Abstract
Reasonable quantification of uncertainty is a major issue of cognitive infocommunications, and logic is a backbone for successful communication. Here, an axiomatic approach to quantum logic, which highlights similarity to and differences to classical logic, is presented. The axiomatic method ensures that applications are not restricted to quantum physics. Based on this, algorithms are developed that assign to an incoming signal a similarity measure to a pattern generated by a set of training signals.
Highlights
Reasonable quantification of uncertainty is a major issue of cognitive infocommunications, and logic is a backbone for successful communication
This paper describes a mathematically rigorous pathway from classical logic to mathematical models that enable a quantification of uncertainty
Such models had been developed and studied especially in the context of quantum physics, where a link to probability is given by Born’s postulate. Both classical logic with and, or, and negation, and the mathematics of quantum mechanics are in the focus of this article
Summary
Reasonable quantification of uncertainty is a major issue of cognitive infocommunications, and logic is a backbone for successful communication. This paper describes a mathematically rigorous pathway from classical logic to mathematical models that enable a quantification of uncertainty. Such models had been developed and studied especially in the context of quantum physics, where a link to probability is given by Born’s postulate. Both classical logic with and, or, and negation, and the mathematics of quantum mechanics are in the focus of this article. The final section of this article describes how to use GramSchmidt processes for representing logical operations by dealing with generating families and constructing orthonormal families which span linear subspaces corresponding to logical expressions
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