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arxiv: 2604.18766 · v1 · submitted 2026-04-20 · 💻 cs.LO

A taxonomy for controlling (in)consistency

Marcelo E. Coniglio , Rafael Ongaratto This is my paper

Pith reviewed 2026-05-10 03:03 UTC · model grok-4.3

classification 💻 cs.LO
keywords paraconsistent logicsLFIsconsistency operatorswap structuresRNmatriceshierarchy of logicscontrolled consistencynegation strength
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The pith

The L_n^k hierarchy introduces a two-dimensional taxonomy of paraconsistent logics that controls degrees of consistency and negation commitment.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper presents the Logics of Controlled Consistency, a hierarchy L_n^k of LFIs that varies along two axes: one allows the consistency operator to behave inconsistently for a fixed number of iterations, and the other strengthens negation. The result is a family of logics that can represent different degrees of paraconsistent commitment, from more skeptical to more dogmatic positions, with known LFIs appearing as special cases. Swap structure semantics establish soundness and completeness for the hierarchy, while RNmatrices give a uniform general semantics. A concrete 5-valued instance called LFI3 is axiomatized separately using Karnaugh maps and twist structures to illustrate the approach beyond three-valued logics.

Core claim

The central claim is that each two-dimensional level L_n^k corresponds to a distinct controlled notion of consistency, inconsistency and negation, so that moving up either axis produces a stronger logic capable of expressing a finer degree of paraconsistent commitment while remaining sound and complete under the swap-structure and RNmatrix semantics.

What carries the argument

The two-dimensional hierarchy L_n^k of Logics of Controlled Consistency, equipped with swap structure semantics and RNmatrix constructions that interpret the consistency and negation operators at each level.

If this is right

  • Each level of the hierarchy corresponds to a distinct philosophical stance on inconsistency, from skepticism to dogmatism.
  • Standard LFIs such as certain da Costa C_n systems arise as particular cases inside the hierarchy.
  • The RNmatrix construction supplies a uniform semantics that scales to the entire family without separate proofs for each level.
  • LFI3 demonstrates how the method extends to five-valued logics beyond the usual three-valued LFIs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Applications that must handle inconsistent data could select an L_n^k level according to how much inconsistency they are willing to tolerate.
  • The two-dimensional parameterization might be used to interpolate between existing paraconsistent systems by varying n and k continuously where possible.
  • Empirical tests on real inconsistent databases could check whether the philosophical distinctions translate into measurable differences in inference behavior.

Load-bearing premise

The swap structure semantics and RNmatrix constructions correctly capture the intended behavior of the consistency and negation operators at every level without hidden constraints or loss of expressiveness.

What would settle it

A concrete model or formula in which raising the iteration bound on inconsistent consistency behavior fails to produce the predicted increase in paraconsistent strength, or in which the 5-valued semantics of LFI3 contradict its proposed axiomatization.

read the original abstract

In this article, the hierarchy of LFIs L$_n^k$, Logics of Controlled Consistency (LCC), is introduced. Inspired by da Costa's original C$_n$ systems, this hierarchy can represent different degrees of paraconsistent commitment and different related notions of consistency, inconsistency, and negation associated with each two-dimensional level of these logics. In one dimension, the logics become increasingly more paraconsistent by allowing the consistency operator to behave inconsistently up to a fixed iteration. In another dimension, the negation is increasingly strengthened. Initially, we present these logics with a swap structure semantics, showing their soundness and completeness. Some well-known LFIs are shown to be particular cases of LCCs. With some examples, we show how these different logics represent different types of paraconsistent commitment: from skepticism to dogmastism, these logics have the multiplicity to represent these different philosophical positions. Furthermore, the development of the hierarchy in a general manner allows pragmatism to take place when considering the different types of paraconsistent commitment. Each level we go up in this direction we get a stronger family of logics. Furthermore, we also present an extension of an LCC, a 5-valued LFI called LFI3, a sublogic of LFI1. LFI3 presents a paradigmatic case for the development of many-valued LFIs that have more than three values. Using a technique that combines Karnaugh Maps and Twist Structures, we give an axiomatization and a semantical account of LFI3. Finally, using RNmatrices, we give a general semantical account of the L$_n^k$ family of logics, and we also prove its soundness and completeness.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript introduces the two-dimensional hierarchy L_n^k of Logics of Controlled Consistency (LCC), extending da Costa's C_n systems. One dimension increases paraconsistency by allowing the consistency operator to behave inconsistently up to a fixed iteration, while the other strengthens the negation. The paper provides swap structure semantics with soundness and completeness, recovers known LFIs as special cases, illustrates philosophical interpretations from skepticism to dogmatism, axiomatizes the 5-valued LFI3 extension using Karnaugh maps and twist structures, and gives a general RNmatrix semantics for the family with soundness and completeness proofs.

