Interpretability in PRA

TL;DR

This paper investigates the interpretability logic of PRA, revealing its unique properties, modal principles, and relationships with other interpretability principles, through modal analysis and frame conditions.

Contribution

It characterizes IL(PRA) by analyzing specific principles like B and Z, and establishes their modal relationships and frame conditions, expanding understanding beyond known interpretability logics.

Findings

IL(PRA) is not ILM or ILP due to PRA's properties.

Principles B and Z are analyzed within IL(PRA) and their interrelations are established.

A frame condition for principle B is proved.

Abstract

In this paper from 2009 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. Moreover, 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.