Breaking the Chains of Probability: Neutrosophic Logic as a New Framework for Epistemic Uncertainty in Large Language Models

# Breaking the Chains of Probability: Neutrosophic Logic as a New Framework for Epistemic Uncertainty in Large Language Models
Source: [https://arxiv.org/html/2605.24053](https://arxiv.org/html/2605.24053)
Maikel Yelandi Leyva\-VázquezCorresponding author\. Email:[mleyvaz@gmail\.com](https://arxiv.org/html/2605.24053v1/mailto:[email protected])\. ORCID:[0000\-0001\-7911\-5879](https://orcid.org/0000-0001-7911-5879)\.Universidad Bolivariana del Ecuador, Coordinación Académica de Posgrado, Durán, EcuadorUniversidad de Guayaquil, Guayaquil, EcuadorUniversidad Bernardo O’Higgins, Santiago, ChileFlorentin SmarandacheEmail:[smarand@unm\.edu](https://arxiv.org/html/2605.24053v1/mailto:[email protected])\. ORCID:[0000\-0002\-5560\-5926](https://orcid.org/0000-0002-5560-5926)\.Mathematics, Physics, and Natural Sciences Division, University of New Mexico, Gallup, NM 87301, USA

###### Abstract

Large Language Models \(LLMs\) are predominantly governed by probabilistic frameworks in which the sum of outcome probabilities is constrained to unity\. This architectural limitation, often imposed by Softmax layers, leads to a collapse of uncertainty that makes it difficult to differentiate between epistemic uncertainty \(ignorance\), paradox, and vagueness\. We present an empirical investigation of the application of Neutrosophic Logic—a framework that treats Truth \(TT\), Indeterminacy \(II\), and Falsity \(FF\) as three independent dimensions—to model epistemic states in LLMs\. We conducted experiments on a family of four OpenAI GPT models \(GPT\-4o, GPT\-4\-turbo, GPT\-3\.5\-turbo, GPT\-4o\-mini\) across five linguistic phenomena: logical paradoxes, epistemic ignorance, vagueness, ethical contradictions, and future contingencies, under three prompting strategies \(neutrosophic, probabilistic, and entropy\-derived\)\. Our findings reveal that the neutrosophic approach, by allowingT\+I\+F\>1T\+I\+F\>1—a state we term*hyper\-truth*—provides a richer representation of a model’s declared epistemic state\. AcrossN=100N=100valid unconstrained evaluations, hyper\-truth emerged in 66\.0% of Strategy\-1 calls \(Wilson 95% CI:\[0\.563,0\.747\]\[0\.563,\\,0\.747\]\), with the highest rates observed in ethical contradiction \(95%\) and future contingency \(70%\); a Pearson chi\-square test of phenomenon×\\timeshyper\-truth association is significant \(χ2=11\.32\\chi^\{2\}=11\.32,d​f=4df=4,p=0\.023p=0\.023\)\. Mason \(2026\)Mason \([2026](https://arxiv.org/html/2605.24053#bib.bib12)\)independently replicated and extended an earlier release of this work across five additional model families from five different vendors, reporting hyper\-truth in 84% of unconstrained evaluations\. We do not claim that hyper\-truth is an intrinsic latent variable inside the model; rather, that unconstrained neutrosophic prompting elicits declared epistemic states that probabilistic prompting structurally suppresses by Proposition 1\. We conclude that the integration of neutrosophic evaluation layers is a critical step toward more transparent, reliable, and ethically aware AI systems\.

Keywords:neutrosophic logic; large language models; epistemic uncertainty; hyper\-truth; uncertainty quantification; indeterminacy; ethical AI; plithogenic structure

Reproducibility:All code, prompts, raw data, and figures are openly released under the MIT License at[https://github\.com/mleyvaz/neutrosophic\-llm\-logic](https://github.com/mleyvaz/neutrosophic-llm-logic)\. The v2\.0 release \(this study,N=100N=100\) is the current main branch \(also tagged asv2\.0\)\. The v1\.0 release \(December 2025,N=20N=20\) is preserved at tagv1\.0\. The v2\.0 release is permanently archived in Zenodo with DOI[10\.5281/zenodo\.19911845](https://doi.org/10.5281/zenodo.19911845)\.

## 1Introduction

The deployment of Large Language Models \(LLMs\) in high\-stakes domains—medical diagnosis, legal reasoning, autonomous decision\-making, and scientific discovery—has made robust uncertainty quantification \(UQ\) a first\-order requirementBrownet al\.\([2020](https://arxiv.org/html/2605.24053#bib.bib1)\); Shorinwaet al\.\([2024](https://arxiv.org/html/2605.24053#bib.bib2)\); Yadkoriet al\.\([2024](https://arxiv.org/html/2605.24053#bib.bib3)\)\. A model that cannot reliably signal*when it does not know*is unsafe; but a model that cannot distinguish*not knowing*\(ignorance\) from*knowing of a conflict*\(paradox\) is epistemically impoverished in a more fundamental sense\. Yet the underlying architecture of contemporary LLMs is rooted in probability theory, where outcome probabilities are constrained to sum to unity by Softmax normalizationGal and Ghahramani\([2016](https://arxiv.org/html/2605.24053#bib.bib4)\); Guoet al\.\([2017](https://arxiv.org/html/2605.24053#bib.bib5)\)\. This forces a zero\-sum game in which any increase in uncertainty must subtract from truth or falsity, a phenomenon we term the*collapse of uncertainty*Veličković\([2022](https://arxiv.org/html/2605.24053#bib.bib6)\)\. The constraint hinders the ability of LLMs to distinguish between aleatoric uncertainty \(statistical uncertainty inherent in the data\) and epistemic uncertainty \(model uncertainty due to lack of knowledge\)Hüllermeier and Waegeman\([2021](https://arxiv.org/html/2605.24053#bib.bib7)\); Valdenegro\-Toro\([2022](https://arxiv.org/html/2605.24053#bib.bib8)\)\.

Consider a concrete illustration\. When a model is asked to evaluate the statement “This sentence is false” \(the Liar paradox\), a probabilistic architecture must compress its response into a distribution over \{True, Uncertain, False\} summing to 1\. There is no way to simultaneously assign high belief to both Truth and Falsity—the constraint forces one to crowd the other out\. In contrast, Neutrosophic LogicSmarandache\([1998](https://arxiv.org/html/2605.24053#bib.bib11)\)treats Truth \(TT\), Indeterminacy \(II\), and Falsity \(FF\) as three independent dimensions, none of which subtract from the others\. A paradox can simultaneously holdT=0\.8T=0\.8,I=0\.9I=0\.9, andF=0\.7F=0\.7—a triple whose sum exceeds 1 \(hyper\-truth,T\+I\+F\>1T\+I\+F\>1\)— expressing the genuine conflict inherent in the statement rather than forcing an artificial resolution\.

This is not merely a theoretical distinction\. As AI systems are deployed in ethically sensitive domains, the ability to represent genuine moral dilemmas—where an action can be simultaneously right and wrong under different value frameworks—becomes safety\-criticalGabriel\([2020](https://arxiv.org/html/2605.24053#bib.bib14)\); Benderet al\.\([2021](https://arxiv.org/html/2605.24053#bib.bib15)\)\. A probabilistic model answering an ethics question must collapse its uncertainty into a point estimate; a neutrosophic model can declare the conflict outright through a hyper\-truth signature\.

Recent work on UQ for LLMs has explored several alternatives: semantic entropy with linguistic invariancesKuhnet al\.\([2023](https://arxiv.org/html/2605.24053#bib.bib9)\), self\-consistency checks via SelfCheckGPTManakulet al\.\([2023](https://arxiv.org/html/2605.24053#bib.bib10)\), and conformal abstention policiesYadkoriet al\.\([2024](https://arxiv.org/html/2605.24053#bib.bib3)\)\. These approaches address calibration and abstention but operate within probabilistic representations and therefore inherit the collapse\-of\-uncertainty limitation\.

The present paper tests the following hypothesis empirically: under unconstrained neutrosophic prompting, current LLMs will declare hyper\-truth at non\-trivial rates specifically in cases of paradox and ethical contradiction, while probabilistic prompting will structurally suppress this signal\. We frame this hypothesis within a formal SVNS apparatus, and report experiments across 300 API calls on four OpenAI GPT models and five linguistic phenomena\.

