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Revisiting the Aerts-Broekaert-Smets quantum model of the liar paradox

This paper provides a pedagogical reconstruction of the Aerts-Broekaert-Smets quantum model of the liar paradox using Dirac notation to clarify that while the unitary dynamics can be represented in a four-dimensional truth-only space, the enlargement of the state space is necessary to account for measurement structures and cognitive memory in revision processes.

Original authors: Massimiliano Sassoli de Bianchi

Published 2026-09-10

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Massimiliano Sassoli de Bianchi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

In the study of how the human mind works, researchers have long been fascinated by the gap between simple logic and complex thought. For decades, scientists have tried to build mathematical models to explain how we make decisions, weigh options, and change our minds. Recently, a field known as quantum cognition has emerged, which borrows tools from physics to describe these mental processes. In this framework, the mind is not seen as a static computer that simply stores facts, but as a dynamic system where the act of asking a question can actually change the state of the information being considered. This approach helps explain why human reasoning sometimes seems contradictory or unstable, especially when faced with self-referential puzzles. One of the most famous of these puzzles is the liar paradox, a sentence that refers to itself in a way that creates an endless loop of truth and falsehood. Understanding how the mind navigates this loop offers a window into the mechanics of deliberation itself.

A researcher has revisited a specific model of this paradox, originally proposed by physicists Aerts, Broekaert, and Smets, to clarify exactly how it works and what it reveals about the nature of thought. Their work focuses on a scenario involving two sentences that refer to each other: one claims the other is false, and the other claims the first is true. When a person tries to determine the truth of these sentences, their mind enters a state of oscillation, flipping back and forth between different truth assignments without ever settling on a stable answer. The original model used a complex mathematical structure to describe this flipping, suggesting that the mind needs extra "space" to keep track of where each thought came from. The new analysis clarifies that this extra space is not required for the internal, rhythmic movement of the cycle itself, which can be fully described in a simpler, four-dimensional space. However, the enlarged space is strictly necessary to accurately model the act of measurement—specifically, the moment a conscious decision initiates the process.

The researcher found that if you only look at the final truth values—whether a sentence is true or false—you can describe the cycle using a smaller, simpler model. In this reduced view, the mind's movement through the paradox looks like a smooth, predictable rotation through four distinct states. However, this simpler view breaks down the moment you try to model the actual act of thinking. When a person makes a choice, such as deciding to believe the first sentence is true, that decision creates a specific starting point that is different from a situation where the same truth value was reached by following a chain of logic. The original, more complex model includes a hidden layer of information that records this origin. Without this record, the model cannot tell the difference between a thought that was freely chosen and one that was forced by the logic of the previous step. This distinction is crucial for the measurement structure because it allows the model to accurately represent how a single decision triggers a specific sequence of mental events, even though the subsequent cycle of oscillation does not inherently depend on this distinction to exist.

By rewriting the model in a clearer mathematical language, the author showed that the "extra" dimensions in the original theory are not just mathematical tricks; they represent a form of cognitive memory. This memory tracks the history of how a belief was formed. In the case of the liar paradox, this history is essential for the measurement process because the same truth value can lead to different future thoughts depending on how it was arrived at. If a person decides a sentence is true, the next step in their thinking might be different than if they simply inferred that it was true from a previous conclusion. The researcher demonstrated that this ability to distinguish between decision and inference is what allows the model to correctly initiate the paradoxical loop. Without this distinction, the model cannot represent the cognitive act that starts the process, even though the oscillation itself can be described without it.

The study also derived a precise description of the energy and movement behind this mental cycle, showing that the mind's progression through the paradox follows a strict, rhythmic pattern. They calculated the exact probabilities of the mind being in any specific state at any given moment, revealing that the system moves through the four stages of the loop in a perfectly timed sequence. This rhythmic movement is not random; it is a deterministic flow that is only interrupted when a new decision is made. The researcher emphasizes that while the internal movement of the mind can be described simply, the act of starting the process requires the more complex, enriched model. This suggests that the complexity of human thought often lies not in the complexity of the ideas themselves, but in the way we keep track of how we arrived at them.

Finally, the author connects this specific puzzle to a broader understanding of how humans deliberate on difficult choices. They suggest that the liar paradox is an extreme example of a common mental process: the back-and-forth of weighing options. In many decisions, a person might tentatively choose one path, only to find that the consequences of that choice make it seem less likely, prompting a switch to another option. Sometimes, this process stabilizes, and the person reaches a firm conclusion. Other times, like in the liar paradox, the process destabilizes itself, leading to an endless cycle of reconsideration. The research indicates that the key to understanding these cycles is recognizing that our minds carry a record of our reasoning history. This record acts as a minimal form of memory, allowing us to distinguish between a choice we made and a conclusion we were forced into. By understanding this mechanism, we gain a clearer picture of why some thoughts get stuck in loops while others move forward to a resolution.

