The quantum model of the two-sentence liar paradox proposed by Aerts, Broekaert, and Smets is an early example of the use of quantum formalism to describe cognitive dynamics. Our reconstruction is primarily pedagogical in intent, but it also leads to a number of clarifications, and to some new observations, concerning the structure of the model. Rewriting the model in Dirac notation, we make explicit the distinction between truth values originating from a decision and from semantic inference, and show that the associated enlargement of the one-sentence state space corresponds to a factorization with respect to which the non-paradoxical configurations are separable while the genuine liar state is entangled. We also derive the Hamiltonian generating the four-state cyclic evolution, express it in compact operator form, and give the resulting transition probabilities in closed form. We emphasize that the liar cycle admits a unitarily equivalent representation in the four-dimensional truth-only space, where the unmeasured liar state is separable, so the dimensional enlargement is not required by the unitary part of the dynamics; it is required, however, by the measurement structure. The enlargement is also required by the dynamics as soon as revision processes are admitted, in which states with identical truth content but different origins have different successors, the origin degree of freedom then acting as a minimal form of cognitive memory. We conclude by discussing the possible relationship between the model and more general deliberative processes.
In human consciousness perceptions are distinct or atomistic events despite being perceived by an apparently undivided inner observer. This paper applies both classical (Boolean) and quantum logic to analysis of the Liar paradox which is taken as a typical example of a self-referential negation in the perception space of an undivided observer. The conception of self-referential paradoxes is a unique ability of the human mind still lacking an explanation on the basis of logic. It will be shown that both classical and quantum logics fail to resolve the paradox because of the particle-like (atomistic) nature of physical events in the moments of perception. The atomistic reality is the only one observed in human consciousness even if, as has been claimed by a number of authors, consciousness arises from a quantum mechanical superposition. However, a computational solution of the paradox can be achieved by embedding quantum registers in a fractal space. The truth values, or q-bit\neigenstates, of the separate parts of the paradox correspond to sub-spaces of the fractal. Since a fractal contains the whole in each part a self-referential negation emerges as the experienced concept in each sub-space. Fractal quantum computation is realized by a cNOT-operation with percolation of the control q-bits into sub-spaces of a nascent fractal. In other words, atomism cannot cope with a self-referential paradox but quantum logic gates embedded in a fractal structure can cope. Remarkably, it will be shown that a mind operating on these algorithmic principles will not inevitably be limited by Turing's halting theorem. I suggest a physical mechanism that can deal with our experience of self-referential paradox. Because it is also shown that this cannot be achieved by any previously suggested classical or quantum mechanical operation, the newly proposed mechanism provides a better model than others for an important aspect of the structure of our minds.\n
Logical paradoxes and inconsistent information pose deep challenges in epistemology and the philosophy of logic. Classical systems typically handle contradictions only through external checks or by altering the logical framework, as in Tarski's hierarchy or paraconsistent logics. We propose a novel approach: a quantum circuit architecture that intrinsically enforces logical consistency during its unitary evolution. By encoding self-referential or contradictory propositions into a quantum state, the circuit uses interference effects to suppress inconsistent outcomes while preserving coherent ones. We demonstrate this with the Liar Paradox ("This statement is false"), showing that the quantum model naturally stabilizes truth values that would be paradoxical classically. The framework builds on orthomodular quantum logic treating logical propositions as subspace projectors and connects to belief revision and cognitive modeling by providing a physical mechanism for coherence restoration in epistemic states. This work bridges formal logic and quantum computation, suggesting that consistency can be embedded as a structural property of reasoning systems rather than imposed externally.
Literary interpretation changes as readers encounter new passages, answer interpretive questions, and revisit earlier material in light of later context. Quantum cognition provides a probability calculus for judgments whose probabilities depend on context and probe order, but the use of Hilbert-space notation alone does not establish nonclassical behavior. This article develops Quantum Semantic Dynamics (QSD) as a constrained quantum-probabilistic model and distinguishes three claim levels: quantum-inspired representation, quantum-probabilistic structure, and intrinsic nonclassicality. The present paper advances only the second. The confirmatory implementation restricts the local semantic state to a two-dimensional space, fixes probe geometry in an independent pilot sample, reconstructs a population-level 2 × 2 density state from three linearly independent binary probe bases, and gives the off-diagonal state components an operational meaning through cross-basis judgments. Contextual evolution is restricted to a one-parameter completely positive trace-preserving contraction toward a preregistered context-target state rather than an arbitrary latent transition. Two probes form the randomized A → B/B → A sequence used for the quantum question-order (QQ) diagnostic; the third is measured in an independent arm for state identification. The paper derives the QQ equality, states its assumptions, and adds an uncertainty-style consistency constraint for the frozen noncommuting probe geometry. QSD is compared with explicitly defined classical comparator classes ranging from a static commutative model to matched sequential latent-state models. Compatibility with QQ or the Hilbert representation is not treated as proof of intrinsic nonclassicality; such a claim would require a separate contextuality analysis against an explicitly defined classical representability criterion. The resulting architecture makes the mathematical status, empirical burden, and rejection conditions of QSD explicit before data collection.
