Strongly complete axiomatizations of "knowing at most" in syntactic structures

Syntactic structures based on standard syntactic assignments model knowledge directly rather than as truth in all possible worlds as in modal epistemic logic, by assigning arbitrary truth values to atomic epistemic formulae. This approach to epistemic logic is very general and is used in several logical frameworks modeling multi-agent systems, but has no interesting logical properties - partly because the standard logical language is too weak to express properties of such structures. In this paper we extend the logical language with a new operator used to represent the proposition that an agent '' knows at most '' a given finite set of formulae and study the problem of strongly complete axiomatization of syntactic structures in this language. Since the logic is not semantically compact, a strongly complete finitary axiomatization is impossible. Instead we present, first, a strongly complete infinitary system, and, second, a strongly complete finitary system for a slightly weaker variant of the language.