  Logic Programs with Ordered Disjunction (LPODs) extend classical logic
programs with the capability of expressing alternatives with decreasing degrees
of preference in the heads of program rules. Despite the fact that the
operational meaning of ordered disjunction is clear, there exists an important
open issue regarding its semantics. In particular, there does not exist a
purely model-theoretic approach for determining the most preferred models of an
LPOD. At present, the selection of the most preferred models is performed using
a technique that is not based exclusively on the models of the program and in
certain cases produces counterintuitive results. We provide a novel,
model-theoretic semantics for LPODs, which uses an additional truth value in
order to identify the most preferred models of a program. We demonstrate that
the proposed approach overcomes the shortcomings of the traditional semantics
of LPODs. Moreover, the new approach can be used to define the semantics of a
natural class of logic programs that can have both ordered and classical
disjunctions in the heads of clauses. This allows programs that can express not
only strict levels of preferences but also alternatives that are equally
preferred. This work is under consideration for acceptance in TPLP.
