Propositional team logics

Abstract

We consider team semantics for propositional logic, continuing In team semantics the truth of a propositional formula is considered in a set of valuations, called a team, rather than in an individual valuation. This offers the possibility to give meaning to concepts such as dependence, independence and inclusion. We associate with every formula ϕ(symbol) based on finitely many propositional variables the set (left open bracket)ϕ(symbol)(right open bracket) of teams that satisfy ϕ(symbol). We define a maximal propositional team logic in which every set of teams is definable as (left open bracket)ϕ(symbol)(right open bracket) for suitable ϕ(symbol). This requires going beyond the logical operations of classical propositional logic. We exhibit a hierarchy of logics between the smallest, viz. classical propositional logic, and the maximal propositional team logic. We characterize these different logics in several ways: first syntactically by their logical operations, and then semantically by the kind of sets of teams they are capable of defining. In several important cases we are able to find complete axiomatizations for these logics.