The Mathematics of Questions

Abstract

I report the existence of exactly one non-trivial solution to the equation $i(A,B)+i(A,\neg B)+i(\neg A,B)+i(\neg A,\neg B)= 0$, where $i(A,B)=\log\frac{P(A\text{ and }B)}{P(A)P(B)}$, and $P(A)$ is the probability of the proposition $A$. The equation specifies an information balance condition between two logical propositions, which is satisfied only by independence and by this new solution. The solution is a new elementary informational relationship between logical propositions, which we denote as $A \sim B$. The $\sim$ relation cannot be expressed as a relationship between probabilities without the use of complex numbers. It can, however, be greatly simplified by expressing each proposition as a combination of a question and an answer, for example, writing, ``All men are mortal'', as (Are all men mortal?, Yes). We will study the mathematics of questions and find out what role the $\sim$ relationship plays inside the algebra. We will find that, like propositions, questions can act on probability distributions. A proposition, $X$, can be given, setting $P(X)$ to 1. The question of $X$ can be raised, setting $P(X)$ to $1/2$. Giving the proposition adds information to the probability distribution, but raising the question takes information away. Introducing questions into probability theory makes it possible to represent subtraction of information as well as addition. We will examine how questions can be related to each other geometrically. Remarkably, the simplest way of orienting questions in space has the same structure as the simplest quantum system -- the two-state system. We will find that the essential mathematical structure of the two-state quantum system can be derived from the mathematics of questions, including non-commutativity, complementarity, wavefunction collapse, the Hilbert space representation and the Born rule, as well as quantum entanglement and non-locality.