Adequate Algebraic Semantics for Extensions of Minimal Logic

The goal of this chapter is to find a representation of j-algebras, convenient for working with logics lying inside the intervals Spec(L₁, L₂). We have to understand the structure of an arbitrary j-algebra A with given upper algebra A⊥ and lower algebra A⊥. The semantic characterization of Glivenko’s logic considered in Section 5.1 prompts the solution to this problem. The desired representation is described in Section 5.2. In Section 5.3 with the help of the obtained representation we characterize the Segerberg logics and demonstrate its effectiveness in this way. Finally, in Section 5.4 we consider the Kripke semantics and define for j-frames analogs of upper and lower algebras associated with a j-algebra.