Formalism for Game-Theoretic Approach to Formal Logic

Unifying Communication Theory, Formal Logic and Game Theory

Victor Morgante

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Abstract

This paper presents a game-theoretic approach to formal logic, where players cooperate to interpret theorems of a theory in a coherent and understandable manner. The objective is to achieve clear communication and a shared understanding of the theorems. We introduce a formalism for this non-adversarial game, including payoff functions, and explore conditions under which cooperation and consensus are likely to occur. Key theorems and propositions are presented, such as the Coherence Theorem, which states that there exists an interpretation allowing all cooperating players to achieve a positive payoff from the theorems of a theory. The Differential Interpretation Game (DIG) is introduced, where players can choose different interpretations that are provable under notation and syntax of a theory, drawing a parallel with the compactness theorem in logic. The stability and convergence of differential interpretations are discussed, highlighting the juxtaposition between local and global coherence in formal logic and game theory. This framework offers insights into the dynamics of interpretation and the role of interpreters in maintaining the integrity of a…