The Liar Paradox

Episode #6 of the course “Philosophical ideas that everyone should know”

The “liar paradox” started when the Greek Epimenides, a Cretan, stated, “All Cretans are liars.” Another, simpler version of this idea is “This sentence is false,” which is false if true, and true if false. Basically, it is impossible to tell whether the sentence or the statement is true or false. This phrasing is also known as the “Epimenides paradox.”

The Liar Paradox is actually a group of related paradoxes that center around truthfulness. These statements are not related to social norms, ethics, or intentions. Instead, they are simply about whether the statement itself is a “truth.”

Each liar paradox has three basic elements. The first is the truth predicate. The truth predicate states or affirms something about the subject of the sentence. Next, the liar paradox contains some form of self-reference (e.g., “I am lying.”). This self-reference can also be in the form of a cycle (where two people speak, both referencing the fact that the other is lying). Finally, the liar paradox also contains some form of capture and release. That is, the truth predicate “captures” the truth or the truth “releases” the truth predicate.

Bertrand Russell states, “The point of philosophy is to start with something so simple that doesn’t seem to be worth talking about and end with something so paradoxical that no one will believe it” (1918). The idea represented by a paradox is essential to mathematics because it allows for scrutiny. The idea involves outlining groups of elements that meet a set of criteria, like the group of all real numbers more than 1 or the group of prime numbers. Then we do operations to find other rules about the elements present in the group. Speaking philosophically, groups are interesting because experts recognize that all math could be completely formulated in group theory by using group rules to understand math in a purely logical way.

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