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Liar's Paradox

From mujun.wiki, the encyclopedia of contradictions.

The Liar's Paradox arises from a statement that asserts its own falsity, such as "This sentence is false." If the sentence is true, then by its own claim it must be false; if it is false, then it must be true. The paradox is among the oldest documented contradictions in classical logic, attributed to Eubulides of Miletus in the 4th century BCE, and it continues to challenge accounts of truth, self-reference, and formal semantics.[1]

Variations include the Epimenides paradox ("All Cretans are liars," uttered by a Cretan), the card paradox, and Yablo's non-self-referential variant.[2] Modern treatment ranges from Tarski's hierarchy of languages to Kripke's fixed-point theory and paraconsistent logic.

1 Statement of the paradox

Consider the sentence L: "L is false." Suppose L is true. Then what L says is the case, namely that L is false — contradiction. Suppose instead L is false. Then it is not the case that L is false; that is, L is true — contradiction.[3]

The paradox can be sharpened by removing token-reflexive language. Quine's revision — "yields falsehood when appended to its own quotation" yields falsehood when appended to its own quotation — eliminates the indexical "this," yet preserves the contradiction.

Truth-value evaluation of the Liar sentence
AssumptionConsequenceResult
L is trueL asserts L is false⊥ contradiction
L is false¬(L is false), i.e. L is true⊥ contradiction
L is neitherViolates bivalence⚠ unstable
L is bothDialetheist reading○ accepted

2 Historical development

Documented references to the paradox span more than two millennia, evolving from a rhetorical curiosity into a foundational concern of mathematical logic.

2.1 Eubulides and the Megarians

Eubulides of Miletus, a 4th-century BCE Megarian, is credited by Diogenes Laërtius as the originator of seven paradoxes, of which the Liar (pseudomenon) is the most enduring.[4] Cicero references the paradox in Academica, treating it as a stock challenge to the Stoic theory of truth.

2.2 Medieval insolubilia

Scholastic logicians, calling such sentences insolubilia, produced detailed taxonomies. Jean Buridan, Thomas Bradwardine, and William of Heytesbury proposed restrictions on signification or self-reference. Buridan's response — that a self-referential sentence implicitly asserts its own truth, and so its falsity makes it simply false — anticipates 20th-century revision theories.

2.3 Modern formalization

Alfred Tarski's 1933 monograph The Concept of Truth in Formalized Languages diagnosed the paradox as a consequence of permitting a language to contain its own truth predicate. Tarski's solution stratifies language into object-language and meta-language levels, blocking the construction of a Liar sentence within a single level.

Saul Kripke's 1975 Outline of a Theory of Truth proposes a fixed-point construction in which truth is partially defined; ungrounded sentences such as the Liar receive no truth value. Graham Priest's dialetheism instead embraces the contradiction, accepting the Liar as both true and false within a paraconsistent logic.[5]

3 Proposed resolutions

  • Tarski hierarchy 1933 Forbid a language from containing its own truth predicate.
  • Kripke fixed point 1975 Treat the Liar as ungrounded and truth-valueless.
  • Revision theory 1982 Truth values revise iteratively without ever stabilizing for the Liar.
  • Dialetheism 1987 Accept some contradictions as true; isolate via paraconsistent logic.
  • Contextualism 1991 Truth predicates shift context across utterances; the Liar exploits the shift.

4 Related contradictions

The Liar belongs to a family of self-referential paradoxes catalogued in mujun.wiki's Self-Reference subcategory.

5 References

  1. 1. Beall, J. C., & Glanzberg, M. (2014). Liar Paradox. Stanford Encyclopedia of Philosophy.
  2. 2. Yablo, S. (1993). Paradox without self-reference. Analysis, 53(4), 251–252.
  3. 3. Sainsbury, R. M. (2009). Paradoxes (3rd ed.). Cambridge University Press.
  4. 4. Diogenes Laërtius. Lives of Eminent Philosophers, Book II, §108.
  5. 5. Priest, G. (2006). In Contradiction (2nd ed.). Oxford University Press.

6 Further reading

  • Tarski, A. (1944). The semantic conception of truth. Philosophy and Phenomenological Research, 4(3), 341–376.
  • Kripke, S. (1975). Outline of a theory of truth. Journal of Philosophy, 72(19), 690–716.
  • Gupta, A., & Belnap, N. (1993). The Revision Theory of Truth. MIT Press.
  • Field, H. (2008). Saving Truth from Paradox. Oxford University Press.