FOIL with constant domains revisited

FOIL is a family of two-sorted first-order modal logics containing both object and intensional variables. Intensional variables are represented by partial functions from worlds to objects and the abstraction operator lambda is used to talk about the object (if any) denoted by an intension in a given world. This paper answers a problem left open in Fitting's (2006) by showing that Fitting's axiomatization of FOIL augmented with infinitely many inductively defined rules, CD(k), k >= 0, allows for the construction of a canonical model that is essentially a constant domains model. Moreover, it is shown that the rules CD(k) are derivable in logics where the symmetry axiom B holds. Hence, Fitting's axiomatisation of FOIL is already complete when the underlying logic imposes symmetric models.