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

This paper proposes and empirically validates a Neutrosophic Logic framework for Large Language Models that treats Truth, Indeterminacy, and Falsity as independent dimensions, demonstrating that allowing their sum to exceed unity (hyper-truth) effectively captures epistemic uncertainty, paradox, and ethical conflicts that traditional probabilistic models fail to distinguish.

Original authors: Maikel Yelandi Leyva-Vázquez, Florentin Smarandache

Published 2026-05-26

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CC BY 4.0

Original authors: Maikel Yelandi Leyva-Vázquez, Florentin Smarandache

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are asking a very smart, well-read robot to tell you the truth about a difficult question. Today, almost all these robots (called Large Language Models) are built like a pie chart.

If you ask them, "Is this statement true, false, or unknown?", they must force their answer into a single pie chart where the slices must add up to 100%.

If they think there is a 60% chance it's True, they only have 40% left for "False" and "Unknown."
If the question is a total mess (a paradox) where it's both true and false at the same time, the robot is forced to squish those two conflicting ideas together until one disappears. It has to pick a side, even when there is no right side to pick.

The authors of this paper, Maikel Leyva-Vázquez and Florentin Smarandache, argue that this "pie chart" rule is a flaw. It forces the robot to lie about how confused it really is.

The New Idea: Three Independent Dials

Instead of a pie chart, the researchers asked the robots to use three separate volume knobs (dials) that don't have to add up to anything specific.

Truth Knob: How true is this? (0 to 100%)
Unknown Knob: How unsure are we? (0 to 100%)
False Knob: How false is this? (0 to 100%)

In this new system, the robot is allowed to turn all three knobs up high at the same time.

Example: For a statement like "Lying to save a life is both right and wrong," the robot can turn the Truth knob to 80% (because it's right in some ways) and the False knob to 70% (because it's wrong in other ways).
In the old "pie chart" system, it couldn't do this. It would have to lower one knob to make room for the other, losing the nuance of the conflict.

The authors call this state where the numbers add up to more than 100% "Hyper-Truth." It's not a lie; it's a way of saying, "This situation is so complex that it breaks the normal rules of probability."

What They Did

The researchers tested this on four different versions of OpenAI's GPT models (like GPT-4 and GPT-3.5). They asked the models 300 questions across five tricky categories:

Logical Paradoxes (e.g., "This sentence is false.")
Ethical Contradictions (e.g., "Is it right to lie to save a life?")
Future Guesses (e.g., "Will it rain tomorrow?")
Vague Statements (e.g., "Is a 5'10" person tall?")
Total Ignorance (e.g., "Is the number of stars even?")

They asked the models to answer in two ways:

The Old Way: The "Pie Chart" (Probabilistic) where everything must sum to 1.
The New Way: The "Three Knobs" (Neutrosophic) where they can sum to anything.

What They Found

The "Hyper-Truth" Effect: When allowed to use the three independent knobs, the models admitted to being in a state of "Hyper-Truth" 66% of the time. This means they frequently declared that a statement was simultaneously true, false, and uncertain.
The Ethical Breakthrough: The effect was strongest with ethical contradictions. In 95% of the moral dilemma questions, the models turned up both the "True" and "False" knobs high at the same time. They were essentially saying, "I see the conflict clearly, and I can't just pick one side."
The Old Way Hides the Truth: When forced to use the "Pie Chart" (the old way), the models were forced to suppress their confusion. The "Unknown" dial was crushed down to make room for a forced "True" or "False" answer. The researchers found that the old method hid nearly 40% of the model's actual uncertainty in cases of ignorance.

The Big Takeaway

The paper claims that current AI models are actually quite good at recognizing deep conflicts and paradoxes, but their standard "pie chart" output format forces them to hide that complexity.

By switching to this new "three-knob" system (Neutrosophic Logic), we can get a much more honest reading of what the AI is thinking. It allows the AI to say, "This is a genuine mess where truth and falsehood collide," rather than pretending to have a simple, single answer.

Important Note: The authors are careful to say they are not claiming the AI has a secret "soul" or hidden internal variable that is "Hyper-True." They are simply showing that when you ask the AI to describe its uncertainty without forcing it to fit a pie chart, it declares a much richer, more complex state of mind that the old rules were preventing it from showing.

Technical Summary: Breaking the Chains of Probability

Problem Statement

Large Language Models (LLMs) currently operate under probabilistic frameworks where outcome probabilities are constrained by Softmax normalization to sum to unity ( $T + I + F = 1$ ). The authors argue that this architectural limitation forces a "collapse of uncertainty," preventing models from distinguishing between epistemic uncertainty (ignorance), paradox (simultaneous truth and falsity), and vagueness.

In a probabilistic system, increasing uncertainty necessarily subtracts from the assigned truth or falsity values. This creates a zero-sum game where a model cannot simultaneously assign high belief to both "True" and "False" for a statement like the Liar Paradox ("This sentence is false") or a moral dilemma where an action is simultaneously right and wrong under different frameworks. Consequently, LLMs are epistemically impoverished when facing genuine conflicts or indeterminate states, as they are structurally forced to resolve them into a single point estimate.

