) <-> -S(b,b).

This is a contradiction!  (It has the form:  p <-> -p; do the truth table.)

Therefore, there is no such barber!