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Studia Logica

, Volume 106, Issue 3, pp 637–670 | Cite as

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

  • Sergei P. Odintsov
  • Stanislav O. Speranski
  • Igor Yu. Shevchenko
Article

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.

Keywords

Independence-friendly logic Game-theoretic semantics Trump semantics Constructivism Realizability Strong negation 

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Notes

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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Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.St. Petersburg State UniversitySaint PetersburgRussia
  3. 3.Novosibirsk State UniversityNovosibirskRussia

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