Consider the following statement:
This whole statement is true if and only if the Earth is flat
Let us denote the whole statement by A. Note that we can then rewrite it a bit more formally as the following logical equivalence:
A is true ⇔ the Earth is flat
There are, obviously, just two possibilities: either A is true or it is false. Suppose A is true, then the left side of the equivalence is true. For the whole equivalence to be true its right side must therefore also be true, i.e. the Earth must be flat. Otherwise, suppose A is false. Then the left side of the equivalence is false. However, for the whole equivalence to be false, the right side must then not be false, i.e. the Earth must be flat.
Consequently, no matter whether A is true or false, the Earth is certainly flat!
For other sophisms check out my other posts.