|Paris is in France - TRUE. The moon is made of cheese - FALSE|
Well the moon isn't made out of cheese and Switzerland is not the most populous country in Europe (its 18th in the rankings). That's two down. How about the other one? Three is definitely prime, and five is greater than three. Certainly Paris is in France and the Himalayas are in Asia.
Gosh, this is infuriating.
Oh hang on, the third false statement must be that the "paragraph contains three false statements"; there are actually only two false statements. Problem solved. Phew.
Satisfied? You shouldn't be.
This explanation seems to have worked, but if you think more carefully, it has unleashed something far more disturbing; a genuine logical paradox:
Suppose that we try to sort the seven sentences in the opening paragraph into two sets; one set containing the true statements and one containing the false statements. Each sentence has to go in one or other set because it must be either true or false but not both (this is called dichotomy which exists as long as we stay out of the world of Fuzzy Logic).
Let's have a go:
...and we've had this problem before.
What is going on here? We have a perfectly innocent looking sentence which can be labelled neither true nor false. Well, this example demonstrates a version of Russell's Paradox, which is named after the British philosopher and logician, Bertrand Russell. He discovered it in 1901 although the Liar Paradox, to which the above situation is very closely related, was known in Ancient Greece.
Russell's statement of the paradox also led to a contradiction when he tried to label a particular statement "true" or "false" and was made in the field of naive set theory, which he was studying in an attempt to clarify the logical foundations of mathematics. He considered the set of all sets that don't contain themselves and asked if this particular set contained itself. The paradox can be written succinctly using set notation:
Can you decipher the symbols in the above statement? Can you convince yourself that it is indeed a paradox? Can you see the parallels between Russell's original version and our example? Can you develop your own version of Russell's paradox?