Beyond Uncertainty: A New Logic for Reliable AI

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Author: Denis Avetisyan

A novel axiomatic possibility theory resolves a long-standing paradox in fuzzy logic, paving the way for more robust and consistent artificial intelligence systems.

This review establishes a formal foundation for modeling epistemic uncertainty using dual possibility and necessity measures, offering an alternative to Dempster-Shafer theory and resolving Zadeh’s Paradox.

Despite decades of research into artificial intelligence, reliably managing uncertainty remains a fundamental challenge, particularly as highlighted by Zadeh’s paradox. This work, ‘Resolving Zadehs Paradox Axiomatic Possibility Theory as a Foundation for Reliable Artificial Intelligence’, proposes a resolution through an axiomatic approach to possibility theory, leveraging dualistic measures of possibility and necessity. By constructing a logically consistent framework, this paper demonstrates how possibility theory surpasses the limitations of Dempster-Shafer theory in resolving conflicting data, offering a pathway towards more robust AI systems. Could this axiomatic foundation finally bridge the gap between formal reasoning and the nuanced logic of natural intelligence?

The Inherent Limitations of Classical Logic

Human reasoning rarely occurs with complete information; instead, it frequently navigates a landscape of ambiguity and partial knowledge. Traditional logical systems, predicated on binary truths – something is either definitively true or false – struggle to effectively process this inherent uncertainty. This limitation arises because real-world data is often imprecise, vague, or simply missing, forcing reliance on assumptions and estimations. Consequently, applying strict logical rules to incomplete datasets can yield inaccurate or unreliable conclusions. The inability of classical logic to gracefully handle these nuances underscores the necessity for alternative frameworks capable of representing and reasoning with degrees of belief, plausibility, or possibility – systems that acknowledge the inherent imperfection of information and strive for robust inference despite it.

Dempster-Shafer Theory and fuzzy logic represent attempts to quantify and reason with epistemic uncertainty – the kind arising from incomplete or imprecise knowledge, rather than inherent randomness. However, both frameworks struggle when tasked with consolidating evidence that directly contradicts itself. While designed to handle ambiguity, the methods employed for combining information often yield results that are overly conservative or even paradoxical when faced with strong disagreement. This arises because the approaches prioritize avoiding incorrect conclusions at the expense of potentially useful inferences, effectively diminishing the impact of supporting evidence when conflicts exist. Consequently, the ability to draw reliable conclusions from real-world data, which is often riddled with conflicting signals, is significantly hampered by these limitations.

Traditional approaches to reasoning under uncertainty, such as Dempster-Shafer Theory and fuzzy logic, often employ a ‘conflict measure’ to assess the degree of disagreement between different sources of evidence. However, this reliance can generate counterintuitive outcomes; as evidence supporting opposing conclusions increases, the conflict measure also rises, paradoxically leading to less confidence in any definitive inference. This behavior stems from the way these theories aggregate evidence, where conflicting information doesn’t necessarily diminish belief in either proposition, but instead amplifies the uncertainty. In contrast, Axiomatic Possibility Theory offers a distinct framework by prioritizing the plausible options – those that don’t directly contradict available evidence – and systematically eliminating the impossible. This approach avoids the pitfalls of escalating conflict measures, providing a more robust and logically consistent method for drawing conclusions from incomplete or ambiguous data, ultimately leading to more reliable inference.

Axiomatic Possibility Theory: A Foundation of Logical Consistency

Axiomatic Possibility Theory (APT) diverges from traditional probabilistic approaches by directly quantifying both the degree to which a proposition is possible and the degree to which it is necessary. Unlike probability, which assigns a single numerical value representing belief, APT utilizes two distinct measures: $𝑃𝑜𝑠(𝐴)$ representing the possibility of proposition A being true, and $𝑁𝑒𝑐(𝐴)$ representing the necessity of proposition A being true. These measures are not constrained by the probabilistic requirement that the sum of probabilities equals one, allowing for the representation of imprecise or incomplete information without forcing artificial normalization. This dual representation provides a more nuanced framework for reasoning under uncertainty, particularly in scenarios where the absence of evidence does not necessarily imply impossibility.