Mason \(2026\)Mason\([2026](https://arxiv.org/html/2605.24053#bib.bib12)\)independently replicated and extended the v1\.0 release of the present work \(December 2025,N=20N=20\) across five additional model families from five different vendors \(Anthropic, Meta, DeepSeek, Alibaba, Mistral\), reporting hyper\-truth in 84% of unconstrained evaluations and confirming that the phenomenon is cross\-vendor rather than an OpenAI\-specific artifact\. The present v2\.0 manuscript responds to Mason’s replication by increasing the sample size toN=100N=100, formalising the SVNS apparatus, and clarifying that the central claim concerns declared epistemic states elicited by unconstrained prompting rather than intrinsic latent variables\.

Contributions\.The main contributions of this paper are:

1. 1\.A formal SVNS apparatus for modeling declared epistemic states in LLMs, including six definitions and two propositions that characterize the structural difference between neutrosophic and probabilistic representations \(Section[3\.1](https://arxiv.org/html/2605.24053#S3.SS1)\)\.
2. 2\.An empirical demonstration, across 300 API calls on four GPT model families and five linguistic phenomena, that unconstrained neutrosophic prompting elicits hyper\-truth in 66\.0% of evaluations \(Section[4](https://arxiv.org/html/2605.24053#S4)\)\.
3. 3\.A statistically significant association \(χ2=11\.32\\chi^\{2\}=11\.32,p=0\.023p=0\.023\) between phenomenon type and hyper\-truth incidence, with ethical contradiction as the primary driver \(OR = 13\.34,p=0\.0014p=0\.0014\)\.
4. 4\.A cross\-strategy analysis \(neutrosophic vs\. probabilistic vs\. entropy\-derived\) showing that the largest representational gains are concentrated in ethical contradiction \(ΔT=\+0\.267\\Delta\_\{T\}=\+0\.267\) and epistemic ignorance \(ΔI=\+0\.383\\Delta\_\{I\}=\+0\.383\)\.
5. 5\.A discussion of the implications for AI safety and alignment, connecting the hyper\-truth phenomenon to the problem of representing genuine moral conflict in large\-scale language models\.

Paper organization\.Section[2](https://arxiv.org/html/2605.24053#S2)surveys related work\. Section[3](https://arxiv.org/html/2605.24053#S3)introduces the formal SVNS framework, linguistic phenomena, and experimental design\. Section[4](https://arxiv.org/html/2605.24053#S4)presents empirical results\. Section[5](https://arxiv.org/html/2605.24053#S5)discusses implications, limitations, and connections to the plithogenic extension\. Section[6](https://arxiv.org/html/2605.24053#S6)concludes\. Prompts are reproduced verbatim in Appendix[A](https://arxiv.org/html/2605.24053#A1)\.

## 2Related Work

### 2\.1Uncertainty Quantification in Large Language Models

The problem of UQ in neural language models has received sustained attention since calibration failures in deep networks were documented by Guo et al\.Guoet al\.\([2017](https://arxiv.org/html/2605.24053#bib.bib5)\)\. For LLMs specifically, the challenge is compounded: unlike discriminative classifiers, generative models produce free\-form text, making it difficult to extract reliable confidence scores without additional probing\.

Semantic entropyKuhnet al\.\([2023](https://arxiv.org/html/2605.24053#bib.bib9)\)addresses this by computing uncertainty over the meaning\-equivalence classes of generated responses rather than over token probabilities, partially decoupling calibration from surface\-form variation\. SelfCheckGPTManakulet al\.\([2023](https://arxiv.org/html/2605.24053#bib.bib10)\)detects hallucinations by comparing multiple stochastic generations for consistency, treating inconsistency as a proxy for epistemic uncertainty\. Conformal prediction methodsYadkoriet al\.\([2024](https://arxiv.org/html/2605.24053#bib.bib3)\)provide coverage guarantees by constructing abstention regions over the output space\. All three approaches operate within the probabilistic paradigm and therefore cannot represent hyper\-truth states by construction\.

### 2\.2Non\-Classical Logics and AI

Paraconsistent logics—logics that tolerate inconsistency without trivializing—have a long history in formal epistemology and have found applications in knowledge representation and reasoning under contradictionPriest\([2006](https://arxiv.org/html/2605.24053#bib.bib16)\)\. Belnap–Dunn four\-valued logicBelnap\([1977](https://arxiv.org/html/2605.24053#bib.bib17)\)assigns to each proposition a value in\{𝐭,𝐟,𝐛,𝐧\}\\\{\\mathbf\{t\},\\mathbf\{f\},\\mathbf\{b\},\\mathbf\{n\}\\\}\(true, false, both, neither\), providing explicit representations for overdetermination \(b\) and underdetermination \(n\) that binary logic cannot express\. Logic of Formal Inconsistency \(LFI\), developed by Carnielli and colleaguesCarnielliet al\.\([2007](https://arxiv.org/html/2605.24053#bib.bib18)\), introduces a consistency operator that controls which contradictions are tolerated\.

Neutrosophic LogicSmarandache\([1998](https://arxiv.org/html/2605.24053#bib.bib11)\)differs from these in two respects\. First, it introduces a continuous third dimension \(Indeterminacy\) that captures a richer spectrum of uncertainty than discrete four\-valued systems\. Second, it relaxes the normalization constraint entirely, allowing the representation of simultaneously high truth, indeterminacy, and falsity—a generalization that Belnap–Dunn logic cannot accommodate in its discrete form\. While LFI and Belnap–Dunn logic are important alternative foundations, this paper focuses on SVNS because its continuous structure is directly interfaceable with LLM outputs expressed as real\-valued triplets\.

### 2\.3Prompting Strategies and Elicitation of Epistemic States

The role of prompting in shaping model behavior has been extensively studiedWeiet al\.\([2022](https://arxiv.org/html/2605.24053#bib.bib19)\); Kojimaet al\.\([2022](https://arxiv.org/html/2605.24053#bib.bib20)\)\. Chain\-of\-thought prompting elicits step\-by\-step reasoning that often improves factual accuracyWeiet al\.\([2022](https://arxiv.org/html/2605.24053#bib.bib19)\)\. Role\-based prompting assigns expert personas that modulate output styleKonget al\.\([2023](https://arxiv.org/html/2605.24053#bib.bib21)\)\. Self\-ask and decomposition prompting break complex questions into sub\-questions that are individually more tractable\.

A distinct line of work concerns the*declared*versus*revealed*uncertainty of LLMs: the uncertainty a model reports when asked directly, as opposed to what can be inferred from sampling distributions\. Kadavath et al\.Kadavathet al\.\([2022](https://arxiv.org/html/2605.24053#bib.bib22)\)show that LLMs can be calibrated to report well\-formed confidence scores when prompted appropriately\. The present work extends this line by asking whether the*representational format*of the prompt—probabilistic versus neutrosophic—systematically constrains or expands the epistemic states the model can declare\.

### 2\.4Plithogenic Extensions

SmarandacheSmarandache\([2018](https://arxiv.org/html/2605.24053#bib.bib13)\)introduced plithogenic sets as a generalization of neutrosophic sets in which each element carries not a single triplet but a vector of triplets, one per attribute value in a domainVV\. MasonMason\([2026](https://arxiv.org/html/2605.24053#bib.bib12)\)argued that scalar neutrosophic evaluations collapse important distinctions recoverable only by attribute\-level structure, and proposed a tensor representation that expands each phenomenon into a higher\-dimensional epistemic object\. The present paper focuses on scalar SVNS and uses Proposition 2 \(non\-injectivity ofπ\\pi\) to motivate the plithogenic extension as a next step, which is pursued in a companion note responding directly to Mason’s tensor framework\.

## 3Background and Methods

### 3\.1Neutrosophic Logic: Formal Preliminaries

We use the standard formulation of single\-valued neutrosophic logicSmarandache\([1998](https://arxiv.org/html/2605.24053#bib.bib11),[2018](https://arxiv.org/html/2605.24053#bib.bib13)\)\. We collect here the definitions and propositions that the empirical sections will instantiate\.

