Ontology is the theory of objects and their ties. It provides criteria for distinguishing different types of objects (concrete and abstract, existent and nonexistent, real and ideal, independent and dependent) and their ties (relations, dependencies and predication).
We can distinguish: a) formal, b) descriptive and c) formalized ontologies.
a) Formal ontology was introduced by Edmund Husserl in his Logical Investigations (1): according to Husserl, its object is the study of the genera of being, the leading regional concepts, i.e., the categories; its true method is the eidetic reduction coupled with the method of categorial intuition. The phenomenological ontology is divided into two: (I) Formal, and (II) Regional, or Material, Ontologies.
The former investigates the problem of truth on three basic levels: (a) Formal Apophantics, or formal logic of judgments, where the a priori conditions for the possibility of the doxic certainty of reason are to be sought, along with (b) the synthetic forms for the possibility of the axiological, and (c) "practical" truths. In other words it is divided into formal logic, formal axiology, and formal praxis.
In contemporary philosophy, formal ontology has been developed in two principal ways. The first approach has been to study formal ontology as a part of ontology, and to analyze it using the tools and approach of formal logic: from this point of view formal ontology examines the logical features of predication and of the various theories of universals. The use of the specific paradigm of the set theory applied to predication, moreover, conditions its interpretation.
This approach is best exemplified by Nino Cocchiarella; according to whom "Formal Ontology is the result of combining the intuitive, informal method of classical ontology with the formal, mathematical method of modern symbolic logic, and ultimately of identifying them as different aspects of one and the same science. That is, where the method of ontology is the intuitive study of the fundamental properties, modes, and aspects of being, or of entities in general, and the method of modern symbolic logic is the rigorous construction of formal, axiomatic systems, formal ontology, the result of combining these two methods, is the systematic, formal, axiomatic development of the logic of all forms of being. As such, formal ontology is a science prior to all others in which particular forms, modes, or kinds of being are studied." (2)
The second line of development returns to its Husserlian origins and analyses the fundamental categories of object, state of affairs, part, whole, and so forth, as well as the relations between parts and the whole and their laws of dependence -- once all material concepts have been replaced by their correlative form concepts relative to the pure 'something'. This kind of analysis does not deal with the problem of the relationship between formal ontology and material ontology." (3).
b) Descriptive ontology concerns the collection of information about the list of objects that can be dependent or independent items (real or ideal).
c) Formalized ontology attempts to constructs a formal codification for the results descriptively acquired at the preceding levels.