Hintikka’s Independence-Friendly Logic Meets Nelson’s Realizability
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.
KeywordsIndependence-friendly logic Game-theoretic semantics Trump semantics Constructivism Realizability Strong negation
Unable to display preview. Download preview PDF.
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.
- 2.Dummett, M., Truth and Other Enigmas, Harvard University Press, 1978.Google Scholar
- 3.Dummett, M., The Logical Basis of Metaphysics, Harvard University Press, 1993.Google Scholar
- 5.Hintikka, J., The Principles of Mathematics Revisited, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511624919.
- 6.Hintikka, J., and G. Sandu, Informational independence as a semantical phenomenon, in J. E. Fenstad et al. (eds.), Logic, Methodology and Philosophy of Science VIII, Studies in Logic and the Foundations of Mathematics 126, North-Holland Publishing Company, 1989, pp. 571–589. doi: 10.1016/S0049-237X(08)70066-1.
- 8.Hodges, W., Some strange quantifiers, in J. Mycielski et al. (eds.), Structures in Logic and Computer Science: A Selection of Essays in Honor of A. Ehrenfeucht, Springer, 1997, pp. 51–65. doi: 10.1007/3-540-63246-8_4.
- 9.Kleene, S. C., Introduction to Metamathematics, North-Holland Publishing Company, 1952.Google Scholar
- 10.Kontinen, J., J. Väänänen and D. Westerståhl (eds.), Special Issue on Dependence and Independence in Logic, Studia Logica 101(2), Springer, 2013.Google Scholar
- 11.Mann, A. L., G. Sandu and M. Sevenster, Independence-Friendly Logic: A Game-Theoretic Approach, Cambridge University Press, 2011.Google Scholar
- 12.Matiyasevich, Yu. V., Hilbert’s Tenth Problem, MIT Press, 1993.Google Scholar
- 14.Osborne, M. J., and A. Rubinstein, A Course in Game Theory, MIT Press, 1994.Google Scholar
- 15.Rogers, H., Jr., Theory of Recursive Functions and Effective Computability, McGraw-Hill Book Company, 1967.Google Scholar
- 16.Väänänen, J., Dependence Logic: A New Approach to Independence Friendly Logic, Cambridge University Press, 2007.Google Scholar
- 17.van Benthem, J., Logic in Games, MIT Press, 2014.Google Scholar
- 19.Wittgenstein, L., Tractatus Logico-Philosophicus, Routledge & Kegan Paul, 1922.Google Scholar
- 20.Yang, F., On Extensions and Variants of Dependence Logic—A Study of Intuitionistic Connectives in the Team Semantics Setting, Ph.D. thesis, University of Helsinki, 2014. Available online at https://helda.helsinki.fi/handle/10138/43011.