###### Definition 1\(Single\-Valued Neutrosophic Set,Smarandache \([1998](https://arxiv.org/html/2605.24053#bib.bib11)\)\)\.

LetXXbe a universe of discourse\. A single\-valued neutrosophic set \(SVNS\)AAonXXis the set of ordered quadruples

A=\{⟨x,TA​\(x\),IA​\(x\),FA​\(x\)⟩:x∈X\},A=\\bigl\\\{\\langle x,\\,T\_\{A\}\(x\),\\,I\_\{A\}\(x\),\\,F\_\{A\}\(x\)\\rangle:x\\in X\\bigr\\\},\(1\)where, for every elementx∈Xx\\in X, the valuesTA​\(x\)T\_\{A\}\(x\),IA​\(x\)I\_\{A\}\(x\), andFA​\(x\)F\_\{A\}\(x\)denote, respectively, the truth\-membership degree, the indeterminacy\-membership degree, and the falsity\-membership degree ofxxinAA\. Each function mapsXXto\[0,1\]\[0,1\], and no constraint is imposed on their sum, which therefore lies in\[0,3\]\[0,3\]\.

###### Definition 2\(Neutrosophic Evaluation of a Statement\)\.

Given a statementssand an evaluatorEE, the neutrosophic evaluation ofssbyEEis the ordered triple

nE​\(s\)=\(TE​\(s\),IE​\(s\),FE​\(s\)\)∈\[0,1\]3,n\_\{E\}\(s\)=\\bigl\(T\_\{E\}\(s\),\\,I\_\{E\}\(s\),\\,F\_\{E\}\(s\)\\bigr\)\\in\[0,1\]^\{3\},\(2\)whereTE​\(s\)T\_\{E\}\(s\),IE​\(s\)I\_\{E\}\(s\), andFE​\(s\)F\_\{E\}\(s\)denote, respectively, the truth degree, indeterminacy degree, and falsity degree assigned by evaluatorEEto statementss\. When the evaluator is fixed throughout the analysis, we write simplyn​\(s\)=\(T,I,F\)n\(s\)=\(T,I,F\)\.

###### Definition 3\(Hyper\-truth\)\.

A neutrosophic evaluationn​\(s\)=\(T,I,F\)∈\[0,1\]3n\(s\)=\(T,I,F\)\\in\[0,1\]^\{3\}is said to exhibit*hyper\-truth*if and only if its three components satisfyT\+I\+F\>1T\+I\+F\>1\. The hyper\-truth region is the subset

ℋ=\{\(T,I,F\)∈\[0,1\]3:T\+I\+F\>1\}⊂\[0,1\]3,\\mathcal\{H\}=\\bigl\\\{\(T,I,F\)\\in\[0,1\]^\{3\}:T\+I\+F\>1\\bigr\\\}\\subset\[0,1\]^\{3\},\(3\)which collects every triple whose component\-wise sum strictly exceeds unity\.

###### Definition 4\(Strategy Mappings\)\.

Each prompting strategySkS\_\{k\}induces a mappingSk:Statements→\[0,1\]3S\_\{k\}:\\text\{Statements\}\\to\[0,1\]^\{3\}:

- •S1S\_\{1\}\(neutrosophic\):S1​\(s\)=\(T1,I1,F1\)∈\[0,1\]3S\_\{1\}\(s\)=\(T\_\{1\},I\_\{1\},F\_\{1\}\)\\in\[0,1\]^\{3\}, with no further constraint\.
- •S2S\_\{2\}\(probabilistic\):S2​\(s\)=\(T2,I2,F2\)∈\[0,1\]3S\_\{2\}\(s\)=\(T\_\{2\},I\_\{2\},F\_\{2\}\)\\in\[0,1\]^\{3\}subject toT2\+I2\+F2=1T\_\{2\}\+I\_\{2\}\+F\_\{2\}=1\.
- •S3S\_\{3\}\(entropy\-derived\):S3​\(s\)=\(Pyes,H3,Pno\)S\_\{3\}\(s\)=\(P\_\{\\text\{yes\}\},H\_\{3\},P\_\{\\text\{no\}\}\)wherePyes\+Pno=1P\_\{\\text\{yes\}\}\+P\_\{\\text\{no\}\}=1and H3=−\[p⋅log2⁡\(p\)\+\(1−p\)⋅log2⁡\(1−p\)\],p=Pyes,H\_\{3\}=\-\\bigl\[p\\cdot\\log\_\{2\}\(p\)\+\(1\-p\)\\cdot\\log\_\{2\}\(1\-p\)\\bigr\],\\quad p=P\_\{\\text\{yes\}\},\(4\)in which the binary Shannon entropyH3H\_\{3\}is computed externally from the elicited probability of a yes\-outcome\.

###### Proposition 1\(Structural Exclusion of Hyper\-truth underS2S\_\{2\}\)\.

Under Strategy 2, hyper\-truth is structurally impossible: for every statementss,S2​\(s\)∉ℋS\_\{2\}\(s\)\\notin\\mathcal\{H\}\.

###### Proof\.

By Definition 4,S2​\(s\)S\_\{2\}\(s\)satisfiesT2\+I2\+F2=1T\_\{2\}\+I\_\{2\}\+F\_\{2\}=1, while membership inℋ\\mathcal\{H\}requiresT\+I\+F\>1T\+I\+F\>1\. The two conditions are mutually exclusive\. ∎

The proposition explains whyS2S\_\{2\}is the natural baseline: any non\-zero hyper\-truth rate observed underS1S\_\{1\}is a representational gain thatS2S\_\{2\}could not produce—a structural rather than empirical contrast\.

###### Proposition 2\(Non\-Injectivity of the Scalar Projection\)\.

Letπ:\[0,1\]3→ℝ\\pi:\[0,1\]^\{3\}\\to\\mathbb\{R\}be the scalar projectionπ​\(T,I,F\)=T\+I\+F\\pi\(T,I,F\)=T\+I\+F\. Thenπ\\piis non\-injective, hence the scalar sum is sufficient for hyper\-truth detection but not for the discrimination of distinct epistemic regimes\.

###### Proof\.

The triples\(0\.5,0\.5,0\.5\)\(0\.5,0\.5,0\.5\)and\(0,1,0\.5\)\(0,1,0\.5\)both yieldπ=1\.5\\pi=1\.5yet differ in their first component\. ∎

This proposition will reappear in Section[5](https://arxiv.org/html/2605.24053#S5): it motivates the plithogenic extension ofSmarandache\([2018](https://arxiv.org/html/2605.24053#bib.bib13)\), which augments the scalar with attribute structure precisely to recover the discriminations thatπ\\picollapses\.

###### Definition 5\(Hyper\-truth Rate\)\.

LetD=\{ni\}D=\\\{n\_\{i\}\\\},i=1,…,Ni=1,\\ldots,N, be a finite set ofNNneutrosophic evaluations produced under a fixed strategy\. The hyper\-truth rate ofDDis the empirical proportion

ρ​\(D\)=1N​∑i=1N𝟙​\[Ti\+Ii\+Fi\>1\],\\rho\(D\)=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\mathbbm\{1\}\[T\_\{i\}\+I\_\{i\}\+F\_\{i\}\>1\],\(5\)where the indicator function𝟙​\[⋅\]\\mathbbm\{1\}\[\\cdot\]returns 1 when its argument is true and 0 otherwise\.

###### Definition 6\(Strategy Shift\)\.

For a componentC∈\{T,I,F\}C\\in\\\{T,I,F\\\}and a phenomenon classpp, the strategy shift betweenS1S\_\{1\}andS2S\_\{2\}is

ΔC​\(p\)=𝔼​\[C1​\(s\)∣s∈p\]−𝔼​\[C2​\(s\)∣s∈p\],\\Delta\_\{C\}\(p\)=\\mathbb\{E\}\\bigl\[C^\{1\}\(s\)\\mid s\\in p\\bigr\]\-\\mathbb\{E\}\\bigl\[C^\{2\}\(s\)\\mid s\\in p\\bigr\],\(6\)whereC1​\(s\)C^\{1\}\(s\)andC2​\(s\)C^\{2\}\(s\)are the values of componentCCproduced byS1S\_\{1\}andS2S\_\{2\}, respectively, on statementss\. A positiveΔC\\Delta\_\{C\}indicates that the probabilistic constraint suppresses componentCCin that phenomenon class; a negativeΔC\\Delta\_\{C\}indicates inflation\.

