The article has one point wrong. The statement "If this sentence is true, then Santa Claus exists." is not equivalent to "If this sentence is false, then Santa Claus doesn't exist." It is equivalent to "If Santa Claus doesn't exist, then this sentence is false".

To break it down into standard boolean algebra, the statement "if A then B" is logically equivalent to saying "not A or B". !A|B = B|!A = "If not B then not A".

Now back to the paradox, using boolean terms. The statement can be rephrased as !A|B: "This sentence is false or Santa Claus exists." Since we know that Santa doesn't exist, the sentence must be false: !(!A|B). That is equivalent to A&!B: "This sentence is true and Santa Claus doesn't exist." Here is where we have the paradox. We just said that the sentence must be false, but that then implies that the sentence is true.

The main mind-bender here is to realize that the negation of "If this sentence is true, then Santa Claus exists." is "This sentence is true and Santa Claus doesn't exist." We are not able to get rid of the statement "this sentence is true" no matter which way we toggle it.