  The $\mathrm{Caus}[-]$ construction takes a base category of ``raw
materials'' and builds a category of higher order causal processes, that is a
category whose types encode causal (a.k.a. signalling) constraints between
collections of systems. Notable examples are categories of higher-order
stochastic maps and higher-order quantum channels. Well-typedness in
$\mathrm{Caus}[-]$ corresponds to a composition of processes being causally
consistent, in the sense that any choice of local processes of the prescribed
types yields an overall process respecting causality constraints. It follows
that closed processes always occur with probability 1, ruling out e.g. causal
paradoxes arising from time loops. It has previously been shown that
$\mathrm{Caus}[\mathcal{C}]$ gives a model of MLL+MIX and BV logic, hence these
logics give sufficient conditions for causal consistency, but they fail to
provide a complete characterisation. In this follow-on work, we introduce graph
types as a tool to examine causal structures over graphs in this model. We
explore their properties, standard forms, and equivalent definitions; in
particular, a process obeys all signalling constraints of the graph iff it is
expressible as an affine combination of factorisations into local causal
processes connected according to the edges of the graph. The properties of
graph types are then used to prove completeness for causal consistency of a new
causal logic that conservatively extends pomset logic. The crucial extra
ingredient is a notion of distinguished atoms that correspond to first-order
states, which only admit a flow of information in one direction. Using the fact
that causal logic conservatively extends pomset logic, we finish by giving a
physically-meaningful interpretation to a separating statement between pomset
and BV.