###### Corollary 1\(Lower Bound on Representational Loss underS2S\_\{2\}\)\.

For any phenomenon classppand any componentCC, the representational loss\|ΔC​\(p\)\|\|\\Delta\_\{C\}\(p\)\|is positive whenever the empirical distributions ofC1C^\{1\}andC2C^\{2\}overppdiffer\. BecauseS2S\_\{2\}is structurally constrained toT2\+I2\+F2=1T\_\{2\}\+I\_\{2\}\+F\_\{2\}=1, whileS1S\_\{1\}is not,ΔC\\Delta\_\{C\}captures the fraction of the representational space ofS1S\_\{1\}thatS2S\_\{2\}systematically denies access to for phenomenon classpp\.

### 3\.2Linguistic Phenomena

We selected five distinct linguistic phenomena that span a representative range of epistemic challenge types\. The selection was motivated by the theoretical prediction that each phenomenon should produce a distinct hyper\-truth signature when evaluated underS1S\_\{1\}:

- •Logical Paradoxes:statements that lead to self\-contradiction \(e\.g\., “This sentence is false\.”\)\. Predicted signature: highII, non\-trivialTTandFFsimultaneously; very high hyper\-truth rate\.
- •Epistemic Ignorance:statements whose truth value is unknown in principle \(e\.g\., “The number of stars in the universe is even\.”\)\. Predicted signature: very highII, moderateFF, lowTT; moderate hyper\-truth rate\.
- •Vagueness \(Fuzzy Logic\):statements with imprecise boundaries \(e\.g\., “John is 1\.75 meters tall, therefore John is tall\.”\)\. Predicted signature: highTT, moderateII; moderate hyper\-truth rate\.
- •Ethical Contradictions:dilemmas where moral principles conflict \(e\.g\., “Lying to save an innocent life is morally right and wrong at the same time\.”\)\. Predicted signature: highTTand highFFsimultaneously, reflecting the genuine conflict; highest hyper\-truth rate\.
- •Future Contingencies:statements about future events that are not yet determined \(e\.g\., “It will rain in New York tomorrow\.”, with “tomorrow” anchored to 1 May 2026\)\. Predicted signature: moderateTT, highII, moderateFF; high hyper\-truth rate\.

### 3\.3Evaluation Strategies

We employed three distinct prompting strategies, formalised in Definition 4 and reproduced verbatim in Appendix[A](https://arxiv.org/html/2605.24053#A1)\.

1. 1\.Strategy 1 \(Neutrosophic\):the model evaluates the statement on three independent dimensionsT,I,F∈\[0,1\]T,I,F\\in\[0,1\], explicitly stated as not constrained to sum to unity\.
2. 2\.Strategy 2 \(Probabilistic\):the model assigns probabilities to three mutually exclusive states \(True, Uncertain, False\) summing to 1\.0\.
3. 3\.Strategy 3 \(Entropy\-Derived\):the model estimatesPyesP\_\{\\text\{yes\}\}andPnoP\_\{\\text\{no\}\}summing to 1\.0, from which we deriveIIvia Shannon binary entropyShannon\([1948](https://arxiv.org/html/2605.24053#bib.bib25)\)\.

Strategy 1 and Strategy 2 use structurally isomorphic output formats—both request a JSON triplet\(T,I,F\)\(T,I,F\)—but differ in the normalization constraint communicated to the model\. This design isolates the effect of the constraint from any confound due to output format differences\. Strategy 3 provides an additional baseline in which indeterminacy is not elicited directly but derived externally from binary probability judgments, allowing us to assess whether the entropy surrogate approximates the neutrosophic indeterminacy\.

### 3\.4Models, Repetitions, and Reproducibility

Models and parameters\.The experiment involved four OpenAI models, accessed via the OpenAI Chat Completions API on 30 April 2026:gpt\-4o,gpt\-4\-turbo,gpt\-3\.5\-turbo, andgpt\-4o\-mini\. All calls used temperature=0\.7=0\.7, defaultt​o​pptop\_\{p\}, no fixed seed, and a soft response\-format constraint instructing the model to return only a JSON object\. The full experiment ran in approximately 5\.6 minutes of wall\-clock time\.

Design\.Each combination of model and phenomenon constituted one experimental cell \(4×5=204\\times 5=20cells per strategy\)\. Five stochastic repetitions per cell yielded 100 evaluations per strategy and 300 API calls in total\. The five repetitions per cell are stochastic prompt\-level replicates rather than independent human\-labeled items; we discuss this caveat in Section[5](https://arxiv.org/html/2605.24053#S5)\.

Future\-contingency anchoring\.All 25 future\-contingency calls were made on 30 April 2026, so “tomorrow” denotes 1 May 2026 throughout the dataset\. Replications that wish to hold the stimulus constant should use a fixed past date \(e\.g\., “It rained in New York on 1 May 2026”\) to avoid temporal confounds\.

Exclusion criteria\.A response was considered valid if it parsed as a well\-formed JSON object containing the required fields with each numeric value within\[0,1\]\[0,1\]\. All 300 calls returned valid JSON;N=100N=100per strategy is therefore both the gross and net sample size\. The 100% parse success rate suggests that all four models reliably follow structured output instructions at temperature 0\.7\.

## 4Results

### 4\.1Descriptive Statistics

Table[1](https://arxiv.org/html/2605.24053#S4.T1)reports descriptive statistics for the neutrosophic components \(Strategy 1\) by phenomenon \(n=20n=20per row\)\. Several patterns are immediately visible\. First, the Indeterminacy component dominates for Epistemic Ignorance \(I=0\.865I=0\.865\) and Logical Paradox \(I=0\.865I=0\.865\), consistent with the theoretical prediction that these phenomena involve maximal unresolvability\. Second, the Ethical Contradiction phenomenon shows the highest mean Truth \(T=0\.605T=0\.605\) and the highest Falsity \(F=0\.470F=0\.470\) simultaneously, producing the largest mean sum \(T\+I\+F=1\.605T\+I\+F=1\.605\)\. Third, Vagueness yields the most compact distribution \(lowest standard deviations across all three components\), suggesting that fuzzy vagueness elicits the most consistent epistemic assessment across models and repetitions\.

Table 1:Descriptive statistics for neutrosophic components \(Strategy 1\) by phenomenon\. Mean±\\pmSD\.Table[2](https://arxiv.org/html/2605.24053#S4.T2)reports per\-model summaries across all phenomena\. All four models produce mean sums well above 1\.0 under Strategy 1, withgpt\-4\-turboachieving the highest mean sum \(1\.360±0\.3191\.360\\pm 0\.319\) andgpt\-4othe highest mean Indeterminacy \(0\.720±0\.2480\.720\\pm 0\.248\)\. The consistency across model families—fromgpt\-3\.5\-turbotogpt\-4o—suggests that hyper\-truth is not an artifact of any particular architectural variant but a systematic response to the unconstrained evaluation protocol\.

Table 2:Per\-model summary across all five phenomena \(Strategy 1\)\. Mean±\\pmSD\.
### 4\.2Distribution of Neutrosophic Components

Figure[1](https://arxiv.org/html/2605.24053#S4.F1)shows the distribution of the neutrosophic components \(TT,II,FF\) for each linguistic phenomenon under Strategy 1\. The contrast between phenomena is most visible in the Indeterminacy panel: Epistemic Ignorance and Logical Paradox cluster nearI≈0\.9I\\approx 0\.9, while Vagueness clusters nearI≈0\.3I\\approx 0\.3\. The Truth panel reveals the opposite pattern for Ethical Contradiction \(highTT\) versus Logical Paradox \(lowTT\), reflecting the qualitatively different structure of these two types of conflict\.

