Quantum Probability Theory and Non-Boolean Logic

Abstract

Since the very inception of quantum theory, the corresponding logic for quantum entities has attracted much attention. The logic underlying the quantum theory is shown to be non-Boolean in character. Boolean logic is a two/valued logic which is used for the description of everyday objects. Modern computers are based on this logic. The existence of an interference term for microscopic entities or quantum entities clearly indicates the existence of three-valued or non-Boolean logic. This is popularly known as quantum logic. It is mathematically shown that a set of propositions which satisfies the different axiomatic structures for the non-Boolean logic generates Hilbert space structures. The quantum probability associated with this type of quantum logic can be applied to decision-making problems in the cognitive domain. It is to be noted that, until now, no quantum mechanical framework is taken as a valid description of the anatomical structures and functions of the brain. This framework of quantum probability is very abstract and devoid of any material content. So it can be applied to any branch of knowledge like biology, social science, etc. Of course, it is necessary to understand the issue of contextualization, i.e., here, in the case of the brain.

The only acceptable point of view appears to be the one that recognizes both sides of reality—the quantitative and the qualitative, the physical and the psychical—as compatible with each other and can embrace them simultaneously.

—Pauli (1955: 208).