 A formal ontology is a mathematical theory of certain entities, formulated in a formal, artificial language, which in turn is based on some logical system like first order logic, or some form of the lambda calculus, or the like. Such a formal ontology will specify axioms about what entities of this kind there are, what their relations among each other are, and so on. Formal ontologies could also only have axioms that state how the things the theory is about, whatever they may be, relate to each other, but no axioms that state that certain things exist. For example, a formal ontology of events won’t say which events there are. That is an empirical question. But it might say under what operations events are closed under, and what structure all the events there are exhibit. Similarly for formal ontologies of the part-whole relation, and others. See (Simons 1987) for a well known book on various formal versions of mereology, the study of parts and wholes.