Fuzzy relational modalities admitting truth-valueless propositions

We introduce fuzzy relational modalities of necessity and possibility designed to handle propositions that can be truth-valueless. We identify several important types of these modalities, corresponding to families of connectives and quantifiers of fuzzy partial logic (Bochvar, Sobociński, Kleene) and show some of their properties. We also present a faithful syntactic translation of the resulting fuzzy partial modal logic to first-order fuzzy partial logic.