Introduction
Modern large language models (LLMs) promise access to knowledge, yet often deliver only a sophisticated illusion of it. The work of Michał P. Karpowicz sheds new light on this phenomenon, proving that hallucination is not an engineering flaw, but a structural necessity. This article analyzes the boundaries of truth in AI systems, moving from classical logic to advanced information theory. You will learn why the mathematical nature of the transformer makes total accuracy impossible and how machine "confabulation" is inextricably linked to creativity and the ability to synthesize information.
Deduction vs. Induction: The Limits of Logic in Statistics
Traditional deduction is a world of logical necessity, where true premises lead to inevitable conclusions. LLMs, however, belong to the world of induction—they operate on probabilities, patterns, and historical data rather than hard evidence. From the perspective of the correspondence theory of truth, a statement is true when it corresponds to reality. Yet AI possesses neither senses nor an ontological map; it operates solely on statistical relationships between tokens.
Within this structure, hallucination is a necessity, not a mistake. The model does not "lie" in the human sense—it simply predicts the most probable sequence of characters. The lack of semantic anchoring in extra-linguistic reality leaves these systems cognitively blind to truth, replacing it with mathematical pattern matching.
Signal Aggregation and Honesty-Enforcement Mechanisms
In LLM architecture, truth is often reduced to pragmatism (utility for the user) or deflationism (a logical label). Karpowicz’s auction theory, however, exposes a deeper problem: the network's internal components bid on signal strength, not truthfulness. Although proper scoring rules are applied to enforce "honesty" (revealing actual probabilistic beliefs), the aggregation process disrupts this harmony.
Instead of coherence—the internal consistency of beliefs—the softmax mechanism favors the dominance of the strongest signal. Consequently, the system does not strive for logical consensus but rewards statistical clout. This leads to situations where the apparent consistency of an answer is merely an artifact of signal amplification rather than the result of reliable information verification.
Jensen’s Gap: The Mathematical Source of Overconfidence
The key to understanding hallucination lies in the log-sum-exp (LSE) function used to combine data in transformers. Its mathematical convexity ensures that the aggregated result is always more "confident" than its components. This phenomenon, known as Jensen’s gap, generates excessive certainty without actual knowledge. The model systematically "adds its own" information, creating synthetic entropy that does not exist in the input data.
This violation of the principle of semantic information conservation causes the model to produce fabricated content with authority. Fascinatingly, this same anomaly drives creativity and cognitive innovation. The difference between hallucination and innovation lies in the user's judgment: when the model "boldly guesses" something accurate, we call it insight; when it errs, we call it hallucination. They are two sides of the same coin.
Summary
Karpowicz’s analysis leads to an inevitable conclusion: hallucination is not an engineering flaw, but the price paid for the ability to generate complex knowledge. No model exists that is simultaneously fully honest, conservative, and useful. AI systems must venture beyond what they "know for sure" to generate valuable responses in an uncertain world.
Hallucination cannot be entirely eliminated, but it can be understood and controlled, much like human intuition. Truth, as performed by machines, is not a property of the sentence but a user's projection onto the text. When we demand truth from a machine, we receive hallucination; when we allow it to hallucinate, we sometimes encounter truth. This is the paradox that defines the limits of modern artificial intelligence.
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