"This sentence is false." might be translated into formal logic as something like this:

Theorem S: ∀x(Cxy → ¬x)

Leading to this proof:

1) ∀x(Cxy → ¬x)2) ∃!x(Cxy)C) ∃!x(¬x)

Which, when translated back into English, would look like this:

Theorem S: For all x, if x is a sentence with content ...