On the local consequence of modal Product logic: standard completeness and decidability
Amanda Vidal
Published: 2023/6/24
Abstract
We study modal extensions of product fuzzy logic in two settings: (i) Kripke models where the accessibility relation itself takes fuzzy values, and (ii) Kripke models with a classical (crisp) accessibility relation. In both cases, the models can be evaluated either over all product algebras or over a single product algebra. In this paper, we focus on the local consequence relation for these four types of modal product logics. We show that reasoning in these modal logics can be reduced to reasoning in propositional product logic. This reduction leads to two main results. First, these logics are standard complete: the corresponding logic defined using all product algebras coincides with the one defined using only the standard product algebra on the interval [0, 1]. Second, we show that these logics are decidable.