Free algebras, states and duality for the propositional Godel(Delta) and Drastic Product logics

In the framework of t-norm based logics, Gödel 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 GΔ, the expansion of G with Baaz–Monteiro connective Δ. 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 GΔ and DP. Finally, we use dual equivalences for the algebraic semantics of both GΔ and DP, to provide a completely combinatorial characterization of states.