The Mathematics of Questions

TL;DR

This work introduces an information-theoretic and erotetic framework in which questions are fundamental objects alongside propositions. It identifies a novel non-trivial solution $A\sim B$ to the information-balance equation $i(A,B)+i(A,\neg B)+i(\neg A,B)+i(\neg A,\neg B)=0$, and develops a rich algebra of pure questions, askable questions, and subjects, including quotient constructions and multiplicative geometric-mean measures. The author then shows that, in the two-state (qubit) case, the geometry of askable questions reproduces the Hilbert-space formalism, the Born rule, and even features such as entanglement and non-locality as natural consequences of raising and combining questions. The approach provides a potential foundational bridge between logic, