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.
The liar paradox is sometimes attributed to Epimenides^{} of Crete, who said "All Cretans are liars"^{[note 1]} (although, 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:
In this case, the paradox does not consist of a single proposition, but a referential cycle of two propositions.
Possible solutions[edit]
- 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?"
- 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.
- 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.
- Various other methods of making the paradox an ill-formed statement, such that it can be said to be not even wrong. (e.g., Tarski's argument that statements about statements have an semantic "level", and most versions of the paradox are mixing the two levels in an invalid manner.^{[citation needed]})
- Various ways of tinkering with what "True" and/or "False" mean, semantically, such that the paradox becomes either "partially true", "neither true nor false", or "both true and false".
- As a variant of the above, a third (or more) value for a statement besides "True" and "False", usually some variation of "Uncertain" or "Invalid".
- You get into an infinite recursion, so asking for the True/False assignment is like asking for the last decimal digit of 1/7.
- Let x = "This statement is False"
- This would mean that [x is False] — ie. that the statement "This statement is False" is, itself, False
- But x = [x is false].
- By substituting "[x is false]" for "x", we get "x = ([x is false] is false)"
- By substituting "([x is false] is false)" for "x", we get "x = ([([x is false] is false) is false] is false)"
- This continues ad infinitum^{[1]}
- Take advantage of an ambiguity in "this statement is false", by claiming it is not exactly clear that "this statement" is self-referential.
- ie. "this statement" could be referring to a completely different statement.
- However, semantic trickery works both ways; we could be more specific — Is the statement "This statement is False", wherein the word "this" refers to the statement, itself, true or false? — but we tend to be lazy.
- Dealing with the two-statement version, "the following statement is true" and "the previous statement is false" both rely on another statement to function, so they cannot have absolute truth values in and of themselves, since a key part of their meaning relies on a separate statement. Think of it like a mathematical statement such as "√x > 2", which is only going to be true, or even defined, for certain values of x.
In mathematics[edit]
The liar paradox in its verbal form is a statement of human language and isn't directly translatable into formal logic that forms the rigorous foundation of modern mathematical proofs. Nonetheless, there are theorems that exploit it by constructing statements of formal logic that refer to themselves indirectly, while on the surface talking about something else.
- Kurt Gödel^{} managed to encode this paradox into number theory, and to conclude that no consistent axiomatic system (ie. a system without contradictions) that includes enough of number theory can be complete (ie. can be a system where every statement is provable or disprovable) and vice versa.
- His proof was constructive: he showed a way, for any given system S satisfying those conditions, to construct a formula G whose explicit meaning is a complicated relationship between some very large natural numbers, but whose implicit meaning is, informally, "I cannot be proven in system S".
- Thus, G becomes a true statement that cannot be proven within S; you can only prove G using a larger or, at least, different axiomatic system.
- If you could prove G, using only the axioms of S, then you would have proven, within S, that G is correct in saying G cannot be proven within S; that is a contradiction.
- The proof of Tarski's undefinability theorem^{} is even closer to the original formulation of the liar's paradox.
- Using a construction similar to Gödel's, it is possible to show that no consistent system that includes enough of number theory can encode its own truth as a logical formula — that is, there is no formula True(x) that is true if and only if x is the Gödel number of a true formula.
- The proof uses reductio ad absurdum: if it was possible, then we could construct a "liar formula" that, informally, states "I am false" — and assuming it to be either true or false leads to a contradiction, which means True(x) cannot exist.
Notes[edit]
- ↑ There exists, however, a non-paradoxical interpretation of this statement. Assuming that "All Cretans are liars" means "All Cretans always lie", if Epimenides lied about it, it means that the negation of the statement — "Not all Cretans always lie" — is true. It can be true if there exists at least one Cretan (who may be Epimenides himself) who does not always lie, and in this case no paradox occurs.