Bell ’ s inequality is violated in classical systems as well as quantum systems by Shiro Ishikawa

Bell’s inequality is usually considered to belong to mathematics and not quantum theory. We think that this complicates understanding of Bell’s theorem. Thus in this paper, contrary to Bell’s spirit (which inherits Einstein’s spirit), we try to discuss Bell’s inequality in the framework of quantum theory with the linguistic Copenhagen interpretation. And we clarify that whether or not Bell’s inequality holds does not depend on whether classical systems or quantum systems, but depend on whether a kind of combined measurements exist or not. And further we assert that Bell’s argument (based on Einstein’s spirit) should be regarded as a philosophical representation of our scientific argument ( based on the linguistic Copenhagen interpretation). Thus, we conclude that our inequality proposed in this paper is the true Bell’s inequality. Key phrases: EPR-paradox, Bell’s inequality, Classical systems, Linguistic Copenhagen interpretation, Quantum Language 1 Review: Quantum language (= Measurement theory ); Quantum language has two formulations, i.e., the C∗-algebraic formulation andW ∗-algebraic formulation, where the former is elementary, handy and somewhat rough. In refs. [20, 21], we discussed Bell’s inequality in the W ∗-algebraic formulation, which is exact, powerful and so mathematically difficult. In this paper, we discussed Bell’s inequality in the C∗-algebraic formulation, which may be expected to be easy to understand. Following refs. [12, 13, 21], we shall review quantum language ( i.e., the linguistic Copenhagen interpretation of quantum mechanics, or measurement theory ), which has the following form: (A) Quantum langage (= measurement theory) = measurement (Axiom 1) + causality (Axiom 2) + linguistic ( Copenhagen ) interpretation (how to use Axioms 1 and 2) We assert that the location of quantum language in the history of world-description is as follows. Parmenides Socrates 0 ©:Greek philosophy Plato Aristotle Schola−−−−−→ sticism 1 © −−→ (monism) Newton (realism) 2 © → relativity theory −−−−−−−−−−→ 3 © → quantum mechanics −−−−−−−−−−→ 4 © −−→ (dualism) Descartes Locke,... Kant (idealism) 6 © −−−−−→ (linguistic view) linguistic philosophy language −−−−−−−−→ (B2) 8 © language −−−−−−→ (B1) 7 ©  5 © −−→ (unsolved) theory of everything (quantum phys.)  10 © −−→ (=MT) quantum language (language) Figure 1: The history of the world-description statistics system theory language −−−−−−−−−−→ (B3) 9 © the linguistic world view ( dualism, idealism ) the realistic world view (monism, realism) 1 KSTS/RR-17/006 October 16, 2017 (Revised December 18, 2017)