Based on ideas of quantum theory of open systems and psychological dual system theory we propose two novel versions of Non-Boolean logic. The first version can be interpreted in our opinion as simplified description of primitive (mythological) thinking and the second one as the toy model of everyday human reasoning in which aside from logical deduction, heuristic elements and beliefs also play the considerable role. Several arguments in favor of the interpretations proposed are adduced and discussed in the paper as well.
Based on ideas of quantum theory of open systems and psychological dual system theory we propose two novel versions of Non-Boolean logic. The first version can be interpreted in our opinion as simplified description of primitive (mythological) thinking and the second one as the toy model of everyday human reasoning in which aside from logical deduction, heuristic elements and beliefs also play the considerable role. Several arguments in favor of the interpretations proposed are adduced and discussed in the paper as well.
Quantum cognition, a growing topic at the interface of quantum physics and cognitive research, provides a new viewpoint on the intricacies of human thought processes. In contrast to classic Boolean algebraic models, which have long dominated cognitive psychology, quantum cognition proposes a new framework influenced by quantum mechanics' algebraic structure. This deviation derives from the understanding that natural notions, based on prototypes, are more closely related to convex sets than to the rigid rules of Boolean algebra. Drawing on the fundamental work of luminaries such as John von Neumann, quantum cognition reveals a unique probability theory based on ortho-algebras, challenging traditional probabilistic reasoning based on Boolean algebras. This change opens up solutions to complex challenges in rationality, logical thinking, and probabilistic reasoning, ushering in a new era of cognitive modelling. Several arguments in the literature support the importance of quantum cognition in explaining cognitive phenomena.
In order to solve the problem of which is the real logic of Quantum Mechanics (QM), we suggest to study the mutual relationships between the physical theory (as a whole, that is both its experimental part and its theoretical part), its mathematics and its logic. That results to be a more general problem than what is considered by traditional studies on Quantum Logic (QL). These studies focus attention on the problem of the relationship between the experimental part of the physical theory and only its mathematics, from which logic is then derived. The new approach, that we suggest, generalises the interpretation that one of us (A. D.) successfully tested on the foundations of classical physical theories. The foundations of a theory are interpreted in terms of two options on the kind of mathematics, either classical one or constructive one, and the kind of logic, either classical logic or intuitionistic logic. It is remarkable that each theory making use of intuitionistic logic – i. e. L. Carnot’s classical mechanics, S. Carnot’s thermodynamics, Lobacevskij’s non-Euclidean geometry is organised in order to solve an universal problem (expressed by a double negated sentence), rather than in an entirely deductive way; hence, a non-classical logic does not belong exclusively to QM. From an historical and foundational analysis about QM, we conclude that in QM it is the law of double negation and not modular law (introduced by Birkhoff and von Neumann without giving physical evidence to it) which fails, i. e. which represents the true difference between Classical Logic (CL) and QL. We have looked for previous works on QL supporting our suggestion; in the past Destouches-Fevrier (1948), Fine (1974), Bell-Hallett (1981), Gauthier (1983), Adelman-Corbett (1995) already defended this thesis on the relevance of double negation law for QL. Under this light, QL results to have the same logic of a classical theory organised in a problematic manner. Hence, QM has to be referred to intuitionistic logic, in the same way every theory which is developed around a fundamental problem, expressed by a double negated sentence, does. In particular, by following the founder of intuitionism, Brouwer, we reject the identification of intuitionistic logic with an entirely deductive system; rather, the logical system of QL, as a theory, is organised in such a way to put a crucial problem to be solved. In agreement with this kind of organisation, among the several QM formulations we choose Heisenberg’s formulation, which was ignored by previous authors on QL. This formulation is aimed to solve the problem of quantum measurement, that originates the uncertainty methodological principle. Moreover, it is remarkable that it makes use of an operational mathematics (discrete matrices only). We interpret its characteristic arguing about its principal problem through double negated sentences; it results to be a moving from arguments belonging to non-classical logic expressed by a double negated sentence to the recognition of a suitable mathematics for QM formulation. In sum, QL shows how it is possible to state the relationship between QM and mathematics. This point has been almost ignored by the most traditional approaches to QL, which rather drew it from an a priori mathematics of QM (commonly, Hilbert space). As a verification of our approach, we show that even both Lobachevskij’s geometry and S. Carnot’s thermodynamics argued in order to achieve mathematical formulas expressing the content of the basic double negated sentences of respective theory. In the same way the recent book on QM by T. F. Jordan formalised his matrix mechanics. 1. Logic in classical physics and in Quantum Mechanics An historical analysis of Quantum Mechanics (QM) shows that there were at the origin of the theory two distinct formulations: Matrix Mechanics (Heisenberg, 1925) and Wave Mechanics (Schroedinger, 1926). Although Dirac subsumed these two formulations in a comprehensive mathematical framework (Hilbert space), we think that the organisation of the principles is different in Matrix Mechanics from that in Wave Mechanics, and this difference implies different logics. So we think that in order to characterise the logic of QM it is necessary to distinguish between this two formulations. In Matrix Mechanics, Heisenberg’s basic idea was to explain atomic spectra not in terms of atomic orbits (as Old Quantum Theory tried to make), but only by means of the study of experimental data; i. e., frequencies and intensities of light both emitted and absorbed by matter. Through a
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.
