  The aim of this paper is to propose a many-valued modal framework to
formalize reasoning with both graded preferences and propositions, in the style
of van Benthem et al.'s classical modal logics for preferences. To do so, we
start from Bou et al.'s minimal modal logic over a finite and linearly ordered
residuated lattice. We then define appropriate extensions on a multi-modal
language with graded modalities, both for weak and strict preferences, and with
truth-constants. Actually, the presence of truth-constants in the language
allows us to show that the modal operators Box and Diamond of the minimal modal
logic are inter-definable. Finally, we propose an axiomatic system for this
logic in an extended language (where the preference modal operators are
definable), and prove completeness with respect to the intended graded
preference semantics.
