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arxiv: 2604.18526 · v1 · submitted 2026-04-20 · 🧮 math.LO

Recognition: unknown

Positive, Negative, and Reliable Information in a First-Order Logic of Evidence and Truth

Abilio Rodrigues, Marcelo E. Coniglio

Pith reviewed 2026-05-10 02:58 UTC · model grok-4.3

classification 🧮 math.LO
keywords first-order logiclogic of evidence and truthsix-valued semanticssoundness and completenesso-extensionsnormal formsreplacement property
0
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The pith

The deductive system of QLETF+ is sound and complete for its six-valued first-order semantics.

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

The paper introduces QLETF+ as the quantified extension of LETF+ to handle first-order statements about positive, negative, and reliable information. It defines models in which each predicate receives an extension, an anti-extension, and an o-extension. The central result is a soundness and completeness proof for the deductive system relative to this six-valued semantics. The logic also satisfies the replacement property and admits conjunctive, disjunctive, and prenex normal forms. A sympathetic reader would care because the system supplies a rigorous way to reason about quantified claims when information can be supported, denied, or confirmed reliable.

Core claim

QLETF+ is a first-order logic of evidence and truth whose models interpret each n-ary predicate by its extension (positive information), anti-extension (negative information), and o-extension (reliable information). The deductive system is sound and complete with respect to these six-valued models, and the logic satisfies the replacement property while admitting conjunctive, disjunctive, and prenex normal forms.

What carries the argument

o-extensions, the sets of n-tuples satisfying the operator oP for each predicate, which isolate reliable information in the six-valued models.

Load-bearing premise

The six-valued semantics with o-extensions correctly captures the intended notions of positive, negative, and reliable information for predicates in a first-order setting.

What would settle it

A formula that is valid in every six-valued model but not derivable in the system, or derivable but invalid in some model, would disprove soundness and completeness.

read the original abstract

In this paper we present the first-order logic QLETF+, a quantified version of the logic LETF+, introduced in Coniglio and Rodrigues (Studia Logica 112:561-606, 2024). QLETF+ exhibits several properties that are not always enjoyed by logics equipped with classicality operators. We show that it satisfies the replacement property and admits conjunctive, disjunctive, and prenex normal forms. Alongside extensions and anti-extensions, as in the previously studied first-order semantics for LETs, we make use here of what we call o-extensions: given an n-ary predicate symbol P, the o-extension of P is the set of n-tuples of individuals that satisfy the predicate oP. We prove the soundness and completeness of the deductive system of QLETF+ with respect to the six-valued first-order 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 / 2 minor

Summary. The manuscript introduces QLETF+, the first-order extension of LETF+, equipped with a six-valued semantics that uses extensions for positive information, anti-extensions for negative information, and o-extensions for reliable information on predicates. It establishes that QLETF+ satisfies the replacement property and admits conjunctive, disjunctive, and prenex normal forms, and proves the soundness and completeness of its deductive system with respect to this semantics via standard canonical-model constructions.

Significance. If the proofs hold, the work supplies a quantified logic for evidence and truth that extends the authors' prior propositional results while adding syntactic normal-form properties useful for automated reasoning. The explicit semantic role of o-extensions provides a clean way to track reliable information, which strengthens the framework's applicability in epistemic and information-based logics.

minor comments (2)
  1. [Abstract] The abstract states that QLETF+ 'exhibits several properties that are not always enjoyed by logics equipped with classicality operators'; a short explicit comparison in §1 or §2 to at least one other system would make this claim easier to verify.
  2. An illustrative example showing how a predicate's extension, anti-extension, and o-extension are assigned in a small domain would clarify the six-valued semantics for readers new to the framework.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript. We are pleased that the contributions of QLETF+—including the replacement property, normal forms, and the soundness/completeness result with respect to the six-valued semantics—are recognized as extending our prior work on LETF+ in a useful way.

Circularity Check

0 steps flagged

Minor self-citation for propositional base; first-order proof self-contained

full rationale

The paper defines QLETF+ as a first-order extension of the propositional LETF+ from the authors' prior 2024 Studia Logica paper. It introduces o-extensions alongside extensions and anti-extensions in the six-valued semantics to capture the target notions of information, then proves soundness and completeness of the deductive system via standard canonical-model constructions for many-valued first-order logics. No step reduces the central metatheoretic claim to a self-definition, fitted parameter renamed as prediction, or load-bearing self-citation chain; the cited prior work supplies only the propositional fragment and does not determine the quantified result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper defines a new logic, so it inherits standard first-order axioms and adds domain-specific rules for evidence operators; no free parameters or invented entities beyond the logic's own connectives and semantics are described in the abstract.

axioms (1)
  • standard math Standard first-order logic axioms plus rules for evidence and truth operators
    Invoked implicitly as the base for QLETF+ deductive system.

