 Kripke’s construction can be applied to a number of different logics, including other many-valued logics such as the ‘Weak Kleene’ logic, and supervaluation logics. See, for instance, Burgess 1986 and McGee 1991 for discussion. Kripke-style constructions engage a fair bit of mathematical subtlety. For an accessible overview of more of the details, see Soames 1999. For a more mathematically rich exposition, see McGee 1991.