Giacomo Sillari a Logical Framework for Convention

In this paper, I provide a logical framework for defining conventions, elaborating on the game-theoretic model proposed by David Lewis. The philosophical analysis of some of the key concepts in Lewis’s model reveals that a modal logic formalization may be a natural one. The paper will develop on the analysis and critique of such concepts as those of common knowledge, indication, and the distinction between epistemic and practical rationality. In particular: (i) the analysis of Lewis’s definition of common knowledge reveals that a suitable formalization can be obtained by adopting an approach analogous to that of awareness structures in modal logic; moreover (ii) the analysis of the notion of indication reveals that the agents may be required to make inductive inferences yielding probabilistic beliefs. I shall stress that such aspects, however, pertain to the sphere of epistemic rationality (i.e., they deal with the justification of the agents’ beliefs) rather than to the sphere of practical rationality. Confounding the two spheres may lead to the wrong conclusion that, in order to make sense of, say, salience as a coordination device, one should incorporate psychological assumptions into an undivided notion of rationality. On the contrary, practical rationality stands as the usual notion of game-theoretic rationality, whereas epistemic rationality incorporates those aspects pointed out in (i) and (ii) above. This attempt to provide a formal framework for Lewis’s theory of convention follows those of Vanderschraaf (1995, 1998) and Cubitt and Sugden (2003). In his work on Lewis, Vanderschraaf provides a characterization of convention as correlated equilibrium, adopting a formal framework close to the set-theoretical one proposed by Aumann (1976). Cubitt and Sugden point out that such a framework does not take into account certain elements that are however present in Lewis’s original theory, and propose a different formal setup altogether. In this paper, I show how a formalization based on modal logic can incorporate those distinctive aspects introduced by David Lewis in Convention.