Property theory is, as I have understood it, first of all characterized by an attempt to approach naive comprehension type-freely and without committing to extensionality.

There is e.g. the work of Raymond Turner in A Theory of Properties in JSL 52 (2) 1987 and his monograph Truth and Modality for Knowledge Representation (Pitman) 1990. There is also Semantics and Property Theory Author(s): Gennaro Chierchia and Raymond Turnerin Linguistics and Philosophy, Vol. 11, No. 3 (Aug., 1988), pp. 261-302, plus works cited.

Are there any important new developments of property theory beyond the work mentioned?