NM-DEKL$^3_\infty$: A Three-Layer Non-Monotone Evolving Dependent Type Logic

We propose ZX-Calculus (Knowledge Evolution Calculus), a conservative extension of Martin-Lof Dependent Type Theory (MLTT) integrating trace-indexed types, presheaf non-monotone semantics, and constructive AGM belief revision. A Coq mechanisation accompanies the paper (34 complete proofs; zero admits for the two central results). (I) Trace types. FinTrace(s0,sn) is an inductive family of typed execution traces. FinTrace and Star(Step) are isomorphic as path types but not judgementally equal; TraceElim exposes the event label e:Event explicitly, giving a more ergonomic interface for event-driven induction. We prove the Trace-Reachability Correspondence, Deterministic Replay, and a canonicity framework via reducibility candidates with a Transport Lemma (RC-elim deferred; all other Core results are Coq-verified). (II) Sheaf semantics. Trace-indexed propositions are contravariant sheaves over the free trace partial-order category Tf. A Separation Theorem (explicit countermodel) distinguishes proof-theoretic monotonicity from semantic non-monotonicity. The term model is an initial CwF (syntactic universal property, not classical completeness). (III) AGM belief revision. We give an explicit constructive partial meet contraction algorithm verified against (C1)-(C4). All eight AGM postulates (R1)-(R8) are theorems. Proofs of R7 and R8 use the Disjunctive Entrenchment Lemma, given a self-contained constructive derivation. (IV) Integration. B^AGM fails the sheaf composition law BP-comp for sequential revision (explicit countermodel, Coq-verified). We introduce Single-Step Revision Systems (SSRS), prove B^AGM is a valid SSRS (Coq-verified), and show this suffices for trace morphisms, retraction characterisation, and revision witnesses. The BP-comp failure reveals a fundamental tension between path-dependent belief revision and functor consistency, not previously identified.

This paper establishes and proves representation theorems for cumulative propositional dependence logic and for cumulative propositional logic with team semantics. Cumulative logics are famously given by System C. For propositional dependence logic, we show that System C entailments are exactly captured by cumulative models from Kraus, Lehmann and Magidor. On the other hand, we show that entailment in cumulative propositional logics with team semantics is exactly captured by cumulative and asymmetric models. For the latter, we also obtain equivalence with cumulative logics based on propositional logic with classical semantics. The proofs will be useful for proving representation theorems for other cumulative logics without negation and material implication.

We present Elenchus, a dialogue system for knowledge base construction grounded in inferentialist semantics, where knowledge engineering is re-conceived as explicitation rather than extraction from expert testimony or textual content. A human expert develops a bilateral position (commitments and denials) about a topic through prover-skeptic dialogue with a large language model (LLM) opponent. The LLM proposes tensions (claims that parts of the position are jointly incoherent) which the expert resolves by retraction, refinement, or contestation. The LLM thus serves as a defeasible derivability oracle whose unreliability is structurally contained by the expert's authority. Our main technical contribution is a mapping from Elenchus dialectical states to material bases in Hlobil and Brandom's NonMonotonic MultiSuccedent (NMMS) logic, satisfying Containment and enabling the elaboration of logical vocabulary that makes explicit the inferential relationships negotiated in the dialectic. We demonstrate the approach on the W3C PROV-O provenance ontology, where a single dialogue session elicits and structures design tensions that a domain expert can articulate, corresponding to decisions documented in a retrospective analysis of the ontology's design. Using pyNMMS, an automated NMMS reasoner, we verify that the structural properties of the resulting material base (nontransitivity, nonmonotonicity, and independence) correspond to specific PROV design rationales, demonstrating end-to-end integration from dialogue through formal reasoning.