A Theory of Inaccessible Information

Abstract

What would be the consequences if there were fundamental limits to our ability to experimentally explore the world? In this work we seriously consider this question. We assume the existence of statements whose truth value is not experimentally accessible. That is, there is no way, not even in theory, to directly test if these statements are true or false. We further develop a theory in which experimentally accessible statements are a union of a fixed minimum number of inaccessible statements. For example, the value of truth of the statements "a" and "b" is not accessible, but the value of truth of the statement "a or b" is accessible. We do not directly assume probability theory, we exclusively define experimentally accessible and inaccessible statements and build on these notions using the rules of classical logic. We find that an interesting structure emerges. Developing this theory, we relax the logical structure to a probabilistic one, obtaining a theory rich in structure that we call "theory of inaccessible information". Surprisingly, the simplest model of theory of inaccessible information is the qubit in quantum mechanics. Along the path for the construction of this theory, we characterise and study a family of multiplicative information measures that we call "inaccessibility measures".