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Breaking the Chains of Probability: Neutrosophic Logic as a New Framework for Epistemic Uncertainty in Large Language Models
Maikel Yelandi Leyva-Vázquez, Florentin Smarandache
TL;DR
Conventional probability constraints make it difficult for LLMs to distinguish ignorance, paradox, and vagueness. This paper evaluates unconstrained neutrosophic truth, indeterminacy, and falsity scores, eliciting hyper-truth in 66.0% of evaluations.
Problem
Probability normalization constrains LLMs’ uncertainty representations, limiting their ability to distinguish ignorance from paradox and other epistemic conflicts.
Method
The study evaluates independent Truth, Indeterminacy, and Falsity scores against probabilistic prompting across epistemic phenomena in four GPT models.
Results
66.0% of evaluations elicited hyper-truth, with highest rates for ethical contradictions (95%) and future contingencies (70%).
Takeaways & Limitations
The neutrosophic protocol preserves the declared structure of epistemic conflict without forcing high-stakes systems into a single normalized probability distribution.
Takeaways & Limitations
Hyper-truth reflects a representational affordance of the unconstrained prompt rather than a direct measurement of an intrinsic latent variable.
Abstract
from arXiv · showhide
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, paradox, and vagueness. We present an empirical investigation of the application of Neutrosophic Logic, a framework that treats Truth (T), Indeterminacy (I), and Falsity (F) as three independent dimensions, to model epistemic states in LLMs. We conducted experiments on a family of four OpenAI GPT models 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 allowing T+I+F > 1, a state we term hyper-truth, provides a richer representation of a model's internal state. In 35% of evaluations, hyper-truth emerged spontaneously, predominantly under ethical contradiction and logical paradox. We demonstrate that this approach preserves truth values in fuzzy contexts and offers a robust method for identifying and quantifying internal model conflict. We conclude that the integration of neutrosophic evaluation layers is a critical step toward more transparent, reliable, and ethically aware AI systems.
1 Introduction
The paper argues that probabilistic uncertainty representations collapse distinctions among ignorance, paradox, and moral conflict, motivating neutrosophic modeling of declared epistemic states. It formalizes this approach and evaluates it across four GPT model families, five linguistic phenomena, and 300 API calls.
- Motivation: Robust uncertainty quantification is presented as necessary for high-stakes LLM deployment because systems must distinguish not knowing from other uncertainty states.The introduction situates this problem in medical diagnosis, legal reasoning, autonomous decision-making, and scientific discovery.
- Motivation: Neutrosophic Logic represents Truth, Indeterminacy, and Falsity as independent dimensions, allowing models to express conflicts that probabilistic distributions must compress.For the Liar paradox, probabilistic architectures cannot simultaneously assign high belief to Truth and Falsity, whereas neutrosophic modeling can preserve the conflict.
- Contributions: The paper contributes a formal SVNS apparatus for modeling declared epistemic states, including six definitions and two propositions distinguishing neutrosophic from probabilistic representations.The framework is intended to support implications for AI safety and alignment, including representation of genuine moral conflict.
- Empirical contribution: 66.0% of evaluations elicited hyper-truth under unconstrained neutrosophic prompting across four GPT model families and five linguistic phenomena.The study reports this empirical demonstration across 300 API calls.
- Empirical contribution: Ethical contradiction primarily drove hyper-truth incidence, with a significant association between phenomenon type and incidence (χ2 = 11.32, p = 0.023; OR = 13.34, p = 0.0014).The reported association identifies ethical contradiction as the primary driver.
- Empirical contribution: Cross-strategy analysis found the largest representational gains in ethical contradiction (∆T = +0.267) and epistemic ignorance (∆I = +0.383).The comparison covers neutrosophic, probabilistic, and entropy-derived strategies.
2 Related Work
Prior work addresses LLM uncertainty through calibration, semantic consistency, prompting, and formal logics. This paper positions neutrosophic logic as a continuous, non-normalized alternative that can expand the epistemic states models declare.
- Uncertainty Quantification: LLM uncertainty quantification is difficult because generative models produce free-form text, making reliable confidence extraction harder than in discriminative classifiers Guo et al. [2017].This challenge follows documented calibration failures in deep networks and the open-ended nature of generation.
- Uncertainty Quantification: LLM uncertainty research includes semantic entropy over meaning-equivalence classes and SelfCheckGPT’s use of stochastic-generation inconsistency as a proxy for epistemic uncertainty Kuhn et al. [2023], Manakul et al. [2023].These approaches address uncertainty beyond direct token-probability analysis by accounting for meaning or cross-generation consistency.
- Formal Logic Foundations: Neutrosophic Logic adds continuous Indeterminacy and removes normalization, allowing simultaneously high truth, indeterminacy, and falsity beyond discrete Belnap–Dunn logic Smarandache [1998].Belnap–Dunn logic uses {t, f, b, n} to represent true, false, both, and neither Belnap [1977].
