Qualitative representations of mereological structures in Artificial Intelligence are generally based on a set of mutually exclusive and collectively exhaustive crisp (RCC5) relations. The amount of information carried by each element in this set of (crisp) relations is determined by the number of bits (N) required to distinguish all members of the set. This paper posits that mereological vagueness can arise if mereological relations are limited in the amount of information they can carry. Consequently, a comprehensive set of vague mereological relations is introduced, where each relation in this set can only carry 0 < n < N bits of information. In this framework, mereological information is treated as quantum information, with quantum bits (qubits) distinguishing relations subject to ontic vagueness. The main contribution is the thorough examination of mereological relations within a 3 qubit quantum system. Quantum bits are not represented as vectors in a Hilbert space but as cells in (granular) partitions. This paper is the companion to the author's quantized-mereology framework and develops its entangled-state layer: the combination of three simultaneous elementary questions, the families of entangled information states, the probability distribution for possible observations and the dynamics induced by sequences of non-simultaneous questions.
Qualitative representations of mereological structures in Artificial Intelligence are generally based on a set of mutually exclusive and collectively exhaustive crisp (RCC5) relations. The amount of information carried by each element in this set of (crisp) relations is determined by the number of bits (N) required to distinguish all members of the set. This paper posits that mereological vagueness can arise if mereological relations are limited in the amount of information they can carry. Consequently, a comprehensive set of vague mereological relations is introduced, where each relation in this set can only carry 0 < n < N bits of information. In this framework, mereological information is treated as quantum information, with quantum bits (qubits) distinguishing relations subject to ontic vagueness. The main contribution is the thorough examination of mereological relations within a 3 qubit quantum system. Quantum bits are not represented as vectors in a Hilbert space but as cells in (granular) partitions. This paper is the companion to the author's quantized-mereology framework and develops its entangled-state layer: the combination of three simultaneous elementary questions, the families of entangled information states, the probability distribution for possible observations and the dynamics induced by sequences of non-simultaneous questions.
This paper investigates quantized mereology, which merges quantum information theory with classical mereology, the formal analysis of the part-whole relationship. It emphasizes that information about crisp mereological relations is captured by classical bits, whereas mereological vagueness involves the use of quantum bits, or qubits, to capture mereological information. Transitioning from classical bits to qubits offers a new understanding of part-whole relations among entities that are subject to mereological vagueness. This paper methodically studies the quantized mereology that arises from the classical RCC5 formalism, one of the most widely studied formalisms in Qualitative Reasoning (QR), a subfield of Artificial Intelligence. The argument is made that this method can be extended to quantize more sophisticated QR formalisms, mirroring how strategies are used in physics to transition from classical to quantum systems. The study explores: (i) the conversion of bits to qubits while understanding that these (qu)bits convey mereological information; (ii) the representation of qubit states via set partitions, acknowledging that set theory aligns more closely with classical logic than with the complex vector spaces employed in quantum physics; (iii) demonstrating that classical QR formalisms can be seen as special cases within quantized formalisms; and (iv) showing that qubits more effectively encapsulate information about vague phenomena.
The successful application of quantum probability in cognitive science has been persistently challenged by the foundational 'why quantum?' question. In this paper, a novel answer is proposed through the development of an ontological grounding for the quantum-like structures observed in human judgment. It is argued that these cognitive structures may be a direct reflection of an objective, information-theoretic vagueness inherent in the macroscopic world. To formalize this, 'Quantized Mereology' is developed - a framework inspired by Rovelli's Relational Quantum Mechanics (RQM), where determinate mereological facts about vague entities like forests and clouds are established relationally through an act of judgment. The framework uses a real-valued Hilbert space as a precise object language, which is given meaning by a partition-theoretic meta-language. A key contribution is a defense of the real-valued structure, in which it is argued that the physical constraints necessitating complex numbers in microphysics constitute a category error when applied to the conceptual systems modeled herein. The result is a generative model for objective vagueness that provides a candidate ontological foundation for quantum cognition.
Abstract The emerging field of quantum mereology considers part-whole relations in quantum systems. Entangled quantum systems pose a peculiar problem in the field, since their total states are not reducible to that of their parts. While there exist several established proposals for modelling entangled systems, like monistic holism or relational holism, there is considerable unclarity, which further positions are available. Using the lambda operator and plural logic as formal tools, we review and develop conceivable models and evaluate their consistency and distinctness. The main result is an exhaustive taxonomy of six distinct and precise models that both provide information about the mereological features as well as about the entangled property. The taxonomy is well-suited to serve as the basis for future systematic investigations.
