Quantum theory is logically inevitable

Abstract

General relativity required the abandonment of Euclidean geometry. Here we show that quantum theory requires the abandonment of classical logic. We show that the Hilbert space representation of quantum theory is logically inevitable. There are no fundamental principles of quantum theory. We find an inevitable generalization of classical logic. The conjunction can take negative values, and reduces to the classical conjunction for order invariant measurements. The expectation of the conjunction is a generalized joint probability that can take negative values. A commutative conjunction leads to the Hilbert space formalism of quantum theory. Quantum theory applies both to microscopic and macroscopic systems. Classicality is represented by non-negativity of the generalized joint probability. We illustrate this by applying the logic to explain puzzling results in quantum cognition.