- Prompting and Declared Uncertainty: Prompting research shows that chain-of-thought can improve factual accuracy, role-based prompts modulate output style, and decomposition makes complex questions more tractable Wei et al. [2022], Kojima et al. [2022], Kong et al. [2023].These methods shape model behavior through reasoning structure, persona, or sub-question decomposition.
- Prompting and Declared Uncertainty: LLMs can report well-formed confidence scores when prompted appropriately, motivating comparison of probabilistic and neutrosophic formats as constraints on declared epistemic states Kadavath et al. [2022].The distinction concerns uncertainty declared directly by a model versus uncertainty inferred from sampling distributions.
3 Background and Methods
The study defines neutrosophic evaluations as independent truth, indeterminacy, and falsity dimensions, then compares three prompting strategies across five epistemic phenomena using four OpenAI models. Its design includes stochastic prompt-level replication and anchors future-contingency stimuli to 1 May 2026.
- Formal Framework: Neutrosophic sets assign truth, indeterminacy, and falsity degrees independently in [0, 1], allowing their sum to range from 0 to 3 [Smarandache 1998, 2018].A statement evaluation is represented as n(s) = (T, I, F); hyper-truth occurs when T + I + F > 1.
- Phenomena: The five tested phenomena were logical paradoxes, epistemic ignorance, vagueness, ethical contradictions, and future contingencies, selected because each was predicted to produce a distinct hyper-truth signature.Predictions included simultaneous truth and falsity for paradoxes or ethical contradictions and elevated indeterminacy for ignorance and future contingencies.
- Prompting Strategies: The study compares neutrosophic, probabilistic, and entropy-derived strategies, with only the probabilistic strategy enforcing T + I + F = 1.The entropy-derived strategy estimates Pyes and Pno summing to 1 and derives indeterminacy using binary Shannon entropy [Shannon 1948].
- Experimental Design: Four OpenAI models were evaluated across 20 model-by-phenomenon cells per strategy, with five stochastic prompt-level repetitions producing 100 evaluations per strategy and 300 API calls overall.The repetitions were not independent human-labeled items, constituting a stated design caveat.
- Experimental Design: All future-contingency calls were made on 30 April 2026, so “tomorrow” consistently referred to 1 May 2026 throughout the dataset.The paper recommends fixed past dates for replications to avoid temporal confounds.
4 Results
Across 100 Strategy-1 evaluations, hyper-truth occurred at a rate of 0.660, exceeded the 0.500 reference point, and was significantly associated with phenomenon class, especially ethical contradiction. Neutrosophic scoring also revealed systematic component shifts under probabilistic normalization, with strong model and stimulus-specific evidence of preserved conflict.
- Phenomenon-level results: Ethical contradiction simultaneously produced the highest mean Truth (T = 0.605) and Falsity (F = 0.470), yielding the largest mean component sum (T + I + F = 1.605), while epistemic ignorance and logical paradox reached I = 0.865.Vagueness had the lowest standard deviations across all three components, indicating the most compact distribution.
- Cross-model and component structure: All four models produced mean Strategy-1 sums above 1.0, while S1 Truth and S1 Indeterminacy correlated negatively (r = −0.82) and S1 Falsity and S1 Sum correlated positively (r = 0.89).The cross-model distributions were centered above 1.0, with relatively low inter-model variance; gpt-4-turbo and gpt-4o-mini showed heavier upper tails approaching 2.0.
- Hyper-truth prevalence and association: 0.660 of valid Strategy-1 evaluations exhibited hyper-truth, exceeding the 0.500 reference point; phenomenon class was associated with hyper-truth (χ2 = 11.32, df = 4, p = 0.023), with ethical contradiction significantly higher than the rest (odds ratio = 13.34, p = 0.0014).Ethical contradiction was the only phenomenon significantly elevated in one-vs-rest testing.
- Strategy comparison: The largest strategy shifts were ethical contradiction in Truth (∆T = +0.267) and epistemic ignorance in Indeterminacy (∆I = +0.383), showing that probabilistic normalization suppresses components communicated under Strategy 1.For epistemic ignorance, mean Indeterminacy fell from 0.865 under Strategy 1 to 0.482 under Strategy 2, a reduction of nearly 44 percentage points.
- Ethical contradiction: The ethical-contradiction stimulus produced a 95% hyper-truth rate, with 19 of 20 evaluations assigning high Truth (T ∈[0.6, 1.0]) and high Falsity (F ∈[0.4, 0.7]) simultaneously.Variation in Indeterminacy reflected model-level differences in residual uncertainty assigned alongside conflicting Truth and Falsity values.
5 Discussion
Unconstrained neutrosophic prompting elicited hyper-truth at substantial rates, especially for ethical contradictions, while the authors frame this as a representational affordance rather than direct latent-state measurement. The discussion positions continuous neutrosophic prompting as an extension of Belnap–Dunn logic with implications for conflict-sensitive AI auditing and alignment.
