Disproof-based Logic Enhancement (DLE)

• Disproof-based Logic Enhancement (DLE) is a contradiction-driven iterative method that refines anonymized candidate answers in multi-agent systems.
• It works by appending detected contradictions back into the query and repeatedly revising the answer until no contradiction remains.
• Empirical results in the GAMA framework show that DLE significantly improves logic-task accuracy and maintains reasoning integrity after anonymization.

Searching arXiv for papers relevant to "Disproof-based Logic Enhancement" and related disproof/refutation logic. arxiv_search query="Disproof-based Logic Enhancement GAMA refutation bilateralism concept bottleneck logic enhancement" max_results=10 Disproof-based Logic Enhancement (DLE) is, in GAMA, the logic-repair module for privacy-preserving multi-agent reasoning: after privacy-preserving anonymization has removed identity-bearing details from a task, DLE repeatedly tests a candidate answer against the question, appends any detected contradiction back into the reasoning context, and iterates until no contradiction remains (Yang et al., 12 Sep 2025). In this sense, DLE is an LLM-mediated iterative contradiction-elimination procedure rather than a symbolic proof system. The acronym is ambiguous in logic research, however: in older proof-theoretic and correspondence-theoretic work, “DLE” usually means Distributive Lattice Expansions, not Disproof-based Logic Enhancement (Greco et al., 2016).

1. Definition, scope, and acronymic disambiguation

In GAMA, Disproof-based Logic Enhancement is introduced as a public-space enhancement module for anonymized tasks. GAMA divides work into a private space, which handles raw sensitive data locally, and a public space, which uses remote or high-performance LLMs only on anonymized content. Within that public space, DLE complements Domain-Rule-based Knowledge Enhancement (DRKE): DRKE addresses the knowledge side of semantic loss under anonymization, while DLE addresses the logic side by repeatedly searching for contradictions in a proposed answer and forcing iterative self-correction until no contradiction remains (Yang et al., 12 Sep 2025).

The motivating claim is that “the process of anonymization can compromise the integrity of task logic.” Replacing names, organizations, or locations with placeholders protects privacy, but can also erase cues needed for reasoning, especially in logic-heavy tasks. DLE is therefore designed not to recover hidden identities, but to preserve solvability by detecting logical inconsistencies between the anonymized question and a candidate answer, revising the answer, reducing hallucinations, and narrowing down the scope of the correct answer (Yang et al., 12 Sep 2025).

This usage should be distinguished from established logical terminology. In the algebraic logic literature, DLE denotes Distributive Lattice Expansions, and in display-calculus work D.LE denotes a display calculus for LE-logics. One source states explicitly: “In this paper, DLE = Distributive Lattice Expansions. It is not used to mean ‘Disproof-based Logic Enhancement’” (Greco et al., 2016). A related line of work on inverse correspondence likewise uses “DLE-logics” to mean normal distributive lattice expansion logics (Conradie et al., 2022). The two senses are unrelated except for the shared acronym.

2. Procedural core: contradiction-driven iterative revision

DLE is defined procedurally as a two-agent iterative reasoning scheme based on disproof. The paper defines disproof in the classical sense: disproof begins by assuming a proposition is true and then finding a contradiction that shows the assumption is false. It then broadens the notion to include outcomes that are “untrue or manifestly absurd and implausible.” Operationally, DLE instantiates this idea through an expert agent that proposes an answer and an assistant agent that analyzes the question-answer pair and identifies contradictions (Yang et al., 12 Sep 2025).

The paper gives the algorithm explicitly:

$\boldsymbol{c}=\emptyset$1

The associated formal updates are

$$\boldsymbol{y}=Expert(\boldsymbol{x}), \qquad \boldsymbol{c}=Assistant(\boldsymbol{x},\boldsymbol{y}),$$

and, when contradiction is found,

$\boldsymbol{x}=\boldsymbol{x}+\boldsymbol{c}.$

The loop stops when $\boldsymbol{c}=\emptyset$ (Yang et al., 12 Sep 2025).

Several technical features follow directly from this specification. First, DLE operates on candidate answers and their consistency with the question, not on raw private data. Second, it works over propositions expressed in natural language—question, answer, and contradiction feedback—rather than over an explicit SAT, CSP, or theorem-proving substrate. Third, its disproof object is a contradiction report produced by the assistant agent; the paper does not define contradiction through symbolic semantics, model checking, or satisfiability criteria. For that reason, DLE is best characterized as iterative contradiction-driven verification and revision, rather than as a formal refutation calculus (Yang et al., 12 Sep 2025).

