Abstract This paper considers the bi-modal logic with both $$\Box $$ □ and $$\Diamond $$ ◊ arising from Kripke models with crisp accessibility whose propositions are valued over the standard Gödel algebra [0, 1]. Since this logic lacks the finite model property, we study the logic $$\textbf{GW}^\textrm{c}$$ GW c relying on witnessed Kripke models where, for each modal formula, there is an assignment where the formula without the modality takes the same value as the modal one. We provide a cut-free sequent calculus and we exploit it to prove that $$\textbf{GW}^\textrm{c}$$ GW c is decidable and meets the finite model property. Finally, we explore a connection between the witnessed models and the well-known bi-relational Kripke semantics.

In this article, we provide a new kind of belief revision operator that we call Moderated Revision. At first glance, it is a non-prioritized operator that combines a basic classical AGM operator with a credibility-limited one. The underlying idea is this: when new observation μ is received, it is accepted but with doubts, i.e., uncertainty. We use a revision operator to model the accepted part and a credibility-limited one to represent uncertainty, whenever necessary. In the presence of uncertainty, a selection of the old knowledge balances the result of the revision through a disjunction, allowing the agent to accept part of the new observation and remain unsettled about the rest.

In this article, we provide the weak version of ensconcement which characterizes an interesting family of Shielded base contractions. In turn, this characterization induces a class of AGM contractions satisfying certain postulates that we reveal here. Finally, we show a connection among the class of contractions given by our weak ensconcement and other kinds of base contraction operators. In doing so, we also point out a flaw in the original theorems that link the epistemic entrenchment with ensconcement (which are well established in the literature), and then we provide two possible solutions.

In this note, we introduce the local version of the operator for belief promotion proposed by Schwind et al. We propose a set of postulates and provide a representation theorem that characterizes the proposal. This family of operators is related to belief promotion in the same way that updating is related to revision, and we provide several results that allow us to show this relationship formally. Furthermore, we also show the relationship of the proposed operator with features of credibility-limited revision theory.

In a previous paper, it was shown that the (minimal) modal logic MŁncwith fuzzy accessibility relations over the finite-valued Łukasiewicz logic Łnand a corresponding multi-modal logic mMŁnc(with a modality □afor each value a in the n-valued Łn-chain) had the same expressive power when the language is extended with truth-constants. In this paper we partially extend these results when replacing the underlying logic Łnby the infinite-valued Łukasiewicz logic (with rational truth constants in the language). We prove that the (standard) tautologies of the modal logic MŁnc(resp. mMŁc) are in fact the common tautologies of all the logics MŁnc(resp. all the logics mMŁn) when letting n vary over N. This fact opens the door to show an alternative proof of the finite model property for these logics and hence their decidability.

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 the Fischer Servi intuitionistic modal logic IK plus the prelinearity axiom with respect to this semantics. We axiomatize also the bi-modal analogues of classical 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 an application of the completeness theorems we obtain a representation theorem for bi-modal Gödel algebras.

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.

Introduces a class of fuzzy closure operators called implicative closure operators, which generalize some notions of fuzzy closure operators already introduced by different authors. We show that implicative closure operators capture some usual consequence relations used in approximate reasoning. We study the relation of the implicative closure operators to other existing fuzzy closure operators as the natural inference operators defined by Boixader and Jacas (1998) and the canonical extension of a classical closure operator defined by Gerla (1996).