Free algebras, states and duality for the propositional GödelΔ and Drastic Product logics

In the framework of t-norm based logics, Godel propositional logic G and drastic product logic DP are strictly connected. In this paper we explore the even stricter relation between DP and the logic G(Delta), the expansion of G with Baaz-Monteiro connective Delta. In particular we provide functional representations of free algebras in the corresponding algebraic semantics. We use then these functional representations to develop a theory of states, that is, finitely additive probability measures, for both G(Delta) and DP. Finally, we use dual equivalences for the algebraic semantics of both G(Delta) and DP, to provide a completely combinatorial characterization of states.