I would argue that the statement 'This statement is false' really is false.

For example, the typical chain of reasoning is:

1. assume the statement is true
2. then, by inference, the statement is false
3. assume the statement is false
4. then, by inference the statement is true.

However, I argue the jump from 3 to 4 is invalid. If the statement is false, interpreting the statement (assuming it's true) creates circular reasoning.

Furthermore, a statement can be false in more ways than one. For example, if

"The golden mountain is made of cheese" is false,

it doesn't follow that

"The golden mountain is *NOT* made of cheese." is true

as the statement may be false in virtue of having no referent. The *NOT* operator can be inserted in multiple places, each having a different meaning.

The statement 'This statement is false' actually has two components to it. Namely:

f(x) = 1 - x (an inverter function)
f(1) == f(0) (an assertion about identity)

Which is negated?

So... if we assume it's true, we can infer than it is false (reductio ad aburdum). If we assume it's false, no inference about it's truth can be made. Therefore, it's false.

I can explain further if anyone's interested.