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arXiv:2512.21914

Logical Consistency as a Dynamical Invariant: A Quantum Model of Self-Reference and Paradox

Nikolaos Cheimarios, Spyridoula Cheimariou

quant-phSubmitted Dec 26, 2025• Announced Dec 29, 2025• 0 citations
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Abstract

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.

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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.
Abstract We propose a scalable quantum circuit architecture for enforcing logical consistency in systems susceptible to contradiction. The model introduces a modular structure in which each contradiction qubit is paired with a corresponding resolution qubit, and a global consistency flag qubit monitors coherence across the system. Logical constraints are embedded directly into the unitary evolution of the circuit using standard quantum gates (CNOT and Toffoli), ensuring that paradox-inducing states are structurally excluded rather than externally filtered. We formally prove that the set of logically coherent configurations corresponds to the fixed-point subspace of the circuit’s unitary operator and demonstrate that the approach generalizes to arbitrary N-qubit systems. As a concrete illustration, we implement a minimal 3-qubit system that models contradiction detection, resolution, and global logical coherence in the context of natural language ambiguity resolution. In this example, unresolved linguistic ambiguity is flagged as a logical inconsistency, while successful disambiguation preserves coherence. Simulations on both ideal quantum circuits and IBM’s quantum hardware validate that only consistent states remain invariant under the computation, while contradictory configurations are actively suppressed.
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.
The classical methods used by recursion theory and formal logic to block paradoxes do not work in quantum information theory. Since quantum information can exist as a coherent superposition of the classical ``yes'' and ``no'' states, certain tasks which are not conceivable in the classical setting can be performed in the quantum setting. Classical logical inconsistencies do not arise, since there exist fixed point states of the diagonalization operator. In particular, closed timelike curves need not be eliminated in the quantum setting, since they would not lead to any paradoxical outcome controllability. Quantum information theory can also be subjected to the treatment of inconsistent information in databases and expert systems. It is suggested that any two pieces of contradicting information are stored and processed as coherent superposition. In order to be tractable, this strategy requires quantum computation.
This report introduces a novel class of reasoning architectures, termed Quantum Circuit Reasoning Models (QCRM), which extend the concept of Variational Quantum Circuits (VQC) from energy minimization and classification tasks to structured logical inference and reasoning. We posit that fundamental quantum mechanical operations, superposition, entanglement, interference, and measurement, naturally map to essential reasoning primitives such as hypothesis branching, constraint propagation, consistency enforcement, and decision making. The resulting framework combines quantum-inspired computation with differentiable optimization, enabling reasoning to emerge as a process of amplitude evolution and interference-driven selection of self-consistent states. We develop the mathematical foundation of QCRM, define its parameterized circuit architecture, and show how logical rules can be encoded as unitary transformations over proposition-qubit states. We further formalize a training objective grounded in classical gradient descent over circuit parameters and discuss simulation-based implementations on classical hardware. Finally, we propose the Quantum Reasoning Layer (QRL) as a differentiable hybrid component for composable reasoning models applicable to scientific, biomedical, and chemical inference domains.
This paper presents an alternative approach to quantum entanglement, one that effectively resolves the logical inconsistencies without leading to logical contradictions. By addressing some of the inconsistencies within quantum mechanics, such as state superposition and non-locality, that challenge classical causal explanations, our method is constructed on the principles of paraconsistent logic. Our aim is to develop a para-consistent framework that supports the features of quantum mechanics while remaining faithful to its fundamental principles. In this pursuit, we scrutinize the philosophical and mathematical foundations of quantum mechanics in relation to classical logic systems. This method is designed to untangle theoretical puzzle spaces and promote coherence in the discussion of quantum theory. Ultimately, our approach offers a potential solution for interpreting quantum mechanics in a more coherent manner.
We present a formulation of quantum circuits where the focus is set on whether a given circuit (made of unitary operators and projective measurements with definite outcomes) does reflect an actually realizable physical experiment. In order to do this, we introduce verifications statements which are purely epistemic assertions indicating whether a outcome is possible at some point and develop our formalism which, in the end, consists in a set of logical rules about verification statements. Finally, we argue that our formalism provides a Lorentz-invariant realistic formulation of quantum circuits and illustrate this by considering a circuit corresponding to Hardy's paradox and showing how our formalism prevents making contradictory assertions regarding our knowledge about the circuit.