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Liar paradox

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The Liar paradox is usually given as "This statement is false".

The truth value of the statement cannot be evaluated because the statement refers to the truth value of itself. Kurt GödelWikipedia managed to encode this paradox into number theory and to conclude that no sufficiently complete axiomatic system can be consistent, and vice versa.

The liar paradox is sometimes attributed to EpimenidesWikipedia of Crete, who said "All Cretans are liars" (although it should be noted that, in the original context (a poem written to advance a theological argument), he probably meant "All Cretans but me are liars about this one thing"). In Titus 1:12-13, Paul makes a passing reference to Epimenides' paradox, in furtherance of a serious argument that (non-Christian) Cretans were "evil beasts". (Although, again, it is not fully paradoxical in Paul's argument, as he seems to have taken it to mean "almost all Cretans are liars".) A similar statement ("I said in my haste, All men are liars.") shows up in Psalm 116:1-19, although, again, the context is not fully within the Liar paradox (in this case, the statement is explicitly an oversimplification, and is easily read as "All men lie sometimes").

An alternative formulation, popular in medieval Europe, is:

Plato: What Socrates is about to say is false.
Socrates: Plato has spoken correctly.

In this case, the paradox does not consist of a single proposition, but a referential cycle of two propositions.

Possible Solutions[edit]

  1. The self-referential aspect renders the entire statement nonsensical and meaningless; stringing words together into a query that is grammatically valid does not mean the resulting query, itself, is valid. Asking if the statement is true or false is like asking "What colour is a loud noise?"
  2. It is a statement to which true and false cannot be applied. Whether the statement is true or false is as undefined as the quotient when dividing by zero.
  3. It has a constantly shifting answer where the temporal delta between true and false reduces to zero resulting in the statement being both true and false at the same time.