This article deals with many-valued modal logics, based only on the necessity operator, over a residuated lattice. We focus on three basic classes, according to the accessibility relation, of Kripke frames: the full class of frames evaluated in the residuated lattice (and so defining the minimum modal logic), the ones only evaluated in the idempotent elements and the ones evaluated in 0 and 1. We show how to expand an axiomatization, with canonical truth-constants in the language, of a finite residuated lattice into one of the modal logic, for each one of the three basic classes of Kripke frames. We also provide axiomatizations for the case of a finite MV chain but this time without canonical truth-constants in the language.

We prove strong completeness of the -version and the 3-version of a Gödel modal logic based on Kripke models where propositions at each world and the accessibility relation are both infinitely valued in the standard Gödel algebra [0,1]. Some asymmetries are revealed: validity in the first logic is reducible to the class of frames having two-valued accessibility relation and this logic does not enjoy the finite model property, while validity in the second logic requires truly fuzzy accessibility relations and this logic has the finite model property. Analogues of the classical modal systems D, T, S4 and S5 are considered also, and the completeness results are extended to languages enriched with a discrete well ordered set of truth constants.

We consider the Gödel bi-modal logic determined by fuzzy Kripke models where both the propositions and the accessibility relation are infinitely valued over the standard Gödel algebra [0,1] and prove strong completeness of Fischer Servi intuitionistic modal logic IK plus the prelinearity axiom with respect to this semantics. We axiomatize also the bi-modal analogues of T, S4, and S5 obtained by restricting to models over frames satisfying the [0,1]-valued versions of the structural properties which characterize these logics. As application of the completeness theorems we obtain a representation theorem for bi-modal Gödel algebras. Gödel Kripke modelsThe language L ✸ (V ar) of propositional bi-modal logic is built from a set V ar of propositional variables, connectives symbols ∨, ∧, →, ⊥, and the modal operators symbols and ✸. Other connectives are defined as usual:Recall that a linear Heyting algebra, or Gödel algebra in the fuzzy literature, is a Heyting algebra satisfying the identity (x ⇒ y) (y ⇒ x) = 1. The variety of these algebras is generated by the standard Gödel algebra [0, 1], the ordered interval with its unique Heyting algebra structure. Let the symbols ·, ⇒, , and denote, respectively, the meet, residuum (implication), and join operations of [0, 1]. 1 Definition 1.1 A Gödel-Kripke model (GK-model) will be a structure M = W, S, e where W is a non-empty set of objects that we call worlds of M, and S : W × W → [0, 1], e : W × V ar → [0, 1] are arbitrary functions. The pair W, S will be called a GK-frame. The function e : W × V ar → [0, 1] associates to each world x a valuation e(x, −) : V ar → [0, 1] which extends to e(x, −) : L ✸ (V ar) → [0, 1] by defining inductively on the construction of the formulas (we utilize the same symbol e to name the extension): e(x, ⊥) := 0 e(x, ϕ ∧ ψ) := e(x, ϕ) · e(x, ψ)

We consider two kinds of similarity-based reasoning and formalise them in a logical setting. In one case, we are led by the principle that conclusions can be drawn even if they are only approximately correct. This leads to a graded approximate entailment, which is weaker than classical entailment. In the other case, we follow the principle that conclusions must remain correct even if the assumptions are slightly changed. This leads to a notion of a graded strong entailment, which is stronger than classical entailment. We develop two logical calculi based on the notions of approximate and of strong entailment, respectively.