On Lockean beliefs that are deductively closed and minimal change

Within the formal setting of the Lockean thesis, an agent belief set is defined in terms of degrees of confidence and these are described in probabilistic terms. This approach is of established interest, notwithstanding some limitations that make its use troublesome in some contexts, like, for instance, in belief change theory. Precisely, Lockean belief sets are not generally closed under (classical) logical deduction. The aim of the present paper is twofold: on one side we provide two characterizations of those belief sets that are closed under classical logic deduction, and on the other we propose an approach to probabilistic update that allows us for a minimal revision of those beliefs, i.e., a revision obtained by making the fewest possible changes to the existing belief set while still accommodating the new information. In particular, we show how we can deductively close a belief set via a minimal revision.

Rotations of Gödel algebras with modal operators

The present paper is devoted to study the effect of connected and disconnected rotations of G\"odel algebras with operators grounded on directly indecomposable structures. The structures resulting from this construction we will present are nilpotent minimum (with or without negation fixpoint, depending on whether the rotation is connected or disconnected) with special modal operators defined on a directly indecomposable algebra. In this paper we will present a (quasi-)equational definition of these latter structures. Our main results show that directly indecomposable nilpotent minimum algebras (with or without negation fixpoint) with modal operators are fully characterized as connected and disconnected rotations of directly indecomposable G\"odel algebras endowed with modal operators.

Querying with Łukasiewicz logic

In this paper we present, by way of case studies, a proof of concept, based on a prototype working on a automotive data set, aimed at showing the potential usefulness of using formulas of {\L}ukasiewicz propositional logic to query databases in a fuzzy way. Our approach distinguishes itself for its stress on the purely linguistic, contraposed with numeric, formulations of queries. Our queries are expressed in the pure language of logic, and when we use (integer) numbers, these stand for shortenings of formulas on the syntactic level, and serve as linguistic hedges on the semantic one. Our case-study queries aim first at showing that each numeric-threshold fuzzy query is simulated by a {\L}ukasiewicz formula. Then they focus on the expressing power of {\L}ukasiewicz logic which easily allows for updating queries by clauses and for modifying them through a potentially infinite variety of linguistic hedges implemented with a uniform syntactic mechanism. Finally we shall hint how, already at propositional level, {\L}ukasiewicz natural semantics enjoys a degree of reflection, allowing to write syntactically simple queries that semantically work as meta-queries weighing the contribution of simpler ones.