A comment on Samson Abramsky’s paper “Relational Databases and Bell’s Theorem”

Abramsky points out a connection between a non-locality phenomena in quantum physics and a non-locality phenomenon in database theory. This seems surprising as one would not expect—a priori—such a connection to exist. The proof of the relevant quantum theoretic phenomenon turns out in Abramsky’s analysis to be a mathematical rather than a physical phenomenon. It is based on the nonexistence, in the mathematical sense, of probability distributions satisfying certain conditions. The probability distributions can even be replaced by 2-valued distributions with the same effect. A similar mathematical non-existence phenomenon exists in database theory. The phenomenon is twofold. On the one hand there is the decomposition problem: how to decompose a large database into smaller parts, which would form an instance of a database schema with the associated dependencies. On the other hand we may start from an instance of a schema and ask whether there is a universal relation. Abramsky calls this contextuality and suggests that a common logic of contextuality is emerging from these examples. We may also ask, is contextuality an even more general phenomenon? If it is, there is all the more reason to take a logical approach to the question. By a logical approach I mean establishing a language, a semantics of the language and an attempt to figure out the logical properties. The characteristic feature of contextuality, be it quantum physics or database theory, is the presence of several tables of data and the question is whether they arise from just one universal table by projection. This co-existence of several limited “realities” is not an unusual theme in the history of ideas, and neither is the question whether there is one ultimate “truth”. This is the case in different areas of humanities for rather obvious reasons, but it exists even in mathematics. For example, in set theory there is discussion about a multiverse position according to which the fact that we have not been able to solve questions such as the Continuum Hypothesis is a consequence of the circumstance that the set theoretical reality of mathematical objects is a multiverse, i.e. a collection of universes, very “close” to each other, some satisfying the Continuum Hypothesis and some not. One may indeed reformulate first order logic so that the semantics is not based on the concept of a model satisfying a sentence, but on the concept