An (almost) fuzzy logic of action and preferences, its quasi-model interpretations, and the problem of its decidability

The logical depiction of the concept of action forms a unique method of grasping its semantic content. It finds its reflection in different proposals of epistemic and deontic-logic systems describing actions as the operational behaviour of an agent as a game played over a given game tree. This perspective establishes some unique connection between actions and the agent's preferences. One of the logical-game-theoretic approaches to their modelling has been elaborated in van Benthem's school in the common area of game-theory, epistemic and fixed-point logic. In reference to some ideas of van Benthem's school - the article aims to propose a new (almost) fuzzy logic system AFLAP for actions and preferences describing a game-theoretic behaviour of an agent over a given game tree. Nevertheless, the backward induction strategy is idealised and exchanged for the Church-Rosser property, and a piece of fuzziness is introduced to the system by considering two unique 'epsilon' relations Move and BI, with their approximation sequences. An abstract model in the form of the so-called quasi-model is proposed for such a system. It forms a multi-level (potentially infinite) construction built due to the pattern recently elaborated and enhanced by Wolter-Kurucz's school. This construction is exploited in the article as a convenient basis for proving the decidability of AFLAP by showing its 'abstract' finite model property. This task is performed by constructing an additional workable quasi-model of the size dependent on the size of the set of subformulae of a given formula of AFLAP. Finally, some R-programming-oriented applications of the earlier theoretic considerations are presented.