Ricardo Oscar Rodríguez

On the Minimum Many-Valued Modal Logic over a Finite Residuated Lattice

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 evaluated in the idempotent elements and the ones only 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.

A modal account of similarity-based reasoning

Abstract One of the goals of a variety of approximate reasoning models is to cope with inference patterns more flexible than those of classical reasoning. Among them, similarity-based reasoning aims at modeling notions of resemblance or proximity among propositions and consequence relations which make sense in such a setting. One way of proceeding is to equip the set of interpretations or possible worlds with a similarity relation S, that is, a reflexive, symmetric, and t-norm-transitive fuzzy relation. We explore a modal approach to similarity-based reasoning, and we define three multi-modal systems with similarity-based Kripke model semantics. A similarity-based Kripke model is a structure 〈W, S, ‖ ‖〉, in which W is the set of possible worlds, ‖ ‖ represents an assignment of possible worlds to atomic formulas, and S is a similarity function S: W × W → G, where G is a subset of the unit interval [0,1] such that 0,1 ϵ G. We provide soundness and completeness results for these systems with respect to some classes of the above structures.

Standard Gödel Modal Logics

We prove strong completeness of the □-version and the ◊-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.

Bi-modal Gödel logic over [0,1]-valued Kripke frames

We consider the Godel bi-modal logic determined by fuzzy Kripke models where both the propositions and the accessibility relation are infinitely valued over the standard Godel 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 Godel algebras.

A Fuzzy Modal Logic for Similarity Reasoning

In this paper we are concerned with the formalization of a similarity-based type of reasoning dealing with expressions of the form approximately ϕ, where ϕis a fuzzy proposition. From a technical point of view we need a fuzzy logic as base logic to deal with the fuzziness of propositions and also we need a modality to account for the notion of approximation or closeness. Therefore we propose a modal fuzzy logic with semantics based on Kripke structures where the accessibility relations are fuzzy similarity relations measuring how similar are the possible worlds.

Logical approaches to fuzzy similarity-based reasoning: an overview

The aim of this paper is to survey a class of logical formalizations of similarity-based reasoning models where similarity is understood as a graded notion of truthlikeness. We basically identify two different kinds of logical approaches that have been used to formalize fuzzy similarity reasoning: syntacticelly-oriented approaches based on a notion of approximate proof, and semantically-oriented approaches based on several notions of approximate entailments. In particular, for these approximate entailments we provide four different formalisations in terms of suitable systems of modal and conditional logics, including for each class a system of graded operators with classical semantics, as well as a system with many-valued operators. Finally, we also explore some nonmonotonic issues of similarity-based reasoning.

Fuzzy approximation relations, modal structures and possibilistic logic

The paper introduces a general axiomatic notion of approximation mapping, a mapping that associates to each crisp proposition p a fuzzy set representing approximately p. It is shown how it can be obtained through fuzzy relations, which are at least reflexive. We study the corresponding multi-modal systems depending on the properties satisfied by the approximate relation. Finally, we show some equivalences between possibilistic logical consequences and global/local logical consequences in the multi-modal systems.This work was funded in part by a CNRS OHLL grant to the Sony Computer Science Laboratory in Paris and the AI Laboratory of the Free University of Brussels (VUB), as well as by an ESF OHMM grant.

Logics for approximate and strong entailments

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.

Graded Similarity-Based Semantics for Nonmonotonic Inferences

We explore which kinds of nonmonotonic inference relations naturally arise when using similarity-based implication and consistency measures to rank propositions à la Gärdenfors and Makinson or to build system of spheres in the sense of Lewis. There is no surprising result, the main interest being to provide a new perspective to nonmonotonic reasoning from a field which has not been traditionally considered.

Decidability of order-based modal logics ☆

Abstract Decidability of the validity problem is established for a family of many-valued modal logics, notably Godel modal logics, where propositional connectives are evaluated according to the order of values in a complete sublattice of the real unit interval [ 0 , 1 ] , and box and diamond modalities are evaluated as infima and suprema over (many-valued) Kripke frames. If the sublattice is infinite and the language is sufficiently expressive, then the standard semantics for such a logic lacks the finite model property. It is shown here, however, that, given certain regularity conditions, the finite model property holds for a new semantics for the logic, providing a basis for establishing decidability and PSPACE-completeness. Similar results are also established for S5 logics that coincide with one-variable fragments of first-order many-valued logics. In particular, a first proof is given of the decidability and co-NP-completeness of validity in the one-variable fragment of first-order Godel logic.