Journal of Philosophical Logic

, Volume 45, Issue 4, pp 381–398 | Cite as

Liar-type Paradoxes and the Incompleteness Phenomena



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.


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


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© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Graduate School of System InformaticsKobe UniversityKobeJapan
  2. 2.Department of Natural SciencesNational Institute of Technology, Kisarazu CollegeKisarazuJapan

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