we can then define a common knowledge function, The equivalence with the syntactic approach sketched above can easily be seen: consider an Aumann structure as the one just defined. We can define a correspondent Kripke structure by taking the same space S, accessibility relations that define the equivalence classes corresponding to the partitions , and a valuation function such that it yields value true to the primitive proposition p in all and only the states s such that , where is the event of the Aumann structure corresponding to the primitive proposition p. It is not difficult to see that the common knowledge accessibility function defined in the previous section corresponds to the finest common coarsening of the partitions for all , which is the finitary characterization of common knowledge also given by Aumann in the 1976 article. Applications Common knowledge was used by David Lewis in his pioneering game-theoretical account of convention. In this sense, common knowledge is a concept still central for linguists and philosophers of language (see Clark 1996) maintaining a Lewisian, conventionalist account of language. Robert Aumann introduced a set theoretical formulation of common knowledge (theoretically equivalent to the one given above) and proved the so-called agreement theorem through which: if two agents have common prior probability over a certain event, and the posterior probabilities are common knowledge, then such posterior probabilities are equal. A result based on the agreement theorem and proven by Milgrom shows that, given certain conditions on market efficiency and information, speculative trade is impossible. The concept of common knowledge is central in game theory. For several years it has been thought that the assumption of common knowledge of rationality for the players in the game was fundamental. It turns out (Aumann and Brandenburger 1995) that, in two-player games, common knowledge of rationality is not needed as an epistemic condition for Nash equilibrium strategies. Computer scientists use languages incorporating epistemic logics (and common knowledge) to reason about distributed systems. Such systems can be based on logics more complicated than simple propositional epistemic logic, see Wooldridge Reasoning about Artificial Agents, 2000 (in which he uses a first-order logic incorporating epistemic and temporal operators) or van der Hoek et al. "Alternating Time Epistemic Logic". In his 2007 book, The Stuff of Thought: Language as a Window into Human Nature, Steven Pinker uses the notion of common knowledge to analyze the kind of indirect speech