Journal of Philosophical Logic

, Volume 45, Issue 4, pp 381–398

# Liar-type Paradoxes and the Incompleteness Phenomena

Article

## Abstract

We define a liar-type paradox as a consistent proposition in propositional modal logic which is obtained by attaching boxes to several subformulas of an inconsistent proposition in classical propositional logic, and show several famous paradoxes are liar-type. Then we show that we can generate a liar-type paradox from any inconsistent proposition in classical propositional logic and that undecidable sentences in arithmetic can be obtained from the existence of a liar-type paradox. We extend these results to predicate logic and discuss Yablo’s Paradox in this framework. Furthermore, we define explicit and implicit self-reference in paradoxes in the incompleteness phenomena.

### Keywords

Gödel’s incompleteness theorem The Liar Paradox Modal logic Arithmetical realization

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