The Origin of the Quantum, Part III: Deviant Logic and Exotic Probability

Classical logic is a system concerned with certain objects that can attain either of two values (usually interpreted as propositions that may be either true or false, commonly denoted 1 or 0 for short), and ways to connect them. Though its origins can be traced back in time to antiquity, and to the Stoic philosopher Chrysippus in particular, its modern form was essentially introduced by the English mathematician and philosopher George Boole (and is thus also known under the name Boolean algebra) in his 1854 book An Investigation of the Laws of Thought, and intended by him to represent a formalization of how humans carry out mental operations. In order to do so, Boole introduced certain connectives and operations, intended to capture the ways a human mind connects and operates on propositions in the process of reasoning.
An elementary operation is that of negation. As the name implies, it turns a proposition into its negative, i.e. from 'it is raining today' to 'it is not raining today'. If we write 'it is raining today' for short as p, 'it is not raining today' gets represented as ¬p, '¬' thus being the symbol of negation.
Two propositions, p and q, can be connected to form a third, composite proposition r in various ways. The most elementary and intuitive connectives are the logical and, denoted by ˄, and the logical or, denoted ˅.
These are intended to capture the intuitive notions of 'and' and 'or': a composite proposition r, formed by the 'and' (the conjunction) of two propositions p and q, i.e. r = p ˄ q, is true if both of its constituent propositions are true -- i.e. if p is true and q is true. Similarly, a composite proposition s, formed by the 'or' (the disjunction) of two propositions p and q, i.e. s = p ˅ q, is true if at least one of its constituent propositions is true, i.e. if p is true or q is true. So 'it is raining and I am getting wet' is true if it is both true that it is raining and that you are getting wet, while 'I am wearing a brown shirt or I am wearing black pants' is true if I am wearing either a brown shirt or black pants -- but also, if I am wearing both! This is a subtle distinction to the way we usually use the word 'or': typically, we understand 'or' to be used in the so-called exclusive sense, where we distinguish between two alternatives, either of which may be true, but not both; however, the logical 'or' is used in the inclusive sense, where a composite proposition is true also if both of its constituent propositions are true.

The Origin of the Quantum, Part II: Incomplete Evidence

In the previous post, we have had a first look at the connections between incompleteness, or logical independence -- roughly, the fact that for any mathematical system, there exist propositions that that system can neither prove false nor true -- and quantumness. In particular, we saw how quantum mechanics emerges if we consider a quantum system as a system only able to answer finitely many questions about its own state; i.e., as a system that contains a finite amount of information. The state of such a system can be mapped to a special, random number, an Ω-number or halting probability, which has the property that any formal system can only derive finitely many bits of its binary expansion; this is a statement of incompleteness, known as Chaitin's incompleteness theorem, equivalent to the more familiar Gödelian version.
In this post, we will exhibit this analogy between incompleteness and quantumness in a more concrete way, explicitly showcasing two remarkable results connecting both notions.
The first example is taken from the paper 'Logical Independence and Quantum Randomness' by Tomasz Paterek et al. Discussing the results obtained therein will comprise the greater part of this post.
The second example can be found in the paper 'Measurement-Based Quantum Computation and Undecidable Logic' by M. Van den Nest and H. J. Briegel; the paper is very interesting and deep, but unfortunately, somewhat more abstract, so I will content myself with just presenting the result, without attempting to explain it very much in-depth.