ParadoxesIII
The liar paradox
'This sentence is false.' If it is true, it is false. If it is false, it is true. A single short sentence seems to break logic.
- Reading time
- 4 min
- Difficulty
- Deep
- Branch
- Logic
- First posed
- Eubulides of Miletus, 4th century BCE; related to Epimenides the Cretan
The puzzle
Consider the sentence: 'This sentence is false.' Is it true or false?
Suppose it is true. Then what it says is the case, and it says it is false. So it is false. Suppose instead it is false. Then what it says is not the case. But it says it is false, so it is not false; it is true. Either way, we reach a contradiction.
This is the liar paradox, credited, like the sorites, to Eubulides of Miletus. A related ancient puzzle involves Epimenides, a Cretan who is supposed to have said that all Cretans are liars. (That version is not a true paradox, since the statement can simply be false if some Cretan once told the truth.)
The liar is short, uses only ordinary words, and seems to show that the most basic principles of logic, that every statement is either true or false and that no statement is both, cannot both hold for a language that can talk about its own truth.
Why it matters
The liar is not a curiosity. It goes to the heart of the concept of truth, which logic, mathematics, science and everyday reasoning all depend on. Any language rich enough to talk about the truth of its own sentences seems vulnerable.
Versions of the same self-referential trick lie behind some of the deepest results of twentieth-century logic: Kurt Gödel's incompleteness theorems, Alan Turing's proof that some problems cannot be solved by any computer, and Alfred Tarski's theorem on the undefinability of truth.
Key terms
- Self-referenceA sentence, set or system referring to itself. Many paradoxes, such as the liar and Russell's, arise from it.
- Principle of bivalenceThe principle that every statement is either true or false, with no third option. Vagueness and the liar paradox put it under pressure.
- Law of non-contradictionThe principle that a statement and its negation cannot both be true at the same time and in the same sense.
- DialetheismThe view that some contradictions are true: some statements, such as the liar sentence, are both true and false.
- Object language and metalanguageThe distinction between a language being talked about (the object language) and the language used to talk about it (the metalanguage). Tarski used it to avoid the liar paradox.
Proposed solutions
Tarski: a hierarchy of languages
No language can contain its own truth predicate. Truth for one language must be defined in a richer metalanguage.
Truth-value gaps
The liar sentence is neither true nor false. It falls into a gap.
Dialetheism: some contradictions are true
The liar sentence is both true and false. Some contradictions are true, and logic can be adapted to accept them without collapse.
Each solution, tested
Tarski: a hierarchy of languages
- How it resolves the paradox
- Alfred Tarski showed in 1933 that, given certain natural assumptions, a consistent language cannot define truth for its own sentences. The solution is a hierarchy: the word 'true' in level 1 applies only to sentences of level 0, 'true' at level 2 to sentences of level 1, and so on. The liar sentence cannot be formed, because 'this sentence' would have to be at a lower level than itself.
- Why it doesn’t fully work
- Natural languages like English are not stratified this way. We happily say 'everything she said is true' without knowing which level her statements are at. The hierarchy seems artificial when applied to ordinary speech.
Truth-value gaps
- How it resolves the paradox
- Saul Kripke showed in 1975 how to build a single language that contains its own truth predicate, by starting with sentences that do not mention truth and gradually extending the predicate. Sentences like the liar never get assigned a value: they are 'ungrounded'. This lets English contain its own truth predicate without contradiction.
- Why it doesn’t fully work
- The strengthened liar: 'This sentence is not true.' If it is in the gap, it is not true, which is what it says, so it seems to be true after all. Gap theories struggle to describe the liar's status without generating a new paradox.
Dialetheism: some contradictions are true
- How it resolves the paradox
- Graham Priest argues that the simplest diagnosis is to take the argument at face value: the liar is both true and false. In classical logic, a single contradiction implies everything ('explosion'), which would be disastrous. But paraconsistent logics block explosion, so a few true contradictions can be contained. Priest suggests other examples, such as the instant of change between two states.
- Why it doesn’t fully work
- Most logicians regard accepting true contradictions as too high a price. If contradictions can be true, it becomes harder to say what is wrong with any argument.
Who argued what
- Tarski: a hierarchy of languages Alfred Tarski
- Truth-value gaps Saul Kripke
- Dialetheism: some contradictions are true Graham Priest, J. C. Beall
Variations
The card
One side of a card says: 'The sentence on the other side is true.' The other side says: 'The sentence on the other side is false.' Neither sentence refers to itself, yet the pair generates the same paradox. Self-reference can be indirect.
Yablo's paradox
Stephen Yablo imagined an infinite sequence of sentences, each saying that all the sentences after it are false. No sentence refers to itself, yet the sequence is still paradoxical. Whether this shows that paradox does not need self-reference is debated.
The truth-teller
'This sentence is true.' No contradiction arises, but nothing determines whether it is true or false. It seems equally consistent either way. Many think this is a sign that something is defective about such sentences.
Go deeper
From the liar to Gödel
In 1931, Kurt Gödel constructed, within arithmetic, a sentence that in effect says 'this sentence is not provable'. If arithmetic is consistent, the sentence is true but unprovable. Gödel's trick replaced 'false' with 'not provable', turning a paradox into a theorem: any consistent formal system strong enough for basic arithmetic contains true statements it cannot prove. Turing used a similar trick to show that no program can decide, for every program, whether it will halt.
Related ideas
Further reading and sources
- EncyclopediaStanford Encyclopedia of Philosophy: Liar Paradox
- WikipediaLiar paradox
- BookGödel, Escher, Bach, Douglas Hofstadter
- BookParadoxes from A to Z, Michael Clark
- BookIn Contradiction, Graham Priest
Open questions
- Can a sentence fail to be either true or false?
- Could some contradictions really be true?
- Is there something wrong with a sentence that talks about itself?