3. Position within GAMA’s privacy-preserving architecture

DLE is one component of a broader privacy-preserving multi-agent system. GAMA’s anonymization pipeline is implemented by AMPP, the Anonymizing Mechanism for Privacy Preservation. AMPP identifies privacy-sensitive entities through MVPI, constructs a privacy box, anonymizes the question before it leaves the private space, and remaps placeholders back into the final answer on return. The paper gives the core equations

$$\boldsymbol{e}_{v_1}=\{e_1,e_2,\ldots,e_N\}=\mathcal{N}_{priv}(x),$$

$$\boldsymbol{e}_{v_2}=\{e_1,e_2,\ldots,e_M\}=\mathcal{A}_{priv}(x),$$

$$\boldsymbol{e}_{priv} = (\boldsymbol{e}_{v_1} \cap \boldsymbol{e}_{v_2}) \cup \mathcal{S}_{fus}((\boldsymbol{e}_{v_1} \cup \boldsymbol{e}_{v_2})- (\boldsymbol{e}_{v_2} \cap \boldsymbol{e}_{v_1})),$$

$$\mathcal{V}=Vocab(\boldsymbol{e}_{priv}), \qquad \boldsymbol{x}^{\prime}=Map(\boldsymbol{x}| \mathcal{V}), \qquad \boldsymbol{y}^{\prime}=Remap(\boldsymbol{y}|\mathcal{V}).$$

The public-space LLMs therefore do not receive the original private content; they receive only $\boldsymbol{x}'$, the anonymized task (Yang et al., 12 Sep 2025).

Within this architecture, DRKE and DLE are complementary rather than redundant.

Module	Function in GAMA	Representative formalism
AMPP	Privacy preservation by anonymization and remapping	$\boldsymbol{x}'=Map(\boldsymbol{x}|\mathcal V)$, $\boldsymbol{y}'=Remap(\boldsymbol{y}|\mathcal V)$
DRKE	Domain-rule-based knowledge enhancement	$\boldsymbol{x}=\boldsymbol{x}+\boldsymbol{c}.$0
DLE	Logic enhancement by contradiction-driven revision	$\boldsymbol{x}=\boldsymbol{x}+\boldsymbol{c}.$1

DRKE analyzes task domains, constructs domain rules, invokes domain expert agents, and fuses their outputs. DLE instead takes a candidate answer, finds contradictions, and iteratively revises until the answer is contradiction-free. The paper does not formalize a single integrated schedule such as “DRKE then DLE,” but it treats both as public-space enhancement mechanisms over anonymized tasks, with DRKE primarily compensating for domain-knowledge loss and DLE primarily compensating for logical degradation (Yang et al., 12 Sep 2025).

A central architectural consequence is that DLE is not a privacy exception or backdoor around AMPP. It operates in the public space on anonymized content only. Its contribution to privacy is therefore indirect: it helps recover utility under anonymization without exposing private entities. The privacy guarantees reported in GAMA are tied primarily to AMPP and anonymization quality rather than to DLE itself (Yang et al., 12 Sep 2025).

4. Empirical profile and task regimes

The empirical case for DLE is strongest on anonymized logic tasks. GAMA is evaluated on Trivia Creative Writing and Logic Grid Puzzle, and on two privacy-focused datasets, Knowledge Privacy Preservation and Logic Privacy Preservation. On Logic Grid Puzzle, the reported scores are Std 57.7, CoT 65.8, Self-Refine 60.0, SPP 68.3, AutoAgents 71.8, and GAMA 75.5. On Logic Privacy Preservation, the reported scores are Std-NER-PRE 64.7, Std-RPR 76.7, Std-MVPI 78.4, AutoAgents-NER-PRE 66.1, AutoAgents-RPR 80.4, AutoAgents-MVPI 81.1, and GAMA 82.0 (Yang et al., 12 Sep 2025).

Evaluation setting	Reported result	Immediate significance
Logic Grid Puzzle	GAMA 75.5 vs AutoAgents 71.8	Higher logic-task performance
Logic Privacy Preservation	GAMA 82.0 vs AutoAgents-MVPI 81.1	Better reasoning after anonymization
DLE ablation on LPP	GAMA 82.0 vs $\boldsymbol{x}=\boldsymbol{x}+\boldsymbol{c}.$2DLE 76.9	Largest direct evidence for DLE

The ablation is the most direct evidence for the module itself. The paper reports GAMA at KPP 54.8 and LPP 82.0, while removing DLE yields KPP 53.2 and LPP 76.9. Removing DLE therefore causes a drop of 1.6 on KPP and 5.1 on LPP. The paper identifies this as the strongest negative effect on the logic-preserving privacy benchmark, exactly where contradiction-based refinement should matter most (Yang et al., 12 Sep 2025).

