Column in Folha: 'Paradoxes are everywhere'

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The phrase "I'm lying," from a previous column , intrigued many readers. If the person is actually lying, then the phrase is true, meaning the person is not lying. If the person is not lying, then the phrase is false, meaning the person is lying. So, is it true or false?

Many readers wrote that it's a paradox, but without explaining what it means. In fact, it's not easy to explain, especially since there are many types. In this case, it's a logical paradox. In our usual reasoning, we use logic based on two rules: the excluded middle – every statement is either true or false – and non-contradiction – a statement cannot be both true and false at the same time. The phrase "I am lying," however, is either simultaneously true and false, or it is neither one nor the other.

This shows that the author of the phrase, the Supreme Leader of the Gödelians of X314 (mentioned in a previous column), does not follow the rules of logic and is therefore insane. Luckily, it's only on their planet that the supreme leader is crazy!

In this example, the logical paradox arises from the fact that the sentence contains a self-reference: it talks about itself. This is a very old idea, dating back at least to ancient Greece, and it admits many variations. "This sentence contradicts itself, but it doesn't!" The commander in the barracks: "Don't do what I'm telling you!" Or even in pairs: "The next sentence is false. The previous sentence is true."

A serious application of self-reference is Russell's paradox, formulated by the British mathematician, philosopher, and writer Bertrand Russell (1872-1970), winner of the Nobel Prize in Literature in 1950. Aiming to resolve the contradictions in mathematical set theory, Russell proposed considering the "set of all sets that are not members of themselves." The question of whether this set is a member of itself or not leads to the same kind of difficulty we encountered earlier with "I am lying." Russell's conclusion is that we must be much more careful in how we define a set to avoid self-references.

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