  We propose a Newtonian semiclassical gravity theory based on the GRW collapse
theory with matter density ontology (GRWm), which we term GRWmN. The theory is
proposed because, as we show, the standard Newtonian semiclassical gravity
theory based on the Schroedinger-Newton equations does not have a consistent
Born rule probability interpretation for gravitationally self-interacting
particles and implies gravitational cat states for macroscopic mass
superpositions. By contrast, we show that GRWmN has a consistent statistical
description of gravitationally self-interacting particles and adequately
suppresses the cat states for macroscopic superpositions. Two possible routes
to experimentally testing GRWmN are also considered. We conclude with a
discussion of possible variants of GRWmN, what a general relativistic extension
would involve, and various objections that might be raised against
semiclassical gravity theories like GRWmN.