The appendix’s case study illustrates the mechanism qualitatively. In an email-processing example, the paper states that DLE progressively narrows down the location of the Massachusetts legislature, “first General Court, second Boston and final Beacon Hill.” This example is consistent with the paper’s broader description of DLE as a procedure that eliminates inadequate or incomplete hypotheses rather than directly deriving a final answer in one pass (Yang et al., 12 Sep 2025).

5. Relation to explicit refutation, logic-enhanced learning, and countermodel methods

DLE as presented in GAMA is not a formal proof/refutation logic. A distinct line of work formalizes proof and refutation as primitive bilateral judgments,

$\boldsymbol{x}=\boldsymbol{x}+\boldsymbol{c}.$3

adds coordination rules

$\boldsymbol{x}=\boldsymbol{x}+\boldsymbol{c}.$4

and proves semantically that a formula cannot be both positively and negatively supported in the same base. That system treats refutation as constructive falsity rather than merely as negation (Barroso-Nascimento et al., 19 Oct 2025). By contrast, GAMA’s DLE uses natural-language contradiction reports generated by an assistant agent, not primitive refutation judgments. This suggests a possible formalization route for future DLE systems, but it is not the method currently implemented in GAMA.

Another neighboring line is logic-enhanced concept-based learning. LogicCBM inserts a differentiable propositional logic layer between concepts and class prediction, supports 16 binary logic operators, and includes negation, XOR, implication, and implication-violation operators such as $\boldsymbol{x}=\boldsymbol{x}+\boldsymbol{c}.$5. The same source is explicit, however, that it does not offer true disproof-oriented reasoning: it has no explicit proof/disproof decomposition, no contradiction-aware loss, no counterexample-driven training, and no formal proof/disproof semantics (Vemuri et al., 8 Dec 2025). Relative to DLE, LogicCBM supplies contradiction-relevant primitives but not a first-class notion of disproof.

Proof-theoretic work on failed derivations provides a different adjacent perspective. In algebraic proof theory for D.LE and LE-logics, if

$\boldsymbol{x}=\boldsymbol{x}+\boldsymbol{c}.$6

the paper constructs a functional frame $\boldsymbol{x}=\boldsymbol{x}+\boldsymbol{c}.$7 such that

$\boldsymbol{x}=\boldsymbol{x}+\boldsymbol{c}.$8

thereby turning non-derivability into a semantic countermodel (Greco et al., 2018). In differential dynamic logic, a usage-aware sequent calculus detects vacuous and over-specified constraints by analyzing which atoms can be generalized or removed while preserving proofs, exposing “proofs for the wrong reasons” without performing explicit disproof search (Dotzel et al., 2023). These approaches are not DLE in GAMA’s sense, but they show that contradiction, refutation, and countermodel construction can be made substantially more formal than prompt-level revision.

6. Limitations and open trajectories

The current DLE formulation is intentionally procedural, and its limitations are correspondingly clear. The paper gives no proof of convergence, no bound on the number of iterations, no formal contradiction semantics, and no guarantee that “no contradiction found” implies correctness. The stopping criterion is only that the assistant cannot identify any new contradiction. The method therefore assumes that the task can be meaningfully evaluated by contradiction detection in natural language, that the assistant agent is capable of recognizing contradictions, and that iterative prompt augmentation with contradiction feedback improves the expert’s next answer. The paper also notes a broader failure mode: if anonymization causes significant semantic loss, contradiction-based refinement may not recover the answer (Yang et al., 12 Sep 2025).

The contrast with more formal refutation frameworks clarifies what is missing. Bilateralist proof/refutation logic shows how disproof can be treated as a primitive judgment equipped with explicit incompatibility constraints and sound-and-complete base-extension semantics (Barroso-Nascimento et al., 19 Oct 2025). LogicCBM, in turn, suggests concrete architectural extensions for disproof-oriented learning, including separate support and refutation channels, contradiction penalties, learning falsifying predicates, counterexample-based training, and distinguishing necessary from sufficient rules (Vemuri et al., 8 Dec 2025). These proposals are not part of GAMA’s DLE, but they indicate how a future DLE program could move from contradiction-guided self-correction toward explicit refutation semantics.

A plausible synthesis is that Disproof-based Logic Enhancement currently names a practical contradiction-driven control loop for anonymized multi-agent reasoning, while adjacent research points toward more formal versions in which refutations are explicit objects, support and refutation are separated architecturally, and failure of derivability yields constructive semantic witnesses. On the evidence presently available, however, DLE denotes the specific GAMA mechanism: $\boldsymbol{x}=\boldsymbol{x}+\boldsymbol{c}.$9 repeated until $\boldsymbol{c}=\emptyset$0 is empty (Yang et al., 12 Sep 2025).