Editing Game semantics (section)

== Quantifiers ==
Foundational considerations of game semantics have been more emphasised by [[Jaakko Hintikka]] and Gabriel Sandu, especially for [[independence-friendly logic]] (IF logic, more recently ''information''-friendly logic), a logic with [[branching quantifier]]s. It was thought that the [[principle of compositionality]] fails for these logics, so that a Tarskian [[truth definition]] could not provide a suitable semantics. To get around this problem, the quantifiers were given a game-theoretic meaning. Specifically, the approach is the same as in classical propositional logic, except that the players do not always have [[perfect information]] about previous moves by the other player. [[Wilfrid Hodges]] has proposed a [[compositional semantics]] and proved it equivalent to game semantics for IF-logics.

More recently {{ill|Shahid Rahman|fr}} and the team of dialogical logic in Lille implemented dependences and independences within a dialogical framework by means of a dialogical approach to [[intuitionistic type theory]] called ''immanent reasoning''.<ref> S. Rahman, Z. McConaughey, A. Klev, N. Clerbout: ''Immanent Reasoning or Equality in Action. A Plaidoyer for the Play level''. Springer (2018). https://www.springer.com/gp/book/9783319911489.<br>
For an application of the dialogical approach to intuitionistic type theory to the ''[[axiom of choice]]'' see S. Rahman and N. Clerbout: ''Linking Games and Constructive Type Theory: Dialogical Strategies, CTT-Demonstrations and the Axiom of Choice''. Springer-Briefs (2015). https://www.springer.com/gp/book/9783319190624.</ref>