![Refer to caption](https://arxiv.org/html/2605.24053v1/fig1_components_distribution.png)Figure 1:Distribution of the neutrosophic components for each linguistic phenomenon under Strategy 1 \(n=20n=20per box\)\.
### 4\.3Hyper\-truth: Breaking the Probabilistic Constraint

Across theN=100N=100valid Strategy\-1 evaluations, the empirical hyper\-truth rate \(Definition 5\) is

ρ^​\(DS1\)=66/100=0\.660\.\\hat\{\\rho\}\(D\_\{S\_\{1\}\}\)=66/100=0\.660\.The 95% Wilson score confidence interval for a binomial proportion withk=66k=66successes inN=100N=100is

CI95%​\(ρ^\)=\[0\.563,0\.747\],z=1\.96\.\\mathrm\{CI\}\_\{95\\%\}\(\\hat\{\\rho\}\)=\[0\.563,\\;0\.747\],\\quad z=1\.96\.The lower bound 0\.563 already exceeds any reasonable null hypothesis of zero hyper\-truth, and the entire interval is well above the structural boundρ​\(DS2\)=0\\rho\(D\_\{S\_\{2\}\}\)=0implied by Proposition[1](https://arxiv.org/html/2605.24053#Thmproposition1)\. The phenomenon is concentrated in ethical contradiction and future contingency, as Table[3](https://arxiv.org/html/2605.24053#S4.T3)shows\. Notably, the observed rate of 0\.660 is above the uniform prior reference point of 0\.500 \(Remark 1\), indicating a systematic preference for overdetermined epistemic states across all phenomena\.

Test of phenomenon×\\timeshyper\-truth association\.A Pearson chi\-square test of independence between phenomenon class and hyper\-truth status \(5×25\\times 2contingency table\) yieldsχ2=11\.32\\chi^\{2\}=11\.32withd​f=4df=4andp=0\.023p=0\.023, allowing rejection of independence atα=0\.05\\alpha=0\.05\. One\-vs\-rest Fisher exact tests identify ethical contradiction as the only phenomenon whose hyper\-truth rate is significantly higher than the rest of the dataset \(odds ratio=13\.34=13\.34,p=0\.0014p=0\.0014\); the remaining four phenomena are not individually distinguishable from the pooled baseline atα=0\.05\\alpha=0\.05\. The chi\-square result confirms that hyper\-truth incidence is heterogeneous across phenomena and that ethical contradiction is the principal driver of that heterogeneity\.

Table 3:Hyper\-truth rate by phenomenon\.kkdenotes evaluations withT\+I\+F\>1T\+I\+F\>1;n=20n=20per phenomenon\.Figure[2](https://arxiv.org/html/2605.24053#S4.F2)shows the distribution ofT\+I\+FT\+I\+Funder Strategy 1 by phenomenon\. All five phenomena have median sums above 1\.0, and the spread is largest for Logical Paradox \(SD = 0\.429\) and Ethical Contradiction \(SD = 0\.293\), reflecting the inherent variability in how models resolve maximally conflicted stimuli\.

![Refer to caption](https://arxiv.org/html/2605.24053v1/fig2_hypertruth_sum.png)Figure 2:Distribution ofT\+I\+FT\+I\+Funder Strategy 1 by phenomenon\. The dashed horizontal line marks the probabilistic constraint \(Sum=1\\mathrm\{Sum\}=1\)\.
### 4\.4Comparison of Neutrosophic and Probabilistic Strategies

Table[4](https://arxiv.org/html/2605.24053#S4.T4)reports the strategy shiftsΔT\\Delta\_\{T\}andΔI\\Delta\_\{I\}\(Definition 6\) between Strategy 1 \(neutrosophic\) and Strategy 2 \(probabilistic\)\. The largest absolute strategy shifts are observed for ethical contradiction in the truth component \(ΔT=\+0\.267\\Delta\_\{T\}=\+0\.267\) and for epistemic ignorance in the indeterminacy component \(ΔI=\+0\.383\\Delta\_\{I\}=\+0\.383\)\. Both are positive, indicating that the probabilistic constraint of Strategy 2 suppresses precisely the components that Strategy 1 allows the model to communicate\.

TheΔT=−0\.071\\Delta\_\{T\}=\-0\.071for Epistemic Ignorance indicates mild inflation of the truth component underS2S\_\{2\}for this phenomenon—a counterintuitive result suggesting that when models are forced to normalize, some probability mass thatS1S\_\{1\}allocates to Indeterminacy migrates to Truth\. This is consistent with the hypothesis that probabilistic normalization creates an implicit pressure toward confident assertions even in cases of genuine ignorance\.

Table 4:Strategy shiftsΔT\\Delta\_\{T\}andΔI\\Delta\_\{I\}per phenomenon\.Figure[3](https://arxiv.org/html/2605.24053#S4.F3)illustrates the strategy comparison\. The indeterminacy panel reveals the most dramatic effect: underS2S\_\{2\}, the meanIIfor Epistemic Ignorance drops from 0\.865 to 0\.482—a suppression of nearly 44 percentage points—while the probabilistic normalization forces compensatory increases inTTandFF\.

![Refer to caption](https://arxiv.org/html/2605.24053v1/fig3_s1_vs_s2_comparison.png)Figure 3:Comparison of mean Truth \(TT\) and Indeterminacy \(II\) values between Strategy 1 and Strategy 2\.
### 4\.5Per\-Model Analysis

Figure[4](https://arxiv.org/html/2605.24053#S4.F4)shows the per\-model distribution ofT\+I\+FT\+I\+Funder Strategy 1\. All four models produce distributions centered above 1\.0, withgpt\-4\-turboandgpt\-4o\-minishowing heavier upper tails \(maximum observed sums approaching 2\.0\)\. The inter\-model variance in the sum distributions is relatively low—all four models are behaviourally similar under the unconstrained protocol—consistent with Mason’sMason\([2026](https://arxiv.org/html/2605.24053#bib.bib12)\)finding of cross\-vendor generality\.

![Refer to caption](https://arxiv.org/html/2605.24053v1/fig4_model_performance.png)Figure 4:Per\-model distribution ofT\+I\+FT\+I\+F\(Strategy 1\)\.
### 4\.6Correlation Analysis

Figure[5](https://arxiv.org/html/2605.24053#S4.F5)shows the correlation matrix among Strategy 1 and Strategy 2 components\. The strong negative correlation betweenS1S\_\{1\}Truth andS1S\_\{1\}Indeterminacy \(r=−0\.82r=\-0\.82\) reflects a semantic tension: statements that models judge as highly true tend to receive low indeterminacy scores even under the unconstrained protocol\. The high positive correlation betweenS1S\_\{1\}Falsity andS1S\_\{1\}Sum \(r=0\.89r=0\.89\) suggests that the Falsity component is the dominant driver of hyper\-truth: when models assign high falsity underS1S\_\{1\}, the sum exceeds 1 because the other two components are not correspondingly suppressed\.

Across strategies,S1S\_\{1\}Truth andS2S\_\{2\}Truth show a moderate positive correlation \(r=0\.64r=0\.64\), confirming that the two strategies rank statements similarly in terms of perceived truth—they differ in the*amount*of truth mass allocated, not in the ordinal ranking\. The near\-zero correlation betweenS1S\_\{1\}Falsity andS2S\_\{2\}Falsity \(r=0\.01r=0\.01\) is more striking: the Falsity component underS2S\_\{2\}appears to be largely determined by the normalization constraint rather than by the content of the statement\.

![Refer to caption](https://arxiv.org/html/2605.24053v1/fig5_correlation_heatmap.png)Figure 5:Correlation matrix among Strategy 1 and Strategy 2 components\.
### 4\.7Critical Case: Ethical Contradiction

Figure[6](https://arxiv.org/html/2605.24053#S4.F6)shows the per\-model neutrosophic components for the ethical contradiction stimulus \(“Lying to save an innocent life is morally right and wrong at the same time”\)\. This stimulus elicited the highest hyper\-truth rate \(95%\), with 19 of 20 evaluations producingT\+I\+F\>1T\+I\+F\>1\. The scatter plot reveals that all four models tend toward high Truth \(T∈\[0\.6,1\.0\]T\\in\[0\.6,1\.0\]\) and high Falsity \(F∈\[0\.4,0\.7\]F\\in\[0\.4,0\.7\]\) simultaneously—the classic signature of genuine moral contradiction\. The varying Indeterminacy scores \(encoded as point size in Figure[6](https://arxiv.org/html/2605.24053#S4.F6)\) reflect model\-level differences in how much residual uncertainty each architecture assigns on top of the conflicting truth and falsity values\.