Quantum information is defined by applying the concepts of ordinary (Shannon) information theory to a quantum sample space consisting of a single framework or consistent family. A classical analogy for a spin-half particle and other arguments show that the infinite amount of information needed to specify a precise vector in its Hilbert space is not a measure of the information carried by a quantum entity with a d-dimensional Hilbert space; the latter is, instead, bounded by log2d bits (one bit per qubit). The two bits of information transmitted in dense coding are located not in one but in the correlation between two qubits, consistent with this bound. A quantum channel can be thought of as a structure or collection of frameworks, and the physical location of the information in the individual frameworks can be used to identify the location of the channel. Analysis of a quantum circuit used as a model of teleportation shows that the location of the channel depends upon which structure is employed; for ordinary teleportation it is not (contrary to Deutsch and Hayden) present in the two bits resulting from the Bell-basis measurement, but in correlations of these with a distant qubit. In neither teleportation nor dense coding does information travel backwards in time, nor is it transmitted by nonlocal (superluminal) influences. It is (tentatively) proposed that all aspects of quantum information can in principle be understood in terms of the (basically classical) behavior of information in a particular framework, along with the framework dependence of this information.
Knowledge representation standardly assumes bivalence: every entity determinately satisfies a predicate or determinately does not. Many domains (vague geographic regions, borderline biological taxa, mereologically indeterminate wholes) resist this assumption. The stabilizer formalism of quantum error correction is presented as a vector-based ontology language for classifying and reasoning about indeterminate predicates, with a dispositional dynamics: systems are represented by state vectors, predicates by Pauli operators, indeterminacy as an object-level property, not a meta-level family of precisifications. It complements, rather than replaces, the model-theoretic representation of classical first-order logic. The real (rebit) fragment is efficiently classically simulable (Gottesman-Knill), so no quantum hardware is required. Three determinacy modes (determinacy (D), gappy indeterminacy (G) and correlated indeterminacy (C)) generate, for three predicates, a complete classification hierarchy, its counts established previously, here derived as orbits of nested symmetry-group actions: 120 information kinds, grouped into 49 semantic kinds, 15 determinacy patterns, 7 information forms and 3 correlation classes. Applied to three mereological predicates, a representative subset of kinds is read explicitly, including correlated-indeterminacy patterns beyond bivalent satisfaction. Two results follow. The first is a violation-cost landscape, induced by a chosen dispositional specification, in which the ideal configuration is itself vague; its same-kind transformations are the normalizer of the unsigned landscape group in the group of local Clifford operations and predicate permutations. The second is a centralizer partition of the interaction space, deciding for each query whether it is revelatory or disturbing. The representability boundary lies between the semantic-kind and information-kind levels.
We reconstruct the explicit formalism of qubit quantum theory from elementary rules on an observer's information acquisition. Our approach is purely operational: we consider an observer O interrogating a system S with binary questions and define S's state as O's ``catalog of knowledge'' about S. From the rules we derive the state spaces for N elementary systems and show that (a) they coincide with the set of density matrices over an N-qubit Hilbert space C2N; (b) states evolve unitarily under the group PSU(2N) according to the von Neumann evolution equation; and (c) O's binary questions correspond to projective Pauli operator measurements with outcome probabilities given by the Born rule. As a by-product, this results in a propositional formulation of quantum theory. Aside from offering an informational explanation for the theory's architecture, the reconstruction also unravels previously unnoticed structural insights. We show that, in a derived quadratic information measure, (d) qubits satisfy inequalities which bound the information content in any set of mutually complementary questions to 1 bit; and (e) maximal sets of mutually complementary questions for one and two qubits must carry precisely 1 bit of information in pure states. The latter relations constitute conserved informational charges which define the unitary groups and, together with their conservation conditions, the sets of pure quantum states. These results highlight information as a ``charge of quantum theory'' and the benefits of this informational approach. This work emphasizes the sufficiency of restricting to an observer's information to reconstruct the theory and completes the quantum reconstruction initiated in a companion paper (P. H\"ohn, arXiv:1412.8323).