Team semantics extends traditional semantics by shifting the interpretation of formulas from individual semantic objects (e.g., assignments, valuations, or worlds) to sets of such objects, commonly referred to as teams. This approach enables the expression of properties, such as variable dependency, that are inaccessible in traditional semantics. Since its introduction by Hodges and subsequent refinement by Väänänen, team semantics has been used to develop expressively enriched logics that retain desirable properties such as compactness.
However, these logics are typically non-substitutional, limiting their algebraic treatment and preventing the development of schematic proof systems. This limitation can be attributed to the flatness principle which is commonly adhered to in many constructions of team semantics for logics.
In this talk, we approach the formation of propositional logic of teams from an algebraic perspective, explicitly discarding the flatness principle. We propose a substitutional logic of teams that is expressive enough to axiomatize key propositional team logics, such as propositional dependence logic.
Our construction parallels the algebraic construction of classical propositional logic from Boolean algebras. The algebras we are using are powersets of Boolean algebras equipped with both internal (pointwise) and external (set-theoretic) operations. The resulting logic is clearly substitutional, and is shown to be sound and complete with respect to a labelled natural deduction system.
If time permits, we will also discuss how we might do to extend this construction to the framework of first-order logic.