Quantum probability theory (QPT) has provided a novel, rich mathematical framework for cognitive modelling, especially for situations which appear paradoxical from classical perspectives. This work concerns the dynamical aspects of QPT, as relevant to cognitive modelling. We aspire to shed light on how the mind's driving potentials (encoded in Hamiltonian and Lindbladian operators) impact the evolution of a mental state. Some existing QPT cognitive models do employ dynamical aspects when considering how a mental state changes with time, but it is often the case that several simplifying assumptions are introduced. What kind of modelling flexibility does QPT dynamics offer without any simplifying assumptions and is it likely that such flexibility will be relevant in cognitive modelling? We consider a series of nested QPT dynamical models, constructed with a view to accommodate results from a simple, hypothetical experimental paradigm on decision-making. We consider Hamiltonians more complex than the ones which have traditionally been employed with a view to explore the putative explanatory value of this additional complexity. We then proceed to compare simple models with extensions regarding both the initial state (e.g. a mixed state with a specific orthogonal decomposition; a general mixed state) and the dynamics (by introducing Hamiltonians which destroy the separability of the initial structure and by considering an open-system extension). We illustrate the relations between these models mathematically and numerically.This article is part of the themed issue 'Second quantum revolution: foundational questions'.
# Abstract — Complex Structure from a Non‑Split Fine Step, and Born Weights as Operational Closure We present a compact extension of the ε‑Projection (“time-as-leakage”) framework that targets a separate foundational “why”: why quantum theory is naturally formulated with complex amplitudes, and what minimal additional assumption fixes the Born weights once a complex structure is present. The starting point is the non‑split cyclic clock architecture (e.g., a
Z48 lift with kernel
Z24), whose fine generator implements an unavoidable toggle+carry action on labels
K=2k+p. After Fourier transforming along the kernel direction
k, each momentum sector reduces to an explicit
2×2 unitary block
g(q) satisfying the exact projective identity
g(q)2=eiq1. This makes the phase degree of freedom structural rather than decorative: the minimal faithful representation requires a rotation plane with an operator
J obeying
J2=−1, i.e., a complex structure (equivalently, complex amplitudes or a real 2D formulation with
SO(2)≅U(1) phase rotations). We then give a worked IR/coarse-graining interpretation showing how repeated discrete fine steps admit a continuous, norm-preserving evolution of the form
U(t)≈e−JHefft (or
e−iHefft in complex notation), with
Heff self-adjoint. Complex structure alone does not yet specify probabilities. To close that gap at the “weights” level (without claiming to solve the full measurement problem), we state a minimal operational assumption: noncontextual additivity of probabilities over mutually exclusive outcomes represented by projectors (or, more generally, POVM effects). Under standard theorems (Gleason-type results, with POVM extensions covering the two-dimensional case), this assumption forces probabilities to take the trace form
p(P)=Tr(ρP), yielding the Born weights
p(P)=∥Pψ∥2 for pure states. The combined message is that (i) projective phase structure and a complex
J arise naturally from the non‑split fine-step clock, and (ii) once one adopts the standard operational identification of “events” with subspaces/effects and requires consistent noncontextual additivity, the Born weights follow as the unique probability calculus. Open work is then cleanly separated: deriving measurement dynamics (records, effective irreversibility) from the projection/open-system sector rather than postulating it.
In human consciousness perceptions are distinct or atomistic events despite being perceived by an apparently undivided inner observer. This paper applies both classical (Boolean) and quantum logic to analysis of the Liar paradox which is taken as a typical example of a self-referential negation in the perception space of an undivided observer. The conception of self-referential paradoxes is a unique ability of the human mind still lacking an explanation on the basis of logic. It will be shown that both classical and quantum logics fail to resolve the paradox because of the particle-like (atomistic) nature of physical events in the moments of perception. I suggest a physical mechanism that can deal with our experience of self-referential paradox. Because it is also shown that this cannot be achieved by any previously suggested classical or quantum mechanical operation, the newly proposed mechanism provides a better model than others for an important aspect of the structure of our minds.