Trial and error mathematics: Dialectical systems and completions of theories

TL;DR

This paper introduces p-dialectical systems, a new class that combines features of dialectical and quasidialectical systems, and demonstrates their enhanced ability to represent completions of first-order theories, including Peano Arithmetic.

Contribution

It defines p-dialectical systems and proves they are more powerful than previous dialectical systems in representing theory completions, including complex theories like Peano Arithmetic.

Findings

p-dialectical systems can represent more theory completions

They coincide with dialectical and quasidialectical systems on certain cases

They can encode completions of Peano Arithmetic

Abstract

This paper is part of a project that is based on the notion of dialectical system, introduced by Magari as a way of capturing trial and error mathematics. In previous work, we investigated the expressive and computational power of dialectical systems, and we compared them to a new class of systems, that of quasidialectical systems, that enrich Magari's systems with a natural mechanism of revision. In the present paper we consider a third class of systems, that of $p$-dialectical systems, that naturally combine features coming from the two other cases. We prove several results about $p$-dialectical systems and the sets that they represent. Then we focus on the completions of first-order theories. In doing so, we consider systems with connectives, i.e. systems that encode the rules of classical logic. We show that any consistent system with connectives represents the completion of a given theory. We prove that dialectical and $q$-dialectical systems coincide with respect to the completions that they can represent. Yet, $p$-dialectical systems are more powerful: we exhibit a $p$-dialectical system representing a completion of Peano Arithmetic which is neither dialectical nor $q$-dialectical.