Diederik Aerts

We analyze the meaning of the violation of the marginal probability law for situations of correlation measurements where entanglement is identified. We show that for quantum theory applied to the cognitive realm such a violation does not lead to the type of problems commonly believed to occur in situations of quantum theory applied to the physical realm. We briefly situate our quantum approach for modeling concepts and their combinations with respect to the notions of “extension” and “intension” in theories of meaning, and in existing concept theories

Excerpt: In 1995, the Leo Apostel Centre in Brussels, Belgium, organised an international conference called ``Einstein meets Magritte''. Nobel prize winner Ilya Prigogine held the opening lecture at the conference, and Heinz von Foerster's lecture was scheduled last... Heinz von Foerster was enchanted by the conference theme and -- in the spirit of surrealist Belgian painter René Magritte -- had chosen an appropriate title for his talk: ``Ceci n'est pas Albert Einstein''. ... [H]e was delighted to grant the organisers the following interview, in which he tells us about an even longer journey -- that of his remarkable life and scientific career.

We put forward the hypothesis that there exist three basic attitudes towards inconsistencies within world views: (1) The inconsistency is tolerated temporarily and is viewed as an expression of a temporary lack of knowledge due to an incomplete or wrong theory. The resolution of the inconsistency is believed to be inherent to the improvement of the theory. This improvement ultimately resolves the contradiction and therefore we call this attitude the ‘regularising’ attitude; (2) The inconsistency is tolerated and both contradicting elements in the theory are retained. This attitude integrates the inconsistency and leads to a paraconsistent calculus; therefore we will call it the paraconsistent attitude. (3) In the third attitude, both elements of inconsistency are considered to be false and the ‘real situation’ is considered something different that can not be described by the theory constructively. This indicates the incompleteness of the theory, and leads us to a paracomplete calculus; therefore we call it the paracomplete attitude. We illustrate these three attitudes by means of two ‘paradoxical’ situations in quantum mechanics, the wave-particle duality and the situation of non locality.

Einstein Meets Magritte: An Interdisciplinary Reflection presents insights of the renowned key speakers of the interdisciplinary Einstein meets Magritte conference (1995, Brussels Free University). The contributions elaborate on fundamental questions of science, with regard to the contemporary world, and push beyond the borders of traditional approaches. All of the articles in this volume address this fundamental theme, but somewhere along the road the volume expanded to become much more than a mere expression of the conference's dynamics. The articles not only deal with several scientific disciplines, they also confront these fields with the full spectrum of contemporary life, and become new science. As such, this volume presents a state-of-the-art reflection of science in the world today, in all its diversity. The contributions are accessible to a large audience of scientists, students, educators, and everyone who wants to keep up with science today.

We propose a theory for modeling concepts that uses the state-context-property theory (SCOP), a generalization of the quantum formalism, whose basic notions are states, contexts and properties. This theory enables us to incorporate context into the mathematical structure used to describe a concept, and thereby model how context influences the typicality of a single exemplar and the applicability of a single property of a concept. We introduce the notion `state of a concept' to account for this contextual influence, and show that the structure of the set of contexts and of the set of properties of a concept is a complete orthocomplemented lattice. The structural study in this article is a preparation for a numerical mathematical theory of concepts in the Hilbert space of quantum mechanics that allows the description of the combination of concepts.

We present a model that allows one to build structures that evolve continuously from classical to quantum, and we study the intermediate situations, giving rise to structures that are neither classical nor quantum. We construct the closure structure corresponding to the collection of eigenstate sets of these intermediate situations, and demonstrate how the superposition principle disappears during the transition from quantum to classical. We investigate the validity of the axioms of quantum mechanics for the intermediate situations

The quantum logical `or' is analyzed from a physical perspective. We show that it is the existence of EPR-like correlation states for the quantum mechanical entity under consideration that make it nonequivalent to the classical situation. Specifically, the presence of potentiality in these correlation states gives rise to the quantum deviation from the classical logical `or'. We show how this arises not only in the microworld, but also in macroscopic situations where EPR-like correlation states are present. We investigate how application of this analysis to concepts could alleviate some well known problems in cognitive science.