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arxiv: 2605.05286 · v2 · pith:WSFJUP5Znew · submitted 2026-05-06 · 💻 cs.LO

Paraconsistent Semantics for Extended Fuzzy Logic Programs via Approximation Fixpoint Theory [Extended Version]

Pascal Kettmann , Hannes Strass , Jesse Heyninck , Jeroen Spaans This is my paper

Pith reviewed 2026-05-25 06:31 UTC · model grok-4.3

classification 💻 cs.LO
keywords paraconsistent semanticsfuzzy logic programsapproximation fixpoint theorynegation as failurestrong negationlogic programmingwell-founded semanticsextended logic programs
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The pith

Approximation fixpoint theory yields coherent paraconsistent semantics for fuzzy logic programs with both negation-as-failure and strong negation.

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

The paper sets out to define semantics for fuzzy logic programs that combine negation-as-failure with strong negation by lifting approximation fixpoint theory to this setting. A sympathetic reader would care because many practical reasoning tasks involve uncertainty (fuzzy degrees), missing evidence, explicit falsehoods, and occasional contradictions, yet still require usable conclusions. The resulting framework produces a family of semantics that stay well-behaved under paraconsistency, meaning contradictions do not force arbitrary conclusions. It also shows that several prior semantics arise as special cases while new ones become available by varying the choice of approximating operators.

Core claim

By lifting approximation fixpoint theory to fuzzy logic programs that contain both negation-as-failure and strong negation, one obtains a uniform way to construct well-behaved semantics; these semantics generalize several existing ones for such programs and generate additional new semantics as well.

What carries the argument

Approximation fixpoint theory applied to the immediate-consequence operator of the fuzzy program, extended to handle both forms of negation while preserving approximation and coherence properties.

If this is right

  • Several known semantics for fuzzy programs with strong negation become special cases inside the new framework.
  • Different choices of approximating operators produce new, previously unstudied semantics for the same class of programs.
  • The semantics remain coherent: contradictions do not force the derivation of arbitrary conclusions.
  • Programs mixing fuzzy degrees with both kinds of negation can be given semantics without requiring separate handling for each negation type.

Where Pith is reading between the lines

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

  • The same construction might be tested on programs that also incorporate probabilistic or weighted facts to see whether coherence is preserved.
  • Practical solvers could be built by computing the fixpoints of the approximating operators on finite fuzzy programs.
  • The approach suggests a route for extending other non-monotonic formalisms to handle both incomplete information and explicit contradictions at once.

Load-bearing premise

The approximating operators for these fuzzy programs continue to satisfy the monotonicity and approximation conditions required by the fixpoint theory even after the two negations and fuzzy truth values are added.

What would settle it

A concrete fuzzy program containing both negation-as-failure and strong negation for which the derived semantics assigns a value that derives both a literal and its strong negation in a way that violates the coherence condition of the framework.

read the original abstract

In logic programming, negation can be interpreted in various ways. Probably best known is the concept of "negation as failure", where "$\mathit{not}\, p$" is true if we have no evidence for $p$. On the other hand, strong negation requires that we have evidence for $p$ being false. Defining semantics for logic programs containing both kinds of negation is a challenging task, and this becomes even more challenging when combining this with other extensions of logic programming, e.g. fuzziness. In this work, we use the framework of approximating fixpoint theory to formulate well-behaved semantics for fuzzy logic programs containing both "by-failure" and strong negation. We show that this framework generalizes several existing semantics as well as giving rise to a host of new semantics.

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

0 major / 0 minor

Summary. The paper uses approximation fixpoint theory to define paraconsistent semantics for fuzzy logic programs that include both negation-as-failure and strong negation. It claims that the resulting framework produces well-behaved semantics, generalizes several existing semantics for such programs, and gives rise to new semantics.

Significance. If the technical claims hold, the work would extend approximation fixpoint theory to handle paraconsistency in fuzzy settings with dual negations, offering a potential unifying framework. The generalization of prior semantics would be a concrete strength, as would any machine-checked or parameter-free aspects of the construction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their review of our manuscript. The provided summary accurately reflects the paper's focus on using approximation fixpoint theory to define paraconsistent semantics for fuzzy logic programs with both negation-as-failure and strong negation. No specific major comments were listed in the report, so there are no individual points requiring point-by-point rebuttal. We remain available to address any questions or provide clarifications on the technical claims, which are supported by the proofs and constructions in the full manuscript.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper applies approximation fixpoint theory to formulate semantics for fuzzy logic programs with both negation-as-failure and strong negation. The abstract and context describe a generalization of prior semantics without any self-definitional constructions, fitted inputs renamed as predictions, or load-bearing self-citations that reduce the central claim to its own inputs by construction. No equations or derivation steps are supplied that exhibit the enumerated circularity patterns, so the framework is treated as self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no free parameters, axioms, or invented entities are described.

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