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

摘要

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.