A causality semantics for Place/Transition nets weighted inhibitor arcs (PTI-nets) is proposed. We extend the standard approach to defining the partial order semantics of Place/Transition nets based on the process semantics given through net unfolding and occurrence nets. To deal with inhibitor arcs at the level of occurrence nets activator arcs (and extra conditions) are used. It is demonstrated how processes corresponding to step sequences of PTI-nets can be constructed and a non-algorithmic, axiomatic, characterisation is given of the processes that can be obtained in this way. In addition, a framework is established allowing to separately discuss behaviour, processes, causality, and their properties before proving that the resulting notions are mutually consistent for the various classes of Petri nets considered. This facilitates an efficient and uniform presentation of our results.