  A substantial generalisation is put forward of the theory of subjective
fiducial inference as it was outlined in earlier papers. In particular, this
theory is extended to deal with cases where the data are discrete or
categorical rather than continuous, and cases where there was important
pre-data knowledge about some or all of the model parameters. The system for
directly expressing and then handling this pre-data knowledge, which is via
what are referred to as global and local pre-data functions for the parameters
concerned, is distinct from that which involves attempting to directly
represent this knowledge in the form of a prior distribution function over
these parameters, and then using Bayes' theorem. In this regard, the individual
attributes of what are identified as three separate types of fiducial argument,
namely the strong, moderate and weak fiducial arguments, form an integral part
of the theory that is developed. Various practical examples of the application
of this theory are presented, including examples involving binomial, Poisson
and multinomial data. The fiducial distribution functions for the parameters of
the models in these examples are interpreted in terms of a generalised
definition of subjective probability that was set out previously.