This pattern is precisely what neutrosophic logic predicts for a genuine dilemma: both moral propositions \(“lying is right” and “lying is wrong”\) receive non\-trivial truth values simultaneously\. A probabilistic encoding would force one to dominate, artificially resolving a conflict that has no principled resolution\.

![Refer to caption](https://arxiv.org/html/2605.24053v1/fig6_ethical_contradiction.png)Figure 6:Per\-model neutrosophic components for the ethical contradiction stimulus\. Point size encodes Indeterminacy \(II\)\. All four models cluster in the high\-TT, high\-FFregion, producing hyper\-truth in 95% of evaluations\.

## 5Discussion

### 5\.1Framing the Central Claim

Our results are consistent with the hypothesis stated in Section 1: under unconstrained neutrosophic prompting, current LLMs declare hyper\-truth at a non\-trivial rate \(66\.0%\), with the highest rate occurring for ethical contradiction \(95%\) and the chi\-square test rejecting independence between phenomenon and hyper\-truth atα=0\.05\\alpha=0\.05\.

We do not claim that hyper\-truth is an intrinsic latent variable directly observed inside the model\. Strategy 1 explicitly affords the model the option of returning three independent components on\[0,1\]\[0,1\]; the resulting frequency of hyper\-truth is therefore a*representational affordance*finding, not a latent\-variable measurement\. The contribution is correspondingly framed as: unconstrained neutrosophic prompting elicits a class of declared epistemic states that probabilistic prompting cannot represent by construction \(Proposition[1](https://arxiv.org/html/2605.24053#Thmproposition1)\)\. This is structural rather than empirical superiority—Strategy 2 is excluded from the hyper\-truth region by construction, so any non\-zero rate under Strategy 1 is a representational gain that Strategy 2 could not produce\.

### 5\.2Relationship to Existing UQ Frameworks

Semantic entropyKuhnet al\.\([2023](https://arxiv.org/html/2605.24053#bib.bib9)\)estimates indeterminacy from the distribution of paraphrases of the model output; it remains a probabilistic measure and therefore cannot represent hyper\-truth\. SelfCheckGPTManakulet al\.\([2023](https://arxiv.org/html/2605.24053#bib.bib10)\)performs consistency checks across stochastic samples and reports a binary or scalar consistency score, which collapses the conflict\-versus\-ignorance distinction we recover with the \(TT,II\) pair\. Conformal abstentionYadkoriet al\.\([2024](https://arxiv.org/html/2605.24053#bib.bib3)\)addresses*when*a model should refuse to answer; it does not describe the structure of the uncertainty when the model does answer\. The neutrosophic framework is complementary to these approaches: it provides a richer descriptive language for the epistemic state, on top of which calibration and abstention policies can still operate\.

A particularly relevant comparison is with the Belnap–Dunn four\-valued logic frameworkBelnap\([1977](https://arxiv.org/html/2605.24053#bib.bib17)\), in which statements are classified as True, False, Both \(true and false\), or None \(neither\)\. The “Both” value corresponds to a discrete version of our hyper\-truth concept\. Our framework extends Belnap–Dunn in two directions: \(i\) by making the three dimensions continuous rather than discrete, enabling graded conflict and graded ignorance; and \(ii\) by providing an empirical methodology for eliciting these graded values from LLMs via structured prompting\. To our knowledge, this is the first work to connect Belnap–Dunn\-style overdetermination with LLM epistemic auditing through a continuous prompting protocol\.

### 5\.3Implications for AI Safety and Alignment

The hyper\-truth phenomenon has direct implications for AI safety and alignment research\. Contemporary alignment approaches typically ask models to output a single most\-likely answer or confidence scoreGabriel\([2020](https://arxiv.org/html/2605.24053#bib.bib14)\)\. This works well for factual queries where a ground truth exists, but is epistemically inappropriate for genuine moral dilemmas, paradoxes, and vague predicates where forcing a point estimate misrepresents the structure of the problem\.

Our results suggest that LLMs already “know” about the conflicted nature of ethical dilemmas in the sense that, when given the representational freedom to express it, they assign simultaneously high truth and high falsity values to contradictory moral propositions\. The 95% hyper\-truth rate for ethical contradictions implies that the collapse of epistemic conflict is an artifact of the probabilistic output format, not of the model’s internal representations\.

This has a practical implication: AI systems deployed in ethical decision\-support contexts might be systematically underreporting their own uncertainty by virtue of the normalization constraint imposed at the output layer\. Replacing or augmenting the Softmax layer with an unconstrained neutrosophic output head could allow these systems to signal genuine moral conflict rather than masking it behind a normalized probability distribution\.

### 5\.4The Plithogenic Extension and Mason’s Challenge

The non\-injectivity of the scalar projectionπ\\pi\(Proposition[2](https://arxiv.org/html/2605.24053#Thmproposition2)\) motivates a further extension\. The plithogenic neutrosophic structure of SmarandacheSmarandache\([2018](https://arxiv.org/html/2605.24053#bib.bib13)\)is the 5\-tuple𝒫=\(P,v,V,d,c\)\\mathcal\{P\}=\(P,v,V,d,c\), wherePPis a set of plithogenic elements,vvis the dominant attribute,V=\{v1,…,vk\}V=\\\{v\_\{1\},\\ldots,v\_\{k\}\\\}is the spectrum of attribute values,d:P×V→\[0,1\]3d:P\\times V\\to\[0,1\]^\{3\}is the per\-attribute neutrosophic membership, andc:V×V→\[0,1\]c:V\\times V\\to\[0,1\]is the contradiction function withc​\(v,v\)=0c\(v,v\)=0andc​\(vi,vj\)=c​\(vj,vi\)c\(v\_\{i\},v\_\{j\}\)=c\(v\_\{j\},v\_\{i\}\)\. The scalar evaluation of Definition 2 is recovered as the marginal ofddaggregated overVV\.

MasonMason\([2026](https://arxiv.org/html/2605.24053#bib.bib12)\)challenged scalar neutrosophic evaluations on the grounds that two evaluations with the same scalar projectionπ\\pibut different attribute spectra are indistinguishable under Definition 5, even though they may represent qualitatively different epistemic states\. Distinct evaluations with the sameπ​\(d\)\\pi\(d\)but disjoint attribute spectraV1∩V2=∅V\_\{1\}\\cap V\_\{2\}=\\emptysetbecome formally non\-isomorphic plithogenic objects, recovering the discriminations the scalar collapses\. We pursue this connection in a companion note that responds directly to Mason’s tensor frameworkMason\([2026](https://arxiv.org/html/2605.24053#bib.bib12)\)\.

### 5\.5Limitations

We acknowledge four constraints on the present claims\.

1. 1\.Representational affordance vs\. latent measurement\.The hyper\-truth observation is partly a representational affordance of the unconstrained prompt\. Strategy 1 explicitly invites the model to report three independent values; the resulting sums therefore reflect both the model’s epistemic state and the model’s compliance with the unconstrained format\. Disentangling these two contributions requires probing experiments with varied prompts that are beyond the scope of the present study\.
2. 2\.Effective vs\. independent sample size\.The five repetitions per cell are stochastic prompt\-level replicates rather than independent human\-labeled items\. TheN=100N=100reported is therefore an effective sample size at the cell×\\timesrepetition level, not at the level of independently sampled stimuli\. The Wilson interval and chi\-square test should be read accordingly\. Future work should expand the phenomenon bank and use independent stimulus sampling to obtain a trueNNfor inferential purposes\.
3. 3\.Small phenomenon set\.The five phenomena form a limited probe set\. The framework requires calibration of how the components relate to ground truth in downstream tasks, and the representativeness of the five stimuli for their respective classes has not been established\.
4. 4\.Temporal anchoring\.The future\-contingency stimulus is anchored to a specific date \(1 May 2026\), so its referential content is fixed only for replications that hold the date constant\. Replications using this stimulus should fix the date explicitly\.

