  The full anhomomorphic logic of coevents $\ascript ^*$ is introduced. Atoms
of $\ascript ^*$ and embeddings of the event set $\ascript$ into $\ascript ^*$
are discussed. The quantum integral over an event $A$ with respect to a coevent
$\phi$ is defined and its properties are treated. Integrals with respect to
various coevents are computed. Reality filters such as preclusivity and
regularity of coevents are considered. A quantum measure $\mu$ that can be
represented as a quantum integral with respect to a coevent $\phi$ is said to
1-generate $\phi$. This gives a stronger reality filter that may produce a
unique coevent called the ``actual reality'' for a physical system. What we
believe to be a more general filter is defined in terms of a double quantum
integral and is called 2-generation. It is shown that ordinary measures do not
1 or 2-generate coevents except in a few simple cases. Examples are given which
show that there are quantum measures that 2-generate but do not 1-generate
coevents. Examples also show that there are coevents that are 2-generated but
not 1-generated. For simplicity only finite systems are considered.
