The "ontological square" and modern type theories
This paper focuses on the connection between “four-category ontologies” (which are based on Aristotle’s ontological square) and modern type-theoretical semantics. Four-category ontologies make a distinction between four types of entities: substantial universals, substantial particulars, accidental universals and accidental particulars. According to B. Smith, “fantology is a doctrine to the effect that the key to the ontological structure of reality is captured syntactically in the ‘Fa’”. Smith argues that predicate logic cannot adequately describe these four types of entities, which are reduced to just two kinds - the general (‘F’) and the particular (‘a’). B. Smith has criticized G. Frege’s predicate logic. He argues that Frege, being the father of modern logic, simultaneously became the father of “fantology” with its ontological commitments. Smith transforms the ontological square to the ontological sextet (which also involves universal and particular events) and proposes a set of predicates for different ontological relations connecting these six types of entities. However, Smith’s approach has a number of limitations: he suggests a theory that describes only predicates of different types as universals. We argue for another formalization for the ontological square’s entities. This approach is based on modern type-theoretical semantics, according to which, the difference between substantial universals and accidental universals can be expressed. In first-order logic the sentences “Socrates is a man” and “Socrates is wise” share the same logical form. However, this fact is not consistent with “ontological square” metaphysics (“being a man” is a substantial universal and “being wise” is an accidental universal). Whereas, according to the type-theoretical approach, relations to accidental universals are expressed by judgments about type (a : A), but relations to accidental universals are expressed by predication (‘Pa’).