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Hintikka’s Independence-Friendly Logic Meets Nelson’s Realizability

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Abstract

Inspired by Hintikka’s ideas on constructivism, we are going to ‘effectivize’ the game-theoretic semantics (abbreviated GTS) for independence-friendly first-order logic (IF-FOL), but in a somewhat different way than he did in the monograph ‘The Principles of Mathematics Revisited’. First we show that Nelson’s realizability interpretation—which extends the famous Kleene’s realizability interpretation by adding ‘strong negation’—restricted to the implication-free first-order formulas can be viewed as an effective version of GTS for FOL. Then we propose a realizability interpretation for IF-FOL, inspired by the so-called ‘trump semantics’ which was discovered by Hodges, and show that this trump realizability interpretation can be viewed as an effective version of GTS for IF-FOL. Finally we prove that the trump realizability interpretation for IF-FOL appropriately generalises Nelson’s restricted realizability interpretation for the implication-free first-order formulas.

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Acknowledgements

The research of S. P. Odintsov was partially supported by the Grants Council (under RF President) for State Aid of Leading Scientific Schools (Grant NSh-6848.2016.1). The research of S. O. Speranski was partially supported by the Alexander von Humboldt Foundation.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, 4 Koptyug Ave., Novosibirsk, Russia, 630090

    Sergei P. Odintsov

  2. St. Petersburg State University, 29B Line 14th, Saint Petersburg, Vasilyevsky Island, Russia, 199178

    Stanislav O. Speranski

  3. Novosibirsk State University, 2 Pirogova St., Novosibirsk, Russia, 630090

    Igor Yu. Shevchenko

Authors
  1. Sergei P. Odintsov
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  2. Stanislav O. Speranski
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  3. Igor Yu. Shevchenko
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Correspondence to Sergei P. Odintsov or Stanislav O. Speranski.

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Presented by Heinrich Wansing

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Odintsov, S.P., Speranski, S.O. & Shevchenko, I.Y. Hintikka’s Independence-Friendly Logic Meets Nelson’s Realizability. Stud Logica 106, 637–670 (2018). https://doi.org/10.1007/s11225-017-9760-x

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