 One of the most rigorous proposals about abstract objects has been developed by Zalta (1983, 1988, and in a series of papers). It is a formal, axiomatic metaphysical theory of objects (both abstract and concrete), and also includes a theory of properties, relations, and propositions. The theory explicitly defines the notion of an abstract object but also implicitly characterizes them using  axioms.[9]  There are three central aspects to the theory: (i) a predicate \(E!\) which applies to concrete entities and which is used to define a modal distinction between abstract and ordinary objects; (ii) a distinction between exemplifying relations and encoding properties (i.e., encoding 1-place relations); and (iii) a comprehension schema that asserts the conditions under which abstract objects exist.