A functional approach to non‐monotonic logic1
Erik J. Sandewall · Computational Intelligence · 1985
Axiom sets and their extensions are viewed as functions from the set of formulas in the language to a set of four truth values, t, f, u for undefined, and k for contradiction. Such functions form a lattice with “contains less information” as the partial order ?, and “combination of several sources of knowledge” as the least‐upper‐bound operation ⊔. Inference rules are expressed as binary relations between such functions. We show that the usual criterium on fixpoints, namely, to be minimal, does not apply correctly in the case of non‐monotonic inference rules. A stronger concept, approachable fixpoints, is introduced and proven to be sufficient for the existence of a derivation of the fixpoint. In addition, the usefulness of our approach is demonstrated by concise proofs for some previously known results about normal default rules.