Maxitive Logic serves as the foundational method for evidence combination within Axiomatic Possibility Theory, offering advantages over Dempster-Shafer Theory. Unlike Dempster-Shafer, which can produce counterintuitive results when evidence conflicts – leading to increased belief in propositions that are mutually exclusive – Maxitive Logic avoids this by combining evidence in a manner that preserves logical consistency. Specifically, it utilizes the maximum of the supporting values, preventing the amplification of conflicting information. This approach directly addresses Zadeh’s paradox, wherein increasing evidence can paradoxically decrease overall belief, by ensuring that combined evidence never exceeds the bounds of possibility and necessity, thereby maintaining a stable and logically sound reasoning framework.

Axiomatic Possibility Theory frames uncertainty through the explicit quantification of possibility and necessity, offering a reasoning framework distinct from probabilistic approaches. Epistemic uncertainty regarding a proposition, 𝐴, is fully captured by an interval defined as [$𝑁𝑒𝑐(𝐴), 𝑃𝑜𝑠(𝐴)$], where $𝑁𝑒𝑐(𝐴)$ represents the degree of necessity – the extent to which 𝐴 is considered necessarily true – and $𝑃𝑜𝑠(𝐴)$ denotes the degree of possibility, indicating the extent to which 𝐴 is considered possibly true. This interval representation provides a complete characterization of the agent’s belief state regarding 𝐴, allowing for consistent reasoning even with incomplete or conflicting information, and avoids the logical inconsistencies present in other uncertainty theories.

Resolving Logical Paradoxes with Possibility Theory

Axiomatic Possibility Theory (APT) resolves ‘Zadeh’s Paradox’ – a known limitation of Dempster-Shafer Theory (DST) – by explicitly accommodating conflicting evidence. DST utilizes normalization techniques which, while producing a single belief value, effectively discard information indicating genuine disagreement between sources. This normalization can lead to counterintuitive results where conflicting evidence is artificially reconciled. In contrast, APT preserves this conflict by representing belief as an interval, allowing for the simultaneous expression of both supporting and opposing evidence. This interval-based approach avoids the forced consensus inherent in DST and provides a more accurate representation of the underlying epistemic state when faced with contradictory information, thereby addressing the core issue highlighted by Zadeh’s Paradox.

Axiomatic Possibility Theory offers a formalized method for aggregating potentially conflicting diagnostic assessments from multiple medical experts. Unlike traditional approaches that average probabilities or rely on majority rule, this theory allows for the explicit representation of disagreement and uncertainty. Each expert provides a degree of possibility – a value between 0 and 1 – to each potential diagnosis, reflecting their belief in its plausibility. These individual assessments are then combined using specific compositional rules defined by the theory, resulting in a collective assessment that accounts for both consensus and divergence. This approach avoids the logical inconsistencies that can arise when combining conflicting probabilistic evidence, providing a rational and consistent basis for medical decision-making, especially in complex cases where expert opinions naturally vary.

Axiomatic Possibility Theory utilizes an ‘Interval Representation’ where knowledge is expressed as a closed interval $[p, q]$, with $p$ and $q$ representing the lower and upper bounds of possibility, respectively. This approach allows for the representation of imprecise knowledge and uncertainty without forcing a single, definitive probability assignment. Critically, this interval-based system preserves conflicting evidence; if expert opinions diverge, the interval will reflect this disagreement rather than attempt to reconcile it through normalization, as is characteristic of Dempster-Shafer Theory. By explicitly representing conflict within the interval, the theory avoids overconfidence in situations where information is incomplete or contradictory, enabling more nuanced and realistic reasoning about possibilities.

Beyond Stability: The Expanding Influence of Possibility Theory

Axiomatic Possibility Theory builds upon the concept of ‘Functional Stability’ by establishing a rigorous mathematical foundation for consistent system behavior, even when confronted with dynamic environments or imperfect data. Unlike traditional approaches that often falter with slight variations in input, this theory defines a set of axioms – fundamental truths – that guarantee predictable outcomes regardless of external disturbances. This isn’t merely about maintaining average performance; it’s about ensuring reliable performance, where the system’s response doesn’t drastically change due to noise or shifting conditions. The framework achieves this by focusing on the preservation of relationships between inputs and outputs, rather than precise numerical values. Consequently, systems designed with this theory demonstrate a remarkable resilience, adapting gracefully to real-world complexities and maintaining operational integrity where more brittle models would fail. This robustness is crucial for applications demanding high levels of dependability, from autonomous robotics to critical infrastructure management.

