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What the liar paradox can reveal about the structure of our minds

Erhard Bieberich

Published Jan 1, 1999• 0 citations
OpenAlex

Abstract

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

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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.

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In 1929 Szilard pointed out that the physics of the observer may play a role in the analysis of experiments. The same year, Bohr pointed out that complementarity appears to arise naturally in psychology where both the objects of perception and the perceiving subject belong to 'our mental content'. Here we argue that the formalism of quantum theory can be derived from two related intuitive principles: (i) inference is a physical process performed by physical systems, observers, which are part of the experimental setup---this implies non-commutativity and imaginary-time quantum mechanics; (ii) experiments must be described from a first-person perspective---this leads to self-reference, complementarity, and real-time quantum dynamics. This approach sheds new light on the foundations of quantum theory and suggests fundamental equations in physics are typically of second order due to the physical nature of the observer. It also suggests some experimental conjectures: (i) the quantum of action could be understood as the result of the additional energy required to transition from unconscious to conscious perception; (ii) humans can observe a single photon of visible light; (iii) self-aware systems and the neural correlates of the self should be composed of two complementary sub-systems, much like the DNA molecule is composed of two strands---this may help explain the double-hemisphere architecture of the brain. Moreover, this approach may help bridge the gap between science and human experience. We discuss the potential implications of these ideas for the modern research programs on consciousness and contemplative science. As side results: (i) we show that message-passing algorithms and stochastic processes can be written in a quantum-like manner; (ii) we provide evidence that non-stoquasticity, a quantum computational resource, may be related to non-equilibrium phenomena.

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Modeling observers as physical systems representing the world from within: Quantum theory as a physical and self-referential theory of inference

John Realpe-Gómez

In 1929 Szilard pointed out that the physics of the observer may play a role in the analysis of experiments. The same year, Bohr pointed out that complementarity appears to arise naturally in psychology where both the objects of perception and the perceiving subject belong to 'our mental content'. Here we argue that the formalism of quantum theory can be derived from two related intuitive principles: (i) inference is a physical process performed by physical systems, observers, which are part of the experimental setup---this implies non-commutativity and imaginary-time quantum mechanics; (ii) experiments must be described from a first-person perspective---this leads to self-reference, complementarity, and real-time quantum dynamics. This approach sheds new light on the foundations of quantum theory and suggests fundamental equations in physics are typically of second order due to the physical nature of the observer. It also suggests some experimental conjectures: (i) the quantum of action could be understood as the result of the additional energy required to transition from unconscious to conscious perception; (ii) humans can observe a single photon of visible light; (iii) self-aware systems and the neural correlates of the self should be composed of two complementary sub-systems, much like the DNA molecule is composed of two strands---this may help explain the double-hemisphere architecture of the brain. Moreover, this approach may help bridge the gap between science and human experience. We discuss the potential implications of these ideas for the modern research programs on consciousness and contemplative science. As side results: (i) we show that message-passing algorithms and stochastic processes can be written in a quantum-like manner; (ii) we provide evidence that non-stoquasticity, a quantum computational resource, may be related to non-equilibrium phenomena.

arXiv:1705.04307•May 11, 2017•4 citations
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
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.
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.
Prevailing theories of consciousness may be characterized as either a physicalist view of mind with material building blocks that grow in complexity unto an emergent conscious experience, or as a dualistic model in which mind-body interaction is taken as the interface of conscious intent and unconscious bodily processing. Roger Penrose supports a model of consciousness that goes beyond dualism by adding a third domain [19]. The Three World model describes interconnected yet independent aspects of consciousness: Physical, Mental & Platonic. These three worlds are grounded in the three axioms of quantum mechanics: measurement, superposition and entanglement. The Mental World corresponds to the superposition principle in which all possible future realities are superposed as potentials before a choice is made. The superposition is analogous to the choices we make everyday. In the Physical World, the measurement principles states that the quantum system must collapse the superposed possibilities into a single actuality. The most peculiar phenomenon in quantum mechanics is entanglement. Quantum systems may be entangled in a timeless and spaceless way such that they will still be connected despite separation in space or time. The Platonic World is akin to entanglement, because mathematics and conceptual forms are unchanging regardless of space or time. Finally, a new model called Fractal Trialism is proposed which describes how there is a nested trialism within each of the three worlds in order to elaborate their interconnectedness. This model describes digital computers, quantum computers and shared experience.
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 1929 Szilard pointed out that the physics of the observer may play a role in the analysis of experiments. The same year, Bohr pointed out that complementarity appears to arise naturally in psychology where both the objects of perception and the perceiving subject belong to 'our mental content'. Here we argue that the formalism of quantum theory can be derived from two related intuitive principles: (i) inference is a physical process performed by physical systems, observers, which are part of the experimental setup---this implies non-commutativity and imaginary-time quantum mechanics; (ii) experiments must be described from a first-person perspective---this leads to self-reference, complementarity, and real-time quantum dynamics. This approach sheds new light on the foundations of quantum theory and suggests fundamental equations in physics are typically of second order due to the physical nature of the observer. It also suggests some experimental conjectures: (i) the quantum of action could be understood as the result of the additional energy required to transition from unconscious to conscious perception; (ii) humans can observe a single photon of visible light; (iii) self-aware systems and the neural correlates of the self should be composed of two complementary sub-systems, much like the DNA molecule is composed of two strands---this may help explain the double-hemisphere architecture of the brain. Moreover, this approach may help bridge the gap between science and human experience. We discuss the potential implications of these ideas for the modern research programs on consciousness and contemplative science. As side results: (i) we show that message-passing algorithms and stochastic processes can be written in a quantum-like manner; (ii) we provide evidence that non-stoquasticity, a quantum computational resource, may be related to non-equilibrium phenomena.
In 1929 Szilard pointed out that the physics of the observer may play a role in the analysis of experiments. The same year, Bohr pointed out that complementarity appears to arise naturally in psychology where both the objects of perception and the perceiving subject belong to 'our mental content'. Here we argue that the formalism of quantum theory can be derived from two related intuitive principles: (i) inference is a physical process performed by physical systems, observers, which are part of the experimental setup---this implies non-commutativity and imaginary-time quantum mechanics; (ii) experiments must be described from a first-person perspective---this leads to self-reference, complementarity, and real-time quantum dynamics. This approach sheds new light on the foundations of quantum theory and suggests fundamental equations in physics are typically of second order due to the physical nature of the observer. It also suggests some experimental conjectures: (i) the quantum of action could be understood as the result of the additional energy required to transition from unconscious to conscious perception; (ii) humans can observe a single photon of visible light; (iii) self-aware systems and the neural correlates of the self should be composed of two complementary sub-systems, much like the DNA molecule is composed of two strands---this may help explain the double-hemisphere architecture of the brain. Moreover, this approach may help bridge the gap between science and human experience. We discuss the potential implications of these ideas for the modern research programs on consciousness and contemplative science. As side results: (i) we show that message-passing algorithms and stochastic processes can be written in a quantum-like manner; (ii) we provide evidence that non-stoquasticity, a quantum computational resource, may be related to non-equilibrium phenomena.