Axiomatizing logics of fuzzy preferences using graded modalities

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.