Ehrenfeucht–Fraïssé (EF) games are a fundamental tool in classical (finite) model theory, providing a combinatorial characterisation of elementary equivalence and a powerful method for proving inexpressibility results.

In this talk, we study extensions of EF games to logics with many-valued semantics, where formulae take values in an underlying algebraic structure such as a Boolean algebra or a commutative semiring, rather than the classical two truth values.

Recent work of Grädel, Brinke, and Mrkonjić shows that in semiring semantics the classical correspondence between EF games and logical indistinguishability breaks down: in general, the standard EF game cannot be both sound and complete for elementary equivalence unless one works over the Boolean semiring, which recovers classical two-valued logic.

This talk reports on work in progress exploring the extent to which these negative results can be overcome, with a primary focus on Boolean-valued models. In this setting, the presence of a well-behaved negation and the resulting logical dualities suggest that EF-style techniques may behave more closely to the classical case.

Our longer-term goal is to extend these ideas to partial Boolean-algebra-valued models, which play an important role in the study of quantum logic and information.