Modal extensions of Łukasiewicz logic for modelling coalitional power

Abstract

Modal logics for reasoning about the power of coalitions capture the notion of effectivity functions associated with game forms. The main goal of coalition logics is to provide formal tools for modeling the dynamics of a game frame whose states may correspond to different game forms. The two classes of effectivity functions studied are the families of playable and truly playable effectivity functions, respectively. In this paper we generalize the concept of effectivity function beyond the yes/no truth scale. This enables us to describe the situations in which the coalitions assess their effectivity in degrees, based on functions over the outcomes taking values in a finite Łukasiewicz chain. Then we introduce two modal extensions of Łukasiewicz finite-valued logic together with many-valued neighborhood semantics in order to encode the properties of many-valued effectivity functions associated with game forms. As our main results we prove completeness theorems for the two newly introduced modal logics.

Rational Pavelka logic: the best among three worlds?

“…6 These initial considerations developed into one of several grand avenues toward formal many-valued and fuzzy logic (with other such considerations provided by Zadeh and Goguen, Gödel, Heyting, or Post; see [26,Section 10.1]) and a fortiori also to, on the one hand, advanced theory of Lukasiewicz logic and MV-algebras-see [9,44,15] for overviews, and on the other, applications and (re-) interpretations that the formal system of fuzzy logic has to offer in the area of reasoning, broadly conceived: here we cannot aim for a complete picture, see e.g. [36,5,13] for some recent developments.…”