The Liar Paradox

Liar ParadoxEpimenides, a Cretan, is said to have made the statement, “All Cretans are liars.” Was his statement true? This is not a perfect paradox, because liars may tell the truth sometimes. If he had said, “Every statement made by a Cretan is false,” a true paradox would be created.

This dialogue illustrates the same paradox:

  • Plato: The next statement by Socrates will be false.
  • Socrates: Plato has spoken truly!

Philip Jourdain proposed a card that reads on the front, “The sentence on the other side of this card is true.” On the back it reads, “The sentence on the other side of this card is false.”

These are variations of the liar paradox, the simplest form of which is the statement, “This sentence is false.” If the statement is true, then it is false. If it is false, then it is true.

This leads to the Pinocchio paradox. If Pinocchio states, “My nose will grow now,” what happens? Or, as Branson Reese would have it:

https://bransonreese.tumblr.com/post/160249161577

These and other paradoxes are explored in Martin Gardner’s book, aha! Gotcha: Paradoxes to Puzzle and Delight.

Aha GotchaThe Truth-Teller Paradox

Consider the statement, “This sentence is true.” This may not seem paradoxical at first, because it does not have the same inconsistency as “This sentence is false.” It may seem to be merely a tautology. However, “This sentence is true” could be true (in which case “This sentence is true” is true) or false (in which case “This sentence is true” is false). Since there are two truth-values that we can consistently apply to the sentence, it is paradoxical. We might say that “This sentence is true” is neither true nor false, but in that case it falsely states that it is true.

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