Paradoxes and Self-Reference
Self-referential paradoxes like the Liar and Russell's paradox have repeatedly forced revisions to logic, set theory, and the theory of truth.
Definition
A logical paradox is an apparently valid argument from apparently true premises to a contradiction or absurdity; many of the deepest such paradoxes arise from self-reference or diagonalization.
Scope
This area covers the major logical and semantic paradoxes and their bearing on the foundations of logic. It treats the semantic paradoxes of truth (the Liar and its kin), the set-theoretic and logical paradoxes (Russell, Burali-Forti) that prompted type theory and axiomatic set theory, the soritical paradoxes of vagueness, and the philosophical interpretation of Godel's incompleteness theorems as the deepest deployment of self-reference. A unifying theme is whether a common diagonal structure underlies them all.
Sub-topics
Core questions
- What is the source of the paradoxes, and do they share a common structure?
- Should the Liar be solved by truth-value gaps, hierarchies, or accepting true contradictions?
- What constraints do the set-theoretic paradoxes place on logic and mathematics?
- What do Godel's theorems show about the limits of formal systems?
Key concepts
- self-reference and diagonalization
- the Liar paradox
- Russell's paradox
- truth-value gaps and gluts
- the inclosure schema
- incompleteness
Key theories
- The inclosure schema
- Priest argues that the paradoxes of self-reference share a single 'inclosure' structure of diagonalization across a boundary, suggesting a uniform diagnosis and, for him, a dialetheic resolution.
- Fixed-point theory of truth
- Kripke constructs a truth predicate by a fixed-point construction over a partially interpreted language, allowing some sentences (including the Liar) to be ungrounded and lack a truth value while retaining a self-applicable truth predicate.
History
Self-referential paradoxes go back to the ancient Liar (Epimenides, Eubulides). Russell's 1901 paradox shook Frege's logicism and motivated type theory and axiomatic set theory; Tarski responded to the Liar with a hierarchy of languages, Godel turned self-reference into the incompleteness theorems (1931), and Kripke and Priest gave influential modern treatments of the semantic paradoxes.
Debates
- Is there a uniform solution to the paradoxes?
- Whether the semantic and set-theoretic paradoxes share a single structure demanding a uniform solution (Russell's vicious-circle principle, Priest's inclosure schema) or whether they require distinct treatments depending on the notions involved.
Key figures
- Bertrand Russell
- Alfred Tarski
- Saul Kripke
- Kurt Godel
- Graham Priest
- Mark Sainsbury
Related topics
Seminal works
- sainsbury2009
- kripke1975
Frequently asked questions
- Why do philosophers care so much about paradoxes?
- Paradoxes are diagnostic: a valid-looking argument to an absurd conclusion shows that one of our apparently obvious assumptions must be wrong. Tracking down which assumption fails has repeatedly reshaped logic, set theory, and the theory of truth, so paradoxes function as stress tests for our most basic concepts.