  We deepen the study of conjoined and disjoined conditional events in the
setting of coherence. These objects, differently from other approaches, are
defined in the framework of conditional random quantities. We show that some
well known properties, valid in the case of unconditional events, still hold in
our approach to logical operations among conditional events. In particular we
prove a decomposition formula and a related additive property. Then, we
introduce the set of conditional constituents generated by $n$ conditional
events and we show that they satisfy the basic properties valid in the case of
unconditional events. We obtain a generalized inclusion-exclusion formula and
we prove a suitable distributivity property. Moreover, under logical
independence of basic unconditional events, we give two necessary and
sufficient coherence conditions. The first condition gives a geometrical
characterization for the coherence of prevision assessments on a family
$\mathscr{F}$ constituted by $n$ conditional events and all possible
conjunctions among them. The second condition characterizes the coherence of
prevision assessments defined on $\mathscr{F}\cup \mathscr{K}$, where
$\mathscr{K}$ is the set of conditional constituents associated with the
conditional events in $\mathscr{F}$. Then, we give a further theoretical result
and we examine some examples and counterexamples. Finally, we make a comparison
with other approaches and we illustrate some theoretical aspects and
applications.