Technical Summary: Revisiting the Aerts-Broekaert-Smets Quantum Model of the Liar Paradox

Problem Statement
The paper addresses the mathematical and conceptual structure of the quantum model of the two-sentence liar paradox originally proposed by Aerts, Broekaert, and Smets (ABS). The liar paradox involves a system of self-referential propositions where truth assignments oscillate indefinitely rather than stabilizing. The original model utilized a quantum formalism to describe this semantic dynamics, embedding the oscillatory behavior in a unitary evolution. However, the original presentation relied heavily on explicit vector and matrix representations, leaving the underlying structural logic—particularly the necessity of the dimensional enlargement from a 2-dimensional to a 4-dimensional local state space per sentence (resulting in a 16-dimensional total Hilbert space)—somewhat opaque. The paper seeks to reconstruct this model to clarify the distinction between truth values arising from direct cognitive decisions versus those arising from semantic inference, and to determine precisely which aspects of the dynamics necessitate the enlarged state space.

Methodology
The author employs a pedagogical reconstruction of the ABS model using Dirac notation and tensor product structures. The methodology involves:

Factorization of State Space: Reinterpreting the 4-dimensional one-sentence space ( $C^4$ ) as a tensor product of two binary degrees of freedom: the truth value ( $H_{tr}$ ) and the origin of that value ( $H_{or}$ ), where the origin is either "decision" ( $d$ ) or "semantic inference" ( $s$ ). This leads to a factorization $H = H_{or} \otimes H_{tr}$ .
Operator Derivation: Deriving the Hamiltonian ( $H$ ) that generates the four-state cyclic evolution of the double liar paradox. This is achieved by diagonalizing the discrete evolution operator ( $U$ ) in a discrete Fourier basis and expressing $H$ in a compact operator form, avoiding the cumbersome matrix manipulations of the original papers.
Comparative Analysis: Contrasting the "extended" representation (16-dimensional, $C^4 \otimes C^4$ ) with a "truth-only" representation (4-dimensional, $C^2 \otimes C^2$ ). The author analyzes the unitary equivalence between these spaces and examine how measurement projectors behave in each.
Extension to Revision Processes: Investigating scenarios where the origin degree of freedom becomes dynamically necessary, specifically in processes involving hypothesis revision where states with identical truth content but different histories require different successors.

Key Contributions and Results

Clarification of Entanglement vs. Separability:
The paper demonstrates that in the extended representation ( $H_{or} \otimes H_{tr}$ ), the non-paradoxical cases (consistent truth assignments) correspond to separable states across the origin-truth partition. In contrast, the genuine liar state is entangled across this partition. Specifically, the liar state is a coherent coupling between the stable correlation structures of the non-paradoxical cases.
Compact Hamiltonian Formulation:
The author derives the Hamiltonian generating the cyclic unitary dynamics in a compact operator form:
$H = \frac{\pi}{4\tau} \left[ 3I - U^2 - (1+i)U - (1-i)U^\dagger \right]$
They also provide closed-form expressions for the transition probabilities $P_n(t)$ between the four states of the cycle, confirming the periodic nature of the evolution with period $4\tau$ .
Necessity of Dimensional Enlargement:
A central finding is that the 16-dimensional Hilbert space is not required for the unitary part of the dynamics. The cyclic evolution can be fully described in a 4-dimensional "truth-only" space ( $C^2 \otimes C^2$ ) via a unitary identification map $\Gamma$ .
- However, the enlargement is strictly required by the measurement structure. In the truth-only space, it is impossible to construct a state such that single-sentence truth projectors map it onto four mutually orthogonal states (the four phases of the cycle). The origin labels ( $d$ vs. $s$ ) are necessary to distinguish the four orthogonal phases initiated by a decision on a single sentence.
Cognitive Memory and Revision:
The paper shows that the origin degree of freedom acts as a minimal form of "cognitive memory." While redundant for the standard four-step cycle, it becomes dynamically necessary when the system admits revision processes. If a cognitive subject revises a hypothesis to avoid a repeating cycle, states with identical truth values but different origins (histories) must have different successors. In such cases, a truth-only representation fails to capture the dynamics, and the system explores an 8-dimensional subspace.
Correction of Historical Inconsistencies:
The author identifies and correct a sign misprint and a basis ordering discrepancy in the Hamiltonian and evolution operators reported in the original historical papers [1, 2], ensuring the derived Hamiltonian commutes with the evolution operator and admits the uniform superposition as a zero-energy ground state.

Significance and Claims
The paper claims to provide a clearer, more transparent structural understanding of the ABS model. Its primary significance lies in disentangling the dynamical requirements from the measurement requirements:

Unitary Dynamics: The oscillatory behavior of the liar paradox is a property of the unitary evolution in a 4-dimensional truth space; the dimensional enlargement is not a dynamical necessity for the cycle itself.
Measurement/Contextuality: The enlargement is a necessity for representing the contextuality of the cognitive interaction. It allows the model to represent the act of "deciding" on a single sentence as a projection that uniquely identifies a specific phase of the inferential cycle, a feat impossible in the reduced truth-only space.
Deliberation: The author suggests the model offers a minimal dynamic mechanism relevant to broader deliberative processes. The distinction between "decision" and "inference" (the origin labels) provides a formal way to model how provisional choices can trigger semantic consequences that either stabilize (self-reinforcing) or destabilize (self-oscillating) a cognitive state. The liar paradox represents the extreme case of self-destabilization where the context is a purely logical injunction, preventing stabilization.

The paper concludes modestly, positioning the work as a pedagogical clarification that makes the conceptual structure of quantum cognition models more explicit, particularly regarding the role of entanglement in semantic dynamics and the representation of cognitive history.

Technical Summary: Revisiting the Aerts-Broekaert-Smets Quantum Model of the Liar Paradox

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