The non-classical nature of quantum theory has led to a number of efforts to formalize the logic of statements whose universe of discourse is the subatomic (see, for example, [3], [6], [8], and [10]).It might be hoped that ideally the fruit of such efforts would be a formalism not totally divorced from the classical logic, but incorporating or incorporated in it in such a manner as to unify the logic of the micro-and the macro-worlds.Indeed, this has been the general direction taken by these efforts, including that which we principally examine and expand upon here.*Josef Jauch, [6], pp.67-89, developed a "propositional calculus of quantum mechanics".He objected to calling a system like his a "logic of quantum mechanics"; however, our use of 'logic' here is not, I think, the use to which Jauch objected.A logic, for our present purposes, is a mathematical formalism having a reasonable interpretation as a theory of deduction; its actual application as normative of argumentation in reality is a matter of empirical judgment.This seems close to the way Jauch wished to use "propositional calculus" but more general, allowing for extension to a quantification theory of quantum mechanics, for example.Although Jauch's presentation of a propositional calculus of quantum mechanics is mathematically interesting and suggestive, there are a number of problems connected with it.It is our purpose to engage some of these problems and to present a rigorously defined formalism intended to come close to doing the job intended by Jauch for his logic.The success or lack of success of this effort as related to actual application of this formalism in the foundations of quantum theory is, I think, a matter for the physicists to decide; the empirical judgment involved in this application is not in the scope of this paper.If a formalism is to have a reasonable interpretation as a theory of deduction, it seems clear that it must include a relation which, when it holds between two elements of the formalism, may be reasonably interpreted as expressing that one of the elements 'follows from' the other, in •The author wishes to thank Robert Piziak for advice and comments in the course of research for this paper.
We discuss the modalities by which we humans should (and in fact, do) compute. That is, we investigate about the logical languages and the computational modes of human reasoning, and the corresponding physical interpretation. In this context, however, the classical world (physical, logical, and computational) does not seem sufficient to give a complete description of the Mind. In fact, the Mind accomplishes different tasks, where it exhibits, alternatively, both classical and quantum features. There are some novelties in two important issues: the long-standing debate on the mind-body relationship, and Turing’s question about a possible identification of the Mind with a computer. The introduction of a quantum metalanguage (QML) for the logic of reasoning is the most important feature to deal with both issues. As far as the first issue is considered, the QML is physically interpreted as a Dissipative Quantum Field Theory (DQFT) of the brain. The corresponding quantum object-language (QOL), which is the logic of reasoning, and is controlled by the QML, is physically interpreted as the Quantum Mechanics of qubits, that is, Quantum Computing (QC). Therefore, the Mind is both language and metalanguage, and the brain is its physical interpretation.
In a earlier work [1, 2, 3, 4] we have put forward the hypothesis that two layers of thought can be identified in the human thought process, which we have called the ‘classical logical ’ layer and the ‘quantum conceptual ’ layer. In the classical logical layer thought processes are given form by an underlying Boolean logical conceptual process, its non deterministic ‘lack of knowledge ’ version modeled by Kolmogorovian probability and in an idealized form by a stochastic process. In the quantum conceptual layer, the thought process gets form under the influence of the totality of the surrounding conceptual landscape. The key event within this layer is that different concepts figure as individual entities, even when they are combinations of other concepts. A phenomenon of conceptual emergence is taking place in the quantum conceptual layer mathematically described by superposition within the quantum formalism used to model this layer. The quantum conceptual thought process is indeterministic in essence; i.e. there is not necessarily an underlying deterministic process. It is however an equally well structured layer than the classical logical one, and we belief that it is mainly the lack of insight about the structure of this quantum conceptual layer that often leads to the idea that the thought processes described in it are mainly unstructured and correspond to intuitive or irrational thought. In the present paper we analyze more in detail the cognitive and mathematical structures of these two layers of thought. More specifically we put forward a concrete