In this paper, we are dealing with the fundamental concepts of Bi-logic proposed by Chilean psychoanalyst Ignacio Matte Blanco in the context of quantum logic, founded by Gareth Birkhoff and John Von Neumann. The main purpose of this paper is to present how a quantum-logical model, represented by the lattice of a closed subspace of Hilbert space, can be used as a computational framework for concepts that are originally described by Sigmund Freud as the fundamental properties of the unconscious psyche.
In this paper, we are dealing with the fundamental concepts of Bi-logic proposed by Chilean psychoanalyst Ignacio Matte Blanco in the context of quantum logic, founded by Gareth Birkhoff and John Von Neumann. The main purpose of this paper is to present how a quantum-logical model, represented by the lattice of a closed subspace of Hilbert space, can be used as a computational framework for concepts that are originally described by Sigmund Freud as the fundamental properties of the unconscious psyche.
A crucial issue both in cognitive and psychoanalytical theories deals with the origin of mental representations. In order to explore this issue, the paper analyzes a pre-logical setting, by considering a formalized approach to the foundations of psychoanalysis in logic, interpreting and integrating the views by Freud, Matte Blanco, Klein and Bion. The formalized approach derives from a quantum model of spin states. A representation of the spin state of a particle in first order logic is abstracted to get a modality interpretable as an abstract projector. The last can be decomposed into a positive, negative and irreal component. The irreal component cannot emerge and, in logic, is absorbed by the two others, giving rise to logical duality. Due to its treatment of undefiniteness and coherence, the paper is meant to contribute to quantum cognition, in its particular sense of affective quantum cognition.
In the present paper, we aim to develop a formal quantum logic theory of the interplay between conscious and unconscious processes of the human mind, a goal that has already been envisaged in quantum cognition; in doing so, we will show how the interplay between formal language and metalanguage allows for characterizing pure quantum states as infinite singletons: in the case of the spin observable, we obtain an equation defining a modality that is then re-interpreted as an abstract projection operator. By including a temporal parameter in the equations and by defining a modal negative operator, we derive an intuitionistic-like negation, for which the non-contradiction law is seen as an equivalent of the quantum uncertainty. Building on the psychoanalytic theory of Bi-Logic by Matte Blanco, we use modalities in interpreting the emergence of conscious representations from an unconscious one, and we demonstrate that this description fits well with Freud's view of the role of negation in mental processes. Psychoanalysis, where affect plays a prominent role in shaping not only conscious, but also unconscious representations, is therefore seen as a suitable model to expand the domain of quantum cognition to the broader field of
§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.
By quoting extensively from unpublished letters written by John von Neumann to Garret Birkhoff during the preparatory phase (in 1935) of their ground-breaking 1936 paper that established quantum logic, the main steps in the thought process leading to the 1936 Birkhoff–von Neumann paper are reconstructed. The reconstruction makes it clear why Birkhoff and von Neumann rejected the notion of quantum logic as the projection lattice of an infinite dimensional complex Hilbert space and why they postulated in their 1936 paper that the quantum propositional system should be isomorphic to an abstract projective geometry. Looking at the paper now I see, that I forgot to say this, which should be said somewhere in the first §: That while common logics did apply to quantum mechanics, if the notion of simultaneous measurability is introduced as an auxiliary notion, we wished to construct a logical system, which applies directly to quantum mechanics – without any extraneous secondary notions like simultaneous measurability. And in order to have such a consequent, one-piece system of logics, we must change the classical class calculus of logics. (J. von Neumann to G. Birkhoff, November 21, 1935)
It has been shown by Birkhoff and v. Neumann (1936) and by Jauch and Piron (1963,1964,1968) that the subspaces of Hilbert space constitute an orthocomplemented quasi-modular lattice Lq, if one considers between two subspaces (elements) a, b the relation a⊆b and the operations a∩b, a∪b, a⊥. Furthermore, since the subspaces can be interpreted as quantum mechanical propositions, and since the operations ∩,∪ ,⊥ have some similarity with the logical operations ⋀ (and), ⋁ (or) and ⌝ (not), the question has been raised already by Birkhoff and v. Neumann, whether the lattice of subspaces of Hilbert space can be interpreted as a propositional calculus, sometimes called quantum logic. There are many kinds of lattices which can be interpreted as a propositional or logical calculus. A Boolean lattice LB of propositions corresponds to the calculus of classical logic and an implicative (Birkhoff, 1961) lattice Li has as a model the calculus of effective (intuitionistic) logic.