Games in algebraic logic
Algebraic logic grew up in the nineteenth century as the first attempt
to formalise the laws of rational thought in an algebraic setting.
Algebras were constructed to correspond to relations of various ranks.
However the problem of representing these algebras as genuine relations
turned out to be very hard [Monk64]. The problem can be simplified by
defining a two player game in which the second player attempts to build
the required representation while the first player attempts to stop her.
In this framework we can see why representability cannot be finitely
axiomatised, while providing an infinite axiom scheme that does define
representability.
Robin Hirsch and Ian Hodkinson