This chapter is offered as a contribution to the logic of down below. We attempt to demonstrate that the nature of human agency necessitates that there actually be such a logic. The ensuing sections develop the suggestion that cognition down below has a structure strikingly similar to the physical structure of quantum states. In its general form, this is not an idea that originates with the present authors. It is known that there exist mathematical models from the cognitive science of cognition down below that have certain formal similarities to quantum mechanics. We want to take this idea seriously. We will propose that the subspaces of von Neumann-Birkhoff lattices are too crisp for modelling requisite cognitive aspects in relation to subsymbolic logic. Instead, we adopt an approach which relies on projections into nonorthogonal density states. The projection operator is motivated from cues which probe human memory.
This chapter is offered as a contribution to the logic of down below. We attempt to demonstrate that the nature of human agency necessitates that there actually be such a logic. The ensuing sections develop the suggestion that cognition down below has a structure strikingly similar to the physical structure of quantum states. In its general form, this is not an idea that originates with the present authors. It is known that there exist mathematical models from the cognitive science of cognition down below that have certain formal similarities to quantum mechanics. We want to take this idea seriously. We will propose that the subspaces of von Neumann-Birkhoff lattices are too crisp for modelling requisite cognitive aspects in relation to subsymbolic logic. Instead, we adopt an approach which relies on projections into nonorthogonal density states. The projection operator is motivated from cues which probe human memory.
Abstract Quantum logics are non-classical logics defined from the mathematical formalism of quantum mechanics. While they are conventionally used to model inferential processes in physics, their scope of application is potentially much broader. We argue that quantum logics can serve as a framework to model human cognition, as their semantics seem able to capture not only how people make inferences about quantum mechanics, but also how they reason in general. We begin by defining quantum logics from an algebraic perspective in a classical first-order setting. Next, we present findings from cognitive science that suggest these logics are apt to characterize human reasoning. We then consider how such a connection between quantum logics and cognition contributes to longstanding philosophical debates about the epistemological status of logic and the problem of adoption. Finally, we discuss how cognitive applications of quantum logics could advance our understanding of human psychology and even quantum foundations.
Experiments in cognitive science and decision theory show that the ways in which people combine concepts and make decisions cannot be described by classical logic and probability theory. This has serious implications for applied disciplines such as information retrieval, artificial intelligence and robotics. Inspired by a mathematical formalism that generalizes quantum mechanics the authors have constructed a contextual framework for both concept representation and decision making, together with quantum models that are in strong alignment with experimental data. The results can be interpreted by assuming the existence in human thought of a double-layered structure, a 'classical logical thought' and a 'quantum conceptual thought', the latter being responsible of the above paradoxes and nonclassical effects. The presence of a quantum structure in cognition is relevant, for it shows that quantum mechanics provides not only a useful modeling tool for experimental data but also supplies a structural model for human and artificial thought processes. This approach has strong connections with theories formalizing meaning, such as semantic analysis, and has also a deep impact on computer science, information retrieval and artificial intelligence. More specifically, the links with information retrieval are discussed in this paper.
§1. Introduction. Our understanding of Nature comes in layers, so should the development of logic. Classic logic is an indispensable part of our knowledge, and its interactions with computer science have recently dramatically changed our life. A new layer of logic has been developing ever since the discovery of quantum mechanics. G. D. Birkhoff and von Neumann introduced quantum logic in a seminal paper in 1936 [1]. But the definition of quantum logic varies among authors (see [2]). How to capture the logic structure inherent in quantum mechanics is very interesting and challenging. Given the close connection between classical logic and theoretical computer science as exemplified by the coincidence of computable functions through Turing machines, recursive function theory, and λ-calculus, we are interested in how to gain some insights about quantum logic from quantum computing. In this note we make some observations about quantum logic as motivated by quantum computing (see [5]) and hope more people will explore this connection. The quantum logic as envisioned by Birkhoff and von Neumann is based on the lattice of closed subspaces of a Hilbert space, usually an infinite dimensional one. The quantum logic of a fixed Hilbert space ℍ in this note is the variety of all the true equations with finitely many variables using the connectives meet, join and negation. Quantum computing is theoretically based on quantum systems with finite dimensional Hilbert spaces, especially the states space of a qubit ℂ2. (Actually the qubit is merely a convenience.
