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arxiv: 2605.05286 · v1 · submitted 2026-05-06 · 💻 cs.LO

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

Pascal Kettmann , Hannes Strass , Jesse Heyninck , Jeroen Spaans
Authors on Pith no claims yet

Pith reviewed 2026-05-08 15:52 UTC · model grok-4.3

classification 💻 cs.LO
keywords fuzzy logic programmingparaconsistent semanticsapproximating fixpoint theorynegation as failurestrong negationextended logic programslogic programming semantics
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The pith

Approximating fixpoint theory defines paraconsistent semantics for fuzzy logic programs containing 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 seeks to give semantics to fuzzy logic programs that include both negation as failure, where a literal is false if no evidence supports it, and strong negation, where evidence must show the literal is false. Combining these with fuzziness makes consistent definitions difficult because inconsistencies can arise in uncertain settings. The authors apply approximating fixpoint theory to construct semantics that remain well-behaved and paraconsistent, handling contradictions without deriving arbitrary conclusions. This generalizes prior semantics for such programs and opens the way to additional ones by varying the choice of approximating operators.

Core claim

Using the framework of approximating fixpoint theory, we formulate well-behaved semantics for fuzzy logic programs containing both negation as failure and strong negation. This framework generalizes several existing semantics as well as giving rise to a host of new semantics.

What carries the argument

Approximating fixpoint theory, which builds semantics from fixed points of approximating operators that handle both forms of negation in the fuzzy setting.

If this is right

  • The new semantics remain paraconsistent, so contradictory information does not force every conclusion.
  • Several previously proposed semantics for extended fuzzy programs become special cases within this single framework.
  • Different choices of approximating operators generate families of new semantics that can be compared systematically.
  • The approach extends naturally to programs that combine fuzziness with the two distinct kinds of negation.

Where Pith is reading between the lines

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

  • The same construction could be tested on other logic programming extensions that mix uncertainty and inconsistency.
  • Practical reasoning systems could adopt these semantics to tolerate both incomplete evidence and conflicting facts.
  • Implementation work could measure how the choice of approximating operator affects the computational cost of finding stable models.

Load-bearing premise

That approximating fixpoint theory can be applied directly to fuzzy programs with both negation as failure and strong negation while automatically producing well-behaved and paraconsistent results without extra restrictions.

What would settle it

A concrete fuzzy logic program mixing negation as failure and strong negation for which the semantics produced by approximating fixpoint theory fails to be paraconsistent or fails to satisfy the well-behaved properties required by the theory.

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 / 3 minor

Summary. The manuscript applies approximation fixpoint theory (AFT) to define paraconsistent semantics for extended fuzzy logic programs that incorporate both negation-as-failure and strong negation. The central claim is that the resulting semantics are well-behaved, generalize several existing approaches, and give rise to new semantics.

Significance. If the technical development holds, the work offers a valuable unified framework for non-monotonic reasoning under uncertainty and inconsistency. The explicit use of AFT to handle the combination of the two negation types while preserving paraconsistency and generalization is a clear strength, as it provides a systematic way to derive and compare multiple semantics rather than treating them in isolation.

minor comments (3)
  1. [Abstract] The abstract asserts that the semantics are 'well-behaved' and 'paraconsistent' without indicating which AFT properties (e.g., consistency or precision) are preserved by the constructed approximating operators; a one-sentence clarification would help readers immediately see the link to the framework.
  2. [§3] In the definition of the extended program (likely §3 or §4), the interaction between the fuzzy truth lattice, the strong negation operator, and the negation-as-failure operator should be stated explicitly, including any monotonicity or approximation conditions required for the AFT fixpoint construction to apply directly.
  3. [§5] The generalization results (claimed in the abstract and presumably proved in §5) would be easier to verify if each existing semantics is recovered as a special case of the new approximating operator with a short table or corollary listing the parameter settings.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The referee's summary correctly captures the central contribution of applying approximation fixpoint theory to obtain well-behaved paraconsistent semantics for extended fuzzy logic programs that combine negation as failure with strong negation. We are pleased that the potential of this unified framework is recognized.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper applies the external framework of approximation fixpoint theory to construct semantics for fuzzy logic programs that include both negation-as-failure and strong negation. The abstract and described approach treat AFT as an independent mathematical tool whose standard properties (approximating operators on a lattice) are used to guarantee well-behavedness and paraconsistency; the resulting semantics are shown to generalize prior work rather than being defined in terms of themselves. No equations or steps reduce by construction to fitted parameters, self-referential definitions, or load-bearing self-citations whose validity depends on the present paper. The derivation chain therefore remains self-contained against the external AFT benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of approximation fixpoint theory as a domain assumption for defining semantics in this extended setting; no free parameters or invented entities are identifiable from the abstract.

axioms (1)
  • domain assumption Approximation fixpoint theory provides a suitable and general framework for formulating well-behaved semantics for fuzzy logic programs with both negation as failure and strong negation.
    Invoked as the basis for defining the new semantics and showing generalization.

pith-pipeline@v0.9.0 · 5443 in / 1195 out tokens · 43916 ms · 2026-05-08T15:52:44.096071+00:00 · methodology

discussion (0)

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Reference graph

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