Axiomatic Possibility Theory gains considerable practical strength through its inherent compatibility with fuzzy logic, a well-established approach to approximate reasoning. This synergy allows the framework to leverage existing techniques for handling imprecise or incomplete information, vastly broadening its potential applications. Rather than requiring a complete overhaul of current AI systems, the theory can be integrated as a layer of robust reasoning on top of existing fuzzy logic controllers and inference engines. This seamless integration facilitates the handling of nuanced data – situations where absolute certainty is unattainable – and empowers systems to make informed decisions even when faced with ambiguity. Consequently, the combined approach moves beyond traditional, brittle probabilistic models, offering a more flexible and adaptable toolkit for building intelligent systems capable of navigating the complexities of the real world.

The development of robust artificial intelligence increasingly demands systems capable of navigating the inherent ambiguities of real-world data, and Possibility Theory offers a compelling alternative to traditional, purely probabilistic approaches. While probabilistic models often struggle with incomplete or unreliable information, this framework excels by representing uncertainty not as a degree of belief, but as a range of plausible possibilities. This allows AI systems built on Possibility Theory to maintain functionality even when faced with noisy or imprecise inputs, creating resilience against the ‘brittleness’ that plagues many existing models. By focusing on what is possible, rather than calculating the likelihood of specific outcomes, this toolkit enables the construction of AI that can reason effectively under conditions of genuine uncertainty, opening doors to more adaptable and reliable intelligent systems.

The pursuit of robust artificial intelligence demands a foundation built upon logical consistency, a principle keenly understood in the resolution of Zadeh’s paradox. This work, grounding possibility theory in a rigorous axiomatic framework, echoes Bertrand Russell’s sentiment: “The point of the question is not whether something is true, but how far it is true.” Just as Russell advocated for nuanced truth, this paper moves beyond simplistic binary evaluations by employing dual measures of possibility and necessity. This nuanced approach, offering a demonstrable alternative to Dempster-Shafer theory, isn’t merely about achieving functional results; it’s about establishing a mathematically sound basis for managing epistemic uncertainty and, consequently, building AI systems worthy of trust.

Beyond Resolution: Charting a Course for Reliable Uncertainty

The resolution of Zadeh’s paradox, achieved through a rigorously axiomatic possibility theory, is not merely a correction of a logical inconsistency. It is an acknowledgement that much of the field has traded mathematical purity for pragmatic expediency. The prevailing reliance on Dempster-Shafer theory, while yielding functional results, has consistently skirted the issue of conflicting evidence and the inherent ambiguity of incomplete information. This work demonstrates that a system built upon logically sound foundations-dual measures of possibility and necessity-offers a path toward truly reliable artificial intelligence, one where confidence isn’t a heuristic, but a provable consequence of the underlying mathematics.

Future work must address the computational complexity that inevitably arises from such rigorous formalism. While elegance dictates completeness, practicality demands efficiency. The challenge lies not in simplifying the mathematics, but in developing algorithms that can exploit its structure to deliver scalable solutions. Furthermore, the inherent subjectivity in defining the fundamental axioms-defining what constitutes possibility and necessity-requires careful consideration. A deeper exploration of the philosophical underpinnings is crucial; otherwise, the system risks becoming a technically sophisticated echo chamber of pre-existing biases.

Ultimately, the pursuit of reliable AI demands a return to first principles. The field has long tolerated approximations and convenient fictions. This work suggests that the price of those conveniences has been a fundamental lack of trustworthiness. The next phase must prioritize demonstrable correctness, even if it necessitates a temporary sacrifice of computational speed. Simplicity, it must be remembered, does not equate to brevity-it demands non-contradiction and logical completeness.

Original article: https://arxiv.org/pdf/2512.05257.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2025-12-08 20:59