Comparing Dialectical Systems: Contradiction and Counterexample in Belief Change (Extended Version)

TL;DR

This paper compares different models of dialectical systems used for belief revision, proving that q-dialectical systems are more powerful than p-dialectical systems, which are more powerful than d-dialectical systems, highlighting their roles in automated reasoning.

Contribution

It proves that q-dialectical systems are strictly more powerful than p-dialectical systems, resolving an open problem and clarifying their roles in belief change.

Findings

Q-dialectical systems are strictly more powerful than p-dialectical systems.

P-dialectical systems are more powerful than d-dialectical systems.

The roles of counterexample and contradiction are complementary in belief revision.

Abstract

Dialectical systems are a mathematical formalism for modeling an agent updating a knowledge base seeking consistency. Introduced in the 1970s by Roberto Magari, they were originally conceived to capture how a working mathematician or a research community refines beliefs in the pursuit of truth. Dialectical systems also serve as natural models for the belief change of an automated agent, offering a unifying, computable framework for dynamic belief management. The literature distinguishes three main models of dialectical systems: (d-)dialectical systems based on revising beliefs when they are seen to be inconsistent, p-dialectical systems based on revising beliefs based on finding a counterexample, and q-dialectical systems which can do both. We answer an open problem in the literature by proving that q-dialectical systems are strictly more powerful than p-dialectical systems, which are themselves known to be strictly stronger than (d-)dialectical systems. This result highlights the complementary roles of counterexample and contradiction in automated belief revision, and thus also in the reasoning processes of mathematicians and research communities.