Against identification of contextuality with violation of the Bell inequalities: lessons from theory of randomness

Abstract: Nowadays contextuality is the hotest topic of quantum foundations and, especially, foundations of quantum information theory. This notion is characterized by the huge diversity of approaches and interpretations. One of the strongest trends in contextual research is to identify contextuality with violation of the Bell inequalities. We call this sort of contextuality {\it Bell test contextuality} (BTC). In this note, we criticize the BTC-approach. It can be compared with an attempt to identify the complex and theoretically nontrivial notion of randomness with a test for randomness (or a batch of tests, as the NIST-test). We advertize {\it Bohr contextuality} -- taking into account all experimental conditions (context). In the simplest case, the measurement context of an observable $A$ is reduced to joint measurement with a compatible observable $B.$ The latter definition was originally considered by Bell in relation to his inequality. We call it {\it joint measurement contextuality} (JMC). Although JMC is based on the use of counterfactuals, by considering it in the general Bohr's framework it is possible to handle JMC on physical grounds. We suggest (similarly to randomness) to certify JMC in experimental data with Bell tests, but only certify and not reduce.