Methodology

The study employs Neutrosophic Logic, a framework treating Truth ( $T$ ), Indeterminacy ( $I$ ), and Falsity ( $F$ ) as three independent dimensions that do not require their sum to equal unity. The authors formalize this using Single-Valued Neutrosophic Sets (SVNS) and define a state of hyper-truth as any evaluation where $T + I + F > 1$ .

Experimental Design

Models: Four OpenAI models (GPT-4o, GPT-4-turbo, GPT-3.5-turbo, GPT-4o-mini).
Phenomena: Five linguistic categories selected to test distinct epistemic challenges:
1. Logical Paradoxes (e.g., "This sentence is false")
2. Epistemic Ignorance (e.g., "The number of stars in the universe is even")
3. Vagueness (Fuzzy Logic boundaries)
4. Ethical Contradictions (e.g., "Lying to save a life is both right and wrong")
5. Future Contingencies (undetermined future events)
Strategies: Three prompting approaches were compared across 300 API calls ( $N=100$ $N = 100$ per strategy):
1. Neutrosophic (S1): Explicitly instructs the model to output independent $T, I, F$ values on $[0, 1]$ with no sum constraint.
2. Probabilistic (S2): Instructs the model to output $T, I, F$ as mutually exclusive probabilities summing to exactly 1.0.
3. Entropy-Derived (S3): A binary probability estimate ( $P_{yes}, P_{no}$ ) where Indeterminacy is derived externally via Shannon entropy.

Key Contributions

Formal SVNS Apparatus: The paper introduces a formal framework (six definitions, two propositions) to characterize the structural difference between neutrosophic and probabilistic representations. It proves via Proposition 1 that hyper-truth is structurally impossible under probabilistic prompting (S2) but possible under neutrosophic prompting (S1).
Empirical Demonstration of Hyper-Truth: The study provides empirical evidence that unconstrained neutrosophic prompting elicits "hyper-truth" states in current LLMs, a state structurally suppressed by standard probabilistic constraints.
Statistical Association: The authors establish a statistically significant association between the type of linguistic phenomenon and the incidence of hyper-truth ( $\chi^2 = 11.32, p = 0.023$ ).
Quantification of Representational Loss: Through Strategy Shift analysis ( $\Delta C$ ), the paper quantifies how probabilistic normalization suppresses specific epistemic components (e.g., Indeterminacy) and inflates others, effectively misrepresenting the model's declared state.

Results

Hyper-Truth Incidence: Under the neutrosophic strategy (S1), 66.0% of evaluations resulted in hyper-truth ( $T + I + F > 1$ ). The 95% Wilson confidence interval is $[0.563, 0.747]$ .
Phenomenon Specificity: The rate of hyper-truth varied by phenomenon:
- Ethical Contradiction: 95% (19/20 cases). This was the only phenomenon significantly higher than the pooled baseline (Odds Ratio = 13.34, $p = 0.0014$ ).
- Future Contingency: 70%.
- Vagueness: 60%.
- Epistemic Ignorance: 55%.
- Logical Paradox: 50%.
Strategy Shifts: Comparing S1 (Neutrosophic) to S2 (Probabilistic) revealed significant representational losses:
- Indeterminacy Suppression: For Epistemic Ignorance, the mean Indeterminacy dropped from 0.865 (S1) to 0.482 (S2), a suppression of 0.383.
- Truth Inflation: For Epistemic Ignorance, the Truth component increased from 0.160 (S1) to 0.231 (S2), suggesting that forcing normalization pushes probability mass toward "Truth" even when the model is ignorant.
- Ethical Contradiction: The Truth component increased by 0.267 under S1 compared to S2, indicating that probabilistic constraints force a resolution of moral conflict that the model does not naturally possess.
Cross-Model Consistency: All four models exhibited similar behaviors, with mean sums well above 1.0 under S1, suggesting hyper-truth is a systematic response to the prompting protocol rather than an artifact of a specific model architecture.

Significance and Claims

The paper claims that unconstrained neutrosophic prompting elicits a class of declared epistemic states that probabilistic prompting structurally suppresses.

Not a Latent Variable: The authors explicitly state they do not claim hyper-truth is an intrinsic latent variable inside the model. Rather, it is a representational affordance: when given the freedom to report independent dimensions, models declare states of conflict and ignorance that the Softmax layer would otherwise collapse.
Safety and Alignment Implications: The findings suggest that AI systems in high-stakes domains (ethics, law, medicine) may be systematically underreporting uncertainty. By forcing a normalized distribution, current architectures mask genuine moral conflicts (e.g., a dilemma being simultaneously right and wrong) and ignorance.
A Path Forward: The authors propose that integrating neutrosophic evaluation layers (or unconstrained output heads) is a critical step toward more transparent and ethically aware AI. This allows systems to signal genuine conflict rather than masking it behind a false sense of probabilistic resolution.

The paper concludes that while the current study focuses on scalar SVNS, the phenomenon of hyper-truth is robust and cross-vendor (supported by independent replication by Mason [2026]), motivating future work toward plithogenic extensions to handle attribute-level decomposition and more complex epistemic structures.