In this paper we study IL(PRA), the interpretability logic of PRA. As PRA is neither an essentially reflexive theory nor finitely axiomatizable, the two known arithmetical completeness results do not apply to PRA: IL(PRA) is not ILM or ILP. IL(PRA) does of course contain all the principles known to be part of IL(All), the interpretability logic of the principles common to all reasonable arithmetical theories. In this paper, we take two arithmetical properties of PRA and see what their consequences in the modal logic IL(PRA) are. These properties are reflected in the so-called Beklemishev Principle B, and Zambella’s Principle Z, neither of which is a part of IL(All). Both principles and their interrelation are submitted to a modal study. In particular, we prove a frame condition for B. morover, we prove that Z follows from a restricted form of B. Finally, we give an overview of the known relationships of IL(PRA) to important other interpetability principles.

The notion of a critical successor [dJV90] has been central to all modal completeness proofs in interpretability logics. In this paper we shall work with an alternative notion, that of an assuring successor. As we shall see, this makes life a lot easier. After a general treatment of assuringness, we shall apply it to obtain a completeness results for ILW, a result first proved by de Jongh and Veltman [dJV99]. In our proof, the gain of assuringness becomes very clear. 1

In this paper, we develop a toolkit to derive principles for the interpetability logic of all reasonable theories by fine-tuning ILM and ILP proofs.

of linear time