Constructive Provability Logic

We present constructive provability logic, an intuitionstic modal logic that validates the Löb rule of Gödel and Löb’s provability logic by permitting logical reflection over provability. Two distinct variants of this logic, CPL and CPL*, are presented in natural deduction and sequent calculus forms which are then shown to be equivalent. In addition, we discuss the use of constructive provability logic to justify stratified negation in logic programming within an intuitionstic and structural proof theory. All theorems presented in this paper are formalized in the Agda proof assistant. An earlier version of this work was presented at IMLA 2011 [Simmons and Toninho 2011].