Significance. If the semantic constructions and proofs hold, the work offers a systematic taxonomy for paraconsistent logics with controllable degrees of consistency and inconsistency. The recovery of known LFIs, the concrete axiomatization of LFI3 as a paradigmatic many-valued LFI, and the uniform RNmatrix account for the entire hierarchy are notable strengths, enabling pragmatic selection among different paraconsistent commitments. This advances the field by providing explicit, generalizable semantics beyond three-valued systems.

major comments (2)
  1. The soundness and completeness arguments for the general L_n^k hierarchy via swap structures (as stated in the abstract and developed in the initial semantic section) require explicit inductive verification or edge-case checks for arbitrary n > 3 and k > 2 to confirm that the intended behavior of the consistency and negation operators is preserved without loss of expressiveness or hidden constraints; this is load-bearing for the central claim that each (n,k) level encodes a distinct degree of paraconsistent commitment.
  2. In the LFI3 section, the axiomatization obtained via Karnaugh maps and twist structures is presented as complete for the 5-valued semantics, but the manuscript should include a direct proof that the axioms are sufficient for the consistency operator at this level (distinct from LFI1) and that the RNmatrix construction uniformly extends without additional constraints.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and constructive comments on our manuscript. We address each major comment below and indicate the revisions to be made.

read point-by-point responses

  1. Referee: The soundness and completeness arguments for the general L_n^k hierarchy via swap structures (as stated in the abstract and developed in the initial semantic section) require explicit inductive verification or edge-case checks for arbitrary n > 3 and k > 2 to confirm that the intended behavior of the consistency and negation operators is preserved without loss of expressiveness or hidden constraints; this is load-bearing for the central claim that each (n,k) level encodes a distinct degree of paraconsistent commitment.

    Authors: The swap-structure semantics and associated soundness/completeness proofs are formulated parametrically for arbitrary n and k, with the inductive argument on formula complexity carrying the parameters through the base and inductive steps. We acknowledge that the presentation would benefit from greater explicitness for higher values. In the revised manuscript we will add a dedicated subsection that performs the induction explicitly for n > 3 and k > 2, including concrete edge-case verifications (e.g., n=4, k=3 and n=5, k=2) to confirm that the consistency and negation operators retain their intended behavior without hidden constraints or loss of expressiveness. This will directly support the claim that each (n,k) level encodes a distinct paraconsistent commitment. revision: yes

  2. Referee: In the LFI3 section, the axiomatization obtained via Karnaugh maps and twist structures is presented as complete for the 5-valued semantics, but the manuscript should include a direct proof that the axioms are sufficient for the consistency operator at this level (distinct from LFI1) and that the RNmatrix construction uniformly extends without additional constraints.

    Authors: The axiomatization of LFI3 is obtained from its 5-valued semantics via Karnaugh maps and twist structures, and completeness with respect to that semantics is already established. To meet the referee’s request we will insert a short lemma that directly verifies the axioms are sufficient to enforce the specific behavior of the consistency operator at the LFI3 level (thereby distinguishing it from LFI1). We will also add an explicit statement and short argument showing that the general RNmatrix construction for the L_n^k hierarchy extends uniformly to LFI3 without imposing further constraints, relying on the existing soundness and completeness results for the family. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained

full rationale

The paper defines the L_n^k hierarchy directly via swap-structure semantics and RNmatrix constructions, supplies independent soundness and completeness proofs for the full family, recovers known LFIs only as special cases after the general semantics are stated, and axiomatizes the LFI3 extension via an explicit Karnaugh-map-plus-twist-structure method whose equations do not presuppose the target results. Philosophical interpretations are presented as illustrative mappings rather than formally derived equivalences, and no load-bearing step reduces a claimed result to a fitted parameter, self-citation, or definitional renaming of its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard background results in paraconsistent logic plus the new semantic constructions; no free parameters are introduced, and no new entities are postulated beyond the hierarchy itself.

axioms (2)
  • domain assumption Standard properties of LFIs and da Costa's C_n systems hold as background
    The paper builds explicitly on da Costa's C_n and known LFIs as special cases.
  • domain assumption Swap structures and RNmatrices provide adequate semantics for the controlled consistency operator
    Invoked to establish soundness and completeness for the entire hierarchy.

pith-pipeline@v0.9.0 · 5607 in / 1440 out tokens · 42594 ms · 2026-05-10T03:03:29.280512+00:00 · methodology

discussion (0)

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Reference graph

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