In quantified modal logic, there are two options when it comes to the interpretation of the quantifiers. The simple approach is to let quantifiers range over all possible objects, not just objects existing in the world or time of evaluation, and use a special predicate to make claims about existence (an existence predicate). This is the constant domain approach. The more complicated approach is to assign a domain of objects to each world or time. This is the varying domain approach.

Assuming that all terms denote, the semantics for atomic formulas on the constant domain approach is obvious: either the denoted object has the denoted property in the world of evaluation, or it hasn’t. On the varying domain approach, there’s a third possibility: the object in question doesn’t exist. Terms (in particular variables) may denote objects not included in the domain of the world of evaluation. In those cases, the question is whether the formula should be evaluated as true, false or undefined. Leaving it undefined will have ramifications for the interpretation of both connectives and quantifiers. Should the negation of a formula whose truth value is undefined also be undefined? What about conjunction, universal quantification and necessitation?

Arguably, the point of doing varying domain semantics is to be able to make existence claims without using any existence predicate. I will argue that the most reasonable way of doing it, therefore, is by evaluating atomic formulas as false whenever terms denote objects not included in the domain of the world of evaluation.