A similarity-based Lukassiewicz's many valued modal logic

Abstract

In our opinion, approximate reasoning is one of the most fascinating branches of I.A., and it has generated an extensive literature. One of the goals of approximate reasoning models is to cope with inference patterns more flexible than those of classical reasoning. Amongst them, similarity-based reasoning aims at modelling notions of resemblance or proximity between propositions and consequence relations which make sense insuch a setting. Considering both approximate reasoning in a general context and a particular one as similarity-based reasoning, uncertainty and vagueness appear as two main notions. We shall associate the term uncertainty with degree of belief regarding a proposition (with itself is crip and may be true or false); on the other hand, vagueness (fuzziness) is associated degree of truth of a proposition (which may be fuzzy, i.e., admits non-extremal degree of truth). Both truth degree of fuzzy proposition and belief degree of crisp proposition are coded by reals from the unit interval [0, 1] (in most cases; we shall not discuss exceptions) but they are handled differently. Our claims are that truth degree should not be mistaken for degree of belief and viceversa and that it is posible to combine them. We shall consider that logical systems corresponding to fuzziness are many-valued logics, whereas systems corresponding to uncertainty are related to various generalizations of modal logics and that presence of both fuzziness and uncertainty gives rise to many-valued modal logics. We shall try to build a logic of both vagueness and uncertainty. With this propose, we explore a modal approach to similarity-based reasoning that is a modal logic over a Rational Pavelka's logic RPL-Iike extension of the infinitely Lukasiewicz's logic. We define a many-valued modal system (which is a many-valued counterpart of classical S5 modal system) with many-valued similarity-based Kripke model semantics.