pith-pipeline@v0.9.0 · 5444 in / 1094 out tokens · 24626 ms · 2026-05-10T02:58:41.635993+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

20 extracted references · 5 canonical work pages

  1. [1]

    Antunes, A

    H. Antunes, A. Rodrigues, W. Carnielli, and M. Coniglio. Valuation semantics for first-order logics of evidence and truth.Journal of Philosophical Logic, 51(5):1141– 1173, 2022. doi: 10.1007/s10992-022-09662-8. 32

    work page doi:10.1007/s10992-022-09662-8 2022

  2. [2]

    N. D. Belnap. How a computer should think. In G. Ryle, editor,Contemporary Aspects of Philosophy. Oriel Press, 1977. Reprinted inNew Essays on Belnap-Dunn Logic, Springer, 2019, pages 35-55

    1977

  3. [3]

    Cant´ u and M

    M. Cant´ u and M. Figallo. On the logic that preserves degrees of truth associated to involutive Stone algebras.Logic Journal of IGPL, 28(5):1000–1020, 2020

    2020

  4. [4]

    Carnielli and A

    W. Carnielli and A. Rodrigues. An epistemic approach to paraconsistency: a logic of evidence and truth.Synthese, 196:3789–3813, 2017. doi: 10.1007/s11229-017-1621-7. URLhttps://rdcu.be/ctJRQ

    work page doi:10.1007/s11229-017-1621-7 2017

  5. [5]

    Carnielli, J

    W. Carnielli, J. Marcos, and S. de Amo. Formal inconsistency and evolutionary databases.Logic and Logical Philosophy, 8(8):115–152, 2000

    2000

  6. [6]

    Carnielli, M.E

    W. Carnielli, M.E. Coniglio, R. Podiacki, and T. Rodrigues. On the way to a wider model theory: completeness theorems for first-order logics of formal inconsistency. The Review Of Symbolic Logic, 7(3):548–578, 2014

    2014

  7. [7]

    Carnielli, M.E

    W. Carnielli, M.E. Coniglio, and A. Rodrigues. Recovery operators, paraconsistency and duality.Logic Journal of the IGPL, 28:624–656, 2019

    2019

  8. [8]

    M. E. Coniglio and A. Rodrigues. From Belnap–Dunn four-valued logic to six-valued logics of evidence and truth.Studia Logica, 112(3):561–606, 2024. First online: 2023-08-09. Preprint:https://bit.ly/47MZ61Z

    2024

  9. [9]

    J. M. Dunn. Intuitive semantics for first-degree entailments and ‘coupled trees’. Philosophical Studies, 29:149–168, 1976

    1976

  10. [10]

    D. Fallis. What is disinformation?Library Trends, 63:401–426, 2015

    2015

  11. [11]

    J. Fetzer. Disinformation: The use of false information.Minds and Machines, 14: 231–240, 2004

    2004

  12. [12]

    Gurevich

    Y. Gurevich. Intuitionistic logic with strong negation.Studia Logica, 36:49–59, 1977. doi: 10.1007/BF02121114

    work page doi:10.1007/bf02121114 1977

  13. [13]

    Marcelino and U

    S. Marcelino and U. Rivieccio. Logics of involutive Stone algebras.Soft Computing, 26(7):3147–3160, 2022

    2022

  14. [14]

    Odintsov and H

    S. Odintsov and H. Wansing. On the methodology of paraconsistent logic. In H. An- dreas and P. Verd´ ee, editors,Logical Studies of Paraconsistent Reasoning in Science and Mathematics. Springer, 2016

    2016

  15. [15]

    Rodrigues and H

    A. Rodrigues and H. Antunes. First-order logics of evidence and truth with constant and variable domains.Logica Universalis, 2022. doi: 10.1007/s11787-022-00306-8

    work page doi:10.1007/s11787-022-00306-8 2022

  16. [16]

    Rodrigues, J

    A. Rodrigues, J. Bueno-Soler, and W. Carnielli. Measuring evidence: a probabilistic approach to an extension of Belnap-Dunn logic.Synthese, 198:5451–5480, 2020

    2020

  17. [17]

    Shramko and H

    Y. Shramko and H. Wansing. The nature of entailment: an informational approach. Synthese, 198(Suppl. 22):5241–5261, 2021. doi: 10.1007/s11229-019-02474-5

    work page doi:10.1007/s11229-019-02474-5 2021

  18. [18]

    A. S. Troelstra and D. van Dalen.Constructivism in Mathematics, volume I. North Holland, 1988. 33

    1988

  19. [19]

    Wansing.The Logic of Information Structures

    H. Wansing.The Logic of Information Structures. Springer, 1993

    1993

  20. [20]

    Wardle and H

    C. Wardle and H. Derakhshan.Information Disorder: Toward an Interdisciplinary Framework for Research and Policy. Council of Europe, 2017. URLhttps://rm. coe.int/information-disorder-report-2017/168076277c. 34

    2017