Game-theoretical interpretation of abelian logic A

Abstract

In the present paper we introduce a variation of Giles's game that captures the semantics of Slaney and Meyer's Abelian logic. This is a variation of the game earlier proposed for the Łukasiewicz infinitely-valued logic. We discuss two possible interpretations of this game. One of the interpretations involves a reference to different types of agents. We also give a brief description of the Abelian logic which as well corresponds to one of the comparative logics proposed by Casari. By different types of agents, we understand agents with diverse cognitive presumptions and capabilities. This reflects the idea that different agents can be encoded by a game (dialogue) semantics and truth (and validity) can be seen as a product of different types of communications between agents, establishing the relation between various types of moves available to the players and the resulting type of rationality. However, the main focus of the paper is concentrated on the technical result concerning the game proposed in the paper. In a separate section, we prove that this game is adequate to the Abelian logic. The game can be extended to the one allowing for the disjunctive strategies. As immediate future research, we suggest proving that Proponent’s winning strategies for some formula $F$ in the game for Abelian logic $\textbf{A}$ with disjunctive strategies correspond to a derivation of the formula $F$ in the hypersequent calculus $\textbf{GA}$.