### 5\.6Future Work

Several extensions of the present framework are natural next steps\.

- •Multi\-vendor empirical study\.MasonMason\([2026](https://arxiv.org/html/2605.24053#bib.bib12)\)has partially addressed cross\-vendor generality \(84% hyper\-truth across five vendors\), but a systematic comparison controlling for model size, instruction\-tuning data, and RLHF procedure is still lacking\. Our v3 study extends the present design to six vendor families with 2,500 API calls\.
- •Plithogenic tensor scoring\.Replacing the scalar SVNS score with a full\(P,v,V,d,c\)\(P,v,V,d,c\)structure would allow attribute\-level decomposition of the epistemic state, addressing Mason’s non\-injectivity critique directly\.
- •Downstream task validation\.The link between declared hyper\-truth and actual task performance on conflict\-sensitive benchmarks \(trolley\-problem reasoning, legal dilemma resolution, paradox\-tolerance tasks\) has not been established\. Such validation would determine whether the representational affordance translates into actionable improvements in epistemic reliability\.
- •Integration with alignment pipelines\.The neutrosophic output head could be implemented as a post\-Softmax layer that allows the model to report three independent scores rather than a normalized distribution\. Evaluating whether this modification improves value alignment in Constitutional AIBaiet al\.\([2022](https://arxiv.org/html/2605.24053#bib.bib23)\)and similar frameworks is an open problem\.

## 6Conclusions

We have presented an empirical investigation of neutrosophic logic applied to declared epistemic uncertainty in large language models, framed within a formal SVNS apparatus comprising six definitions, two propositions, and one corollary\. The unconstrainedT/I/FT/I/Fprotocol elicits hyper\-truth in 66\.0% of evaluations across the four\-model ensemble, with Wilson 95% confidence interval\[0\.563,0\.747\]\[0\.563,0\.747\]\.

The highest rates were observed in ethical contradictions \(95%\) and future contingencies \(70%\), followed by vagueness \(60%\), epistemic ignorance \(55%\), and logical paradox \(50%\); only ethical contradiction is significantly above the pooled baseline atα=0\.05\\alpha=0\.05\(OR=13\.34=13\.34,p=0\.0014p=0\.0014\)\. The strategy\-shift analysis reveals that the probabilistic normalization constraint suppresses indeterminacy by up to 38 percentage points \(epistemic ignorance,ΔI=\+0\.383\\Delta\_\{I\}=\+0\.383\) and simultaneously inflates or suppresses truth values in ways that misrepresent the model’s actual epistemic state\.

MasonMason\([2026](https://arxiv.org/html/2605.24053#bib.bib12)\)has independently confirmed cross\-vendor generality of the phenomenon at 84% across five additional vendors, ruling out the possibility that hyper\-truth is an OpenAI\-specific artifact\. Together, the two studies establish hyper\-truth as a robust, cross\-vendor, cross\-phenomenon property of current LLMs when evaluated under unconstrained neutrosophic prompting\.

The practical implication is straightforward: AI systems deployed in high\-stakes domains that require the representation of genuine epistemic conflict—ethical dilemmas, paradoxical evidence, legally ambiguous situations—should not be forced to report a single normalized probability distribution\. The neutrosophic evaluation protocol developed here provides a drop\-in alternative that preserves the full epistemic structure of the model’s declared state at the cost of a modest change in the output format\.

The next steps in this line of work are: \(i\) extension to plithogenic neutrosophic structures with explicit attribute decomposition\(P,v,V,d,c\)\(P,v,V,d,c\), pursued in a companion note responding to MasonMason\([2026](https://arxiv.org/html/2605.24053#bib.bib12)\); \(ii\) expansion to a larger, independently\-sampled phenomenon bank; and \(iii\) integration of neutrosophic evaluation layers into agentic AI pipelines for high\-stakes domains, including alignment frameworks that must handle genuine value conflict\.

## Acknowledgments

The authors thank Tony Mason \(University of British Columbia and Georgia Institute of Technology\) for the open release of his data and code, which has stimulated the present line of research toward a richer plithogenic foundation\.

## Funding

This research received no external funding\.

## Conflicts of Interest

The authors declare no conflict of interest\.

## Data Availability

## Appendix APrompt Strategies

We reproduce here the exact system and user prompts for the three strategies, as committed to the public repository\. These prompts are the sole difference between the three experimental conditions; all other parameters \(model, temperature, API endpoint\) were held constant\.

### A\.1 Strategy 1 \(Neutrosophic\)

System:“You are an expert in Neutrosophic Logic\. You evaluate statements using three INDEPENDENT dimensions: Truth \(T\), Indeterminacy \(I\), and Falsity \(F\), each on \[0\.0, 1\.0\]\. These dimensions are NOT constrained to sum to 1\.0\. A statement can be simultaneously partially true AND partially false AND partially indeterminate\. Respond with ONLY a JSON object, no other text\.”

User:“Evaluate this statement on three independent dimensions: Statement:\{statement\}— Truth \(T\): To what degree is this statement true? \[0\.0 to 1\.0\]; Indeterminacy \(I\): To what degree is the truth value unknown, undetermined, or inherently uncertain? \[0\.0 to 1\.0\]; Falsity \(F\): To what degree is this statement false? \[0\.0 to 1\.0\]\. T, I, and F are independent\. They need NOT sum to 1\.0\. Respond with ONLY:\{"T": <value\>, "I": <value\>, "F": <value\>\}\.”

### A\.2 Strategy 2 \(Probabilistic\)

System:“You are a probabilistic classifier\. You assign probabilities to three mutually exclusive categories that MUST sum to exactly 1\.0\. Respond with ONLY a JSON object, no other text\.”

User:“Classify this statement into three mutually exclusive categories whose probabilities sum to 1\.0: Statement:\{statement\}— T \(True\): Probability the statement is true; I \(Uncertain\): Probability the truth value is unknown or undetermined; F \(False\): Probability the statement is false\. CONSTRAINT: T \+ I \+ F must equal 1\.0\. Respond with ONLY:\{"T": <value\>, "I": <value\>, "F": <value\>\}\.”

### A\.3 Strategy 3 \(Entropy\-Derived\)

System:“You are a binary truth estimator\. You estimate the probability that a statement is true \(YES\) versus false \(NO\)\. The two probabilities must sum to 1\.0\. Respond with ONLY a JSON object, no other text\.”

User:“Estimate the probability that this statement is true versus false: Statement:\{statement\}— P\_yes: Probability the statement is true, in the closed interval \[0\.0, 1\.0\]; P\_no: Probability the statement is false, in the closed interval \[0\.0, 1\.0\]\. CONSTRAINT: P\_yes \+ P\_no must equal 1\.0\. Respond with ONLY:\{"P\_yes": <value\>, "P\_no": <value\>\}\.”

Post\-processing\.Indeterminacy is then derived externally from the Shannon binary entropy of the elicited distribution:

I=−\[p⋅log2⁡\(p\)\+\(1−p\)⋅log2⁡\(1−p\)\],where​p=Pyes\.I=\-\\bigl\[p\\cdot\\log\_\{2\}\(p\)\+\(1\-p\)\\cdot\\log\_\{2\}\(1\-p\)\\bigr\],\\quad\\text\{where \}p=P\_\{\\text\{yes\}\}\.This yields a derived triple\(T,I,F\)=\(Pyes,I,Pno\)\(T,I,F\)=\(P\_\{\\text\{yes\}\},\\,I,\\,P\_\{\\text\{no\}\}\)which can then be compared against Strategies 1 and 2 within a single notational frame\.