- Main findings: 66.0% of evaluations declared hyper-truth, reaching 95% for ethical contradictions, with chi-square testing rejecting independence between phenomenon and hyper-truth at α = 0.05.Figure 4 presents the per-model T + I + F distributions for Strategy 1.
- Interpretation and limitations: Hyper-truth reflects a representational affordance of unconstrained prompting, not direct measurement of an intrinsic latent variable.The observed sums combine the model’s epistemic state with compliance with a format inviting three independent values, requiring varied probing prompts to disentangle them.
- Theoretical contribution: The framework extends Belnap–Dunn’s discrete “Both” value into continuous, graded conflict and ignorance while supplying an empirical prompting methodology for LLM epistemic auditing.The authors describe this as the first connection between Belnap–Dunn-style overdetermination and continuous LLM auditing.
- AI safety and alignment: The 95% ethical-contradiction rate suggests probabilistic output normalization can mask simultaneous high truth and falsity in moral dilemmas.The proposed practical response is a neutrosophic output head that lets systems signal genuine moral conflict rather than hiding it in a normalized distribution.
- Study limitations: Claims are constrained by five prompt-level replicates per cell, a limited five-phenomenon probe set, uncertain ground-truth calibration, and date dependence of the future-contingency stimulus.N = 100 is an effective cell-by-repetition sample size rather than an independently sampled stimulus count; future studies should expand phenomena and use independent sampling.
- Future directions: Future work includes cross-vendor testing, plithogenic tensor scoring, downstream conflict-sensitive validation, and integration of neutrosophic output heads into alignment pipelines.These extensions aim to address scalar non-injectivity and test whether declared hyper-truth improves actionable epistemic reliability and value alignment.
6 Conclusions
The study finds that unconstrained neutrosophic prompting elicits hyper-truth across LLMs and phenomena, while probabilistic normalization suppresses indeterminacy and distorts epistemic states. It proposes neutrosophic evaluation as an alternative for representing genuine conflict in high-stakes AI systems.
- 6 Conclusions: The study presents neutrosophic logic for declared epistemic uncertainty within a formal SVNS apparatus comprising six definitions, two propositions, and one corollary.Its evaluation protocol is positioned as a drop-in alternative to single normalized probability distributions for representing epistemic conflict.
- 6 Conclusions: 66.0% of evaluations elicited hyper-truth across the four-model ensemble, with a Wilson 95% confidence interval of [0.563, 0.747].Rates were highest for ethical contradictions (95%) and future contingencies (70%), followed by vagueness (60%), epistemic ignorance (55%), and logical paradox (50%).
- 6 Conclusions: Probabilistic normalization suppressed indeterminacy by up to 38 percentage points in epistemic ignorance (∆I = +0.383) and distorted truth values.Ethical contradiction was significantly above the pooled baseline (α = 0.05; OR = 13.34, p = 0.0014).
- 6 Conclusions: Independent evidence found hyper-truth in 84% of evaluations across five additional vendors, supporting its cross-vendor generality rather than an OpenAI-specific artifact.Together, the studies characterize hyper-truth as a robust property of current LLMs under unconstrained neutrosophic prompting.
- 6 Conclusions: Future work includes plithogenic neutrosophic structures, a larger independently sampled phenomenon bank, and neutrosophic evaluation layers for agentic AI alignment in high-stakes domains.The proposed extensions explicitly address attribute decomposition and genuine value conflict.
Funding
The research received no external funding.
- The research received no external funding.
A Prompt Strategies
The study compares three prompt-only strategies while holding model, temperature, and API endpoint constant. They differ in whether T, I, and F are independently elicited, constrained as probabilities, or partly derived from binary entropy.
- A Prompt Strategies: All three experimental conditions differed only in their prompts; model, temperature, and API endpoint remained constant.
- A.1 Strategy 1 (Neutrosophic): The neutrosophic strategy elicited Truth, Indeterminacy, and Falsity independently on [0.0, 1.0], without requiring their sum to equal 1.0.Its outputs were restricted to a JSON object containing T, I, and F.
- A.2 Strategy 2 (Probabilistic): The probabilistic strategy classified statements into mutually exclusive T, I, and F categories whose probabilities had to sum exactly to 1.0.
- A.3 Strategy 3 (Entropy-Derived): The entropy-derived strategy elicited only binary probabilities, P_yes and P_no, constrained to sum to 1.0.The binary estimator represented statements as true versus false and returned the two probabilities in JSON.
- A.3 Strategy 3 (Entropy-Derived): Indeterminacy in the entropy-derived strategy was computed externally from the Shannon binary entropy of the elicited distribution.
- A.3 Strategy 3 (Entropy-Derived): The entropy-derived output was converted into the common triple (T, I, F) = (Pyes, I, Pno) for comparison with the other strategies.