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Liar-type Paradoxes and the Incompleteness Phenomena

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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.

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Notes

  1. It is proved in [7] that the sentence ∀x ξ(x) is provably equivalent to the Gödel sentence of T in PA, and proofs of the second incompleteness theorem based on Priest’s formalization were given in [7, 12]. See also [16].

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Authors and Affiliations

  1. Graduate School of System Informatics, Kobe University, 1-1 Rokkodai, Nada, Kobe, 657-8501, Japan

    Makoto Kikuchi

  2. Department of Natural Sciences, National Institute of Technology, Kisarazu College, 2-11-1 Kiyomidai-Higashi, Kisarazu, Chiba, 292-0041, Japan

    Taishi Kurahashi

Authors
  1. Makoto Kikuchi
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  2. Taishi Kurahashi
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Correspondence to Makoto Kikuchi.

Additional information

This work was supported by JSPS KAKENHI Grant Numbers 24540125,12J00654,26887045. The authors would like to thank Hidenori Kurokawa for valuable discussions.

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Kikuchi, M., Kurahashi, T. Liar-type Paradoxes and the Incompleteness Phenomena. J Philos Logic 45, 381–398 (2016). https://doi.org/10.1007/s10992-015-9378-2

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