This statement is false.
This statement is true.

The above are self referential statements, which are considered taboo due to their being known to cause logical problems. The first statement is obviously paradoxical. It cannot be classified as either true or false. The second seems ambiguous. Declaring it to be either true or false seems valid, but I like to think of it as true.

Circular definitions are another taboo. The Barber of Seville paradox is due to a circular definition. Modern logic requires some undefined primitive terms to avoid this particular type of paradox.

Hofstadter has published some cute self referential statements which are more amusing than paradoxical. For example.

This English sentence is difficult to translate into French.

For a while I was a Russian sentence, but now I am an English sentence.

The above follow all the rules of English grammar, but seem difficult to grok completely. I do not have his book handy and do not remember any of his others.

I have friends who got into fierce arguments over the first of the above. One insisted on a literal translation (Cette sentence Englais . . .), the other wanted to convey the senses with his translation (Cette sentence Francais . . .).

There are various paradoxes which can be created by improper operations on infinites series. For example the following series converges.

1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 . . .

Paradoxical reults can be obtained by expressing it as the difference of two diverging series (n improper operation).

(1 + 1/3 + 1/5 . . .) - (1/2 + 1/4 + 1/6. . .)