Comparing Dialectical Systems: Contradiction and Counterexample in Belief Change (Extended Version)
This resolves an open problem in formal models of belief revision, with implications for automated agents and reasoning processes in mathematics and research communities, though it is incremental in nature.
The paper tackled the problem of comparing the power of different dialectical systems for belief change, proving that q-dialectical systems are strictly more powerful than p-dialectical systems, which are stronger than d-dialectical systems.
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.