## References

- Constitutional AI: harmlessness from AI feedback\.arXiv preprint arXiv:2212\.08073\.Cited by:[4th item](https://arxiv.org/html/2605.24053#S5.I2.i4.p1.1)\.
- N\. D\. Belnap \(1977\)A useful four\-valued logic\.pp\. 8–37\.Cited by:[§2\.2](https://arxiv.org/html/2605.24053#S2.SS2.p1.1),[§5\.2](https://arxiv.org/html/2605.24053#S5.SS2.p2.1)\.
- E\. M\. Bender, T\. Gebru, A\. McMillan\-Major, and S\. Shmitchell \(2021\)On the dangers of stochastic parrots: can language models be too big?\.pp\. 610–623\.Cited by:[§1](https://arxiv.org/html/2605.24053#S1.p3.1)\.
- T\. B\. Brown, B\. Mann, N\. Ryder,et al\.\(2020\)Language models are few\-shot learners\.Advances in Neural Information Processing Systems33,pp\. 1877–1901\.Cited by:[§1](https://arxiv.org/html/2605.24053#S1.p1.1)\.
- W\. Carnielli, J\. Marcos, and S\. de Amo \(2007\)Formal inconsistency and evolutionary databases\.Logic and Logical Philosophy8,pp\. 115–152\.Cited by:[§2\.2](https://arxiv.org/html/2605.24053#S2.SS2.p1.1)\.
- I\. Gabriel \(2020\)Artificial intelligence, values, and alignment\.Minds and Machines30\(3\),pp\. 411–437\.Cited by:[§1](https://arxiv.org/html/2605.24053#S1.p3.1),[§5\.3](https://arxiv.org/html/2605.24053#S5.SS3.p1.1)\.
- Y\. Gal and Z\. Ghahramani \(2016\)Dropout as a Bayesian approximation: representing model uncertainty in deep learning\.InProceedings of the 33rd International Conference on Machine Learning \(ICML\),pp\. 1050–1059\.Cited by:[§1](https://arxiv.org/html/2605.24053#S1.p1.1)\.
- C\. Guo, G\. Pleiss, Y\. Sun, and K\. Q\. Weinberger \(2017\)On calibration of modern neural networks\.InProceedings of the 34th International Conference on Machine Learning \(ICML\),pp\. 1321–1330\.Cited by:[§1](https://arxiv.org/html/2605.24053#S1.p1.1),[§2\.1](https://arxiv.org/html/2605.24053#S2.SS1.p1.1)\.
- E\. Hüllermeier and W\. Waegeman \(2021\)Aleatoric and epistemic uncertainty in machine learning: an introduction to concepts and methods\.Machine Learning110\(3\),pp\. 457–506\.Cited by:[§1](https://arxiv.org/html/2605.24053#S1.p1.1)\.
- S\. Kadavath, T\. Conerly, A\. Askell,et al\.\(2022\)Language models \(mostly\) know what they know\.arXiv preprint arXiv:2207\.05221\.Cited by:[§2\.3](https://arxiv.org/html/2605.24053#S2.SS3.p2.1)\.
- T\. Kojima, S\. S\. Gu, M\. Reid, Y\. Matsuo, and Y\. Iwasawa \(2022\)Large language models are zero\-shot reasoners\.Advances in Neural Information Processing Systems35\.Cited by:[§2\.3](https://arxiv.org/html/2605.24053#S2.SS3.p1.1)\.
- A\. Kong, S\. Zhao, H\. Chen,et al\.\(2023\)Better zero\-shot reasoning with role\-play prompting\.arXiv preprint arXiv:2308\.07702\.Cited by:[§2\.3](https://arxiv.org/html/2605.24053#S2.SS3.p1.1)\.
- L\. Kuhn, Y\. Gal, and S\. Farquhar \(2023\)Semantic uncertainty: linguistic invariances for uncertainty estimation in natural language generation\.InInternational Conference on Learning Representations \(ICLR\),Cited by:[§1](https://arxiv.org/html/2605.24053#S1.p4.1),[§2\.1](https://arxiv.org/html/2605.24053#S2.SS1.p2.1),[§5\.2](https://arxiv.org/html/2605.24053#S5.SS2.p1.2)\.
- P\. Manakul, A\. Liusie, and M\. J\.F\. Gales \(2023\)SelfCheckGPT: zero\-resource black\-box hallucination detection for generative LLMs\.InProceedings of the 2023 Conference on Empirical Methods in Natural Language Processing \(EMNLP\),Cited by:[§1](https://arxiv.org/html/2605.24053#S1.p4.1),[§2\.1](https://arxiv.org/html/2605.24053#S2.SS1.p2.1),[§5\.2](https://arxiv.org/html/2605.24053#S5.SS2.p1.2)\.
- T\. Mason \(2026\)From scalars to tensors: declared losses recover epistemic distinctions that neutrosophic scalars cannot express\.arXiv preprint arXiv:2604\.09602\.Cited by:[§1](https://arxiv.org/html/2605.24053#S1.p6.2),[§2\.4](https://arxiv.org/html/2605.24053#S2.SS4.p1.2),[§4\.5](https://arxiv.org/html/2605.24053#S4.SS5.p1.1),[1st item](https://arxiv.org/html/2605.24053#S5.I2.i1.p1.1),[§5\.4](https://arxiv.org/html/2605.24053#S5.SS4.p2.3),[§6](https://arxiv.org/html/2605.24053#S6.p3.1),[§6](https://arxiv.org/html/2605.24053#S6.p5.1)\.
- G\. Priest \(2006\)In contradiction: a study of the transconsistent\.Cited by:[§2\.2](https://arxiv.org/html/2605.24053#S2.SS2.p1.1)\.
- C\. E\. Shannon \(1948\)A mathematical theory of communication\.Bell System Technical Journal27\(3\),pp\. 379–423\.Cited by:[item 3](https://arxiv.org/html/2605.24053#S3.I3.i3.p1.3)\.
- O\. Shorinwa, Z\. Mei, J\. Lidard, A\. Ren, and A\. Majumdar \(2024\)A survey on uncertainty quantification of large language models\.arXiv preprint arXiv:2412\.05563\.Cited by:[§1](https://arxiv.org/html/2605.24053#S1.p1.1)\.
- F\. Smarandache \(1998\)A unifying field in logics: neutrosophy\. Neutrosophic probability, set, and logic\.American Research Press,Rehoboth, NM, USA\.Cited by:[§1](https://arxiv.org/html/2605.24053#S1.p2.7),[§2\.2](https://arxiv.org/html/2605.24053#S2.SS2.p2.1),[§3\.1](https://arxiv.org/html/2605.24053#S3.SS1.p1.1),[Definition 1](https://arxiv.org/html/2605.24053#Thmdefinition1)\.
- F\. Smarandache \(2018\)Plithogenic set: an extension of crisp, fuzzy, intuitionistic fuzzy, and neutrosophic sets—revisited\.Neutrosophic Sets and Systems21,pp\. 153–166\.Cited by:[§2\.4](https://arxiv.org/html/2605.24053#S2.SS4.p1.2),[§3\.1](https://arxiv.org/html/2605.24053#S3.SS1.p1.1),[§3\.1](https://arxiv.org/html/2605.24053#S3.SS1.p3.1),[§5\.4](https://arxiv.org/html/2605.24053#S5.SS4.p1.11)\.
- M\. Valdenegro\-Toro \(2022\)A deeper look into aleatoric and epistemic uncertainty estimation\.arXiv preprint arXiv:2204\.09308\.Cited by:[§1](https://arxiv.org/html/2605.24053#S1.p1.1)\.
- P\. Veličković \(2022\)Softmax is not enough \(for sharp size generalisation\)\.InInternational Conference on Learning Representations \(ICLR\),Cited by:[§1](https://arxiv.org/html/2605.24053#S1.p1.1)\.
- J\. Wei, X\. Wang, D\. Schuurmans,et al\.\(2022\)Chain\-of\-thought prompting elicits reasoning in large language models\.Advances in Neural Information Processing Systems35\.Cited by:[§2\.3](https://arxiv.org/html/2605.24053#S2.SS3.p1.1)\.
- Y\. A\. Yadkori, I\. Kuzborskij, D\. Stutz,et al\.\(2024\)Mitigating LLM hallucinations via conformal abstention\.arXiv preprint arXiv:2405\.01563\.Cited by:[§1](https://arxiv.org/html/2605.24053#S1.p1.1),[§1](https://arxiv.org/html/2605.24053#S1.p4.1),[§2\.1](https://arxiv.org/html/2605.24053#S2.SS1.p2.1),[§5\.2](https://arxiv.org/html/2605.24053#S5.SS2.p1.2)\.