. We put forward a new take on the logic of quantum mechanics, following Schrödinger's point of view that it is composition which makes quantum theory what it is, rather than its particular propositional structure due to the existence of superpositions, as proposed by Birkhoff and von Neumann. This gives rise to an intrinsically quantitative kind of logic, which truly deserves the name ‘logic’ in that it also models meaning in natural language, the latter being the origin of logic, that it supports automation, the most prominent practical use of logic, and that it supports probabilistic inference. The physics and the logic of quantum-ish logic. In 1932 John von Neumann formalized Quantum Mechanics in his book “Mathematische Grundlagen der Quantenmechanik”. This was effectively the official birth of the quantum mechanical formalism which until now, some 75 years later, has remained the same. Quantum theory underpins so many things in our daily lives including chemical industry, energy production and information technology, which arguably makes it the most technologically successful theory of physics ever. However, in 1935, merely three years after the birth of his brainchild, von Neumann wrote in a letter to American mathematician Garrett Birkhoff: “I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space no more.” (sic)—for more details see [73]. Soon thereafter they published a paper entitled “The Logic of Quantum Mechanics” [13]. Their ‘quantum logic’ was cast in order-theoretic terms, very much in the spirit of the then reigning algebraic view of logic, with the distributive law being replaced with a weaker (ortho)modular law. This resulted in a research community of quantum logicians [68, 71, 47, 30]. However, despite von Neumann's reputation, and the large body of research that has been produced in the area, one does not find a trace of this activity neither in the mainstream physics, mathematics, nor logic literature. Hence, 75 years later one may want to conclude that this activity was a failure. What went wrong? The mathematics of it. Let us consider the raison d'être for the Hilbert space formalism.
The authors provide an overview of the results they have obtained over the last decade on the identification of quantum structures in cognition and, more specifically, in the formalization and representation of natural concepts. They firstly discuss the quantum foundational reasons that led them to investigate the mechanisms of formation and combination of concepts in human reasoning, starting from the empirically observed deviations from classical, logical, and probabilistic structures. They then develop their quantum-theoretic perspective in Fock space which allows successful modeling of various sets of cognitive experiments collected by different scientists, including themselves. In addition, they formulate a unified explanatory hypothesis for the presence of quantum structures in cognitive processes, and discuss their recent discovery of further quantum aspects in concept combinations, namely, ‘entanglement’ and ‘indistinguishability.’ They finally illustrate the outlook for future research.
This paper examines the structural discrepancy between the mathematical formulations of physics—particularly those of quantum mechanics—and the conditions under which human beings recognize existence and establish observation. The fundamental equations of quantum mechanics do not require sensory distinctions such as whether a particle is “moving” or “at rest”; the same formalism allows both dynamic and static states. In contrast, human perceptual structures presuppose continuous change or motion for the recognition of existence and the establishment of observation. A completely unchanging entity does not readily emerge within the sensory world. Focusing on this discrepancy, this paper argues that mathematical equations describe the consistency of states rather than the conditions of existential recognition or generative processes. Furthermore, concepts such as “will” and “choice” do not belong to the third-person domain described by mathematical formalism, but instead function as first-person preconditions. Accordingly, this paper does not claim that there exist phenomena that cannot be expressed mathematically; rather, it clarifies that there are layers of reality that mathematics does not, by design, attempt to describe. This distinction does not indicate a limitation of mathematics, but instead reveals a layered structure of world-description, offering a new perspective for connecting physics, cognitive science, and philosophy.