  It is well know that basic conditionals satisfy some desirable basic logical
and probabilistic properties, such as the compound probability theorem, but
checking the validity of these becomes trickier when we switch to compound and
iterated conditionals. We consider de Finetti's notion of conditional as a
three-valued object and as a conditional random quantity in the betting
framework. We recall the notions of conjunction and disjunction among
conditionals in selected trivalent logics. First, in the framework of specific
three-valued logics we analyze the notions of iterated conditioning introduced
by Cooper-Calabrese, de Finetti and Farrell, respectively. We show that the
compound probability theorem and other basic properties are not preserved by
these objects, by also computing some probability propagation rules. Then, for
each trivalent logic we introduce an iterated conditional as a suitable random
quantity which satisfies the compound prevision theorem and some of the
desirable properties. We also check the validity of two generalized versions of
Bayes' Rule for iterated conditionals. We study the p-validity of generalized
versions of Modus Ponens and two-premise centering for iterated conditionals.
Finally, we observe that all the basic properties are satisfied only by the
iterated conditional mainly developed in recent papers by Gilio and Sanfilippo
in the setting of conditional random quantities.
