  Quantum theory is indeterministic, but not completely so. When a system is in
a pure state there are properties it possesses with certainty, known as actual
properties. The actual properties of a quantum system (in a pure state) fully
determine the probability of finding the system to have any other property. We
call this feature intermediate determinism. In dimensions of at least three,
the intermediate determinism of quantum theory is guaranteed by the structure
of its lattice of properties. This observation follows from Gleason's theorem,
which is why it fails to hold in dimension two. In this work we extend the idea
of intermediate determinism from properties to measurements. Under this
extension intermediate determinism follows from the structure of quantum
effects for separable Hilbert spaces of any dimension, including dimension two.
Then, we find necessary and sufficient conditions for a general probabilistic
theory to obey intermediate determinism. We show that, although related, both
the no-restriction hypothesis and a Gleason-type theorem are neither necessary
nor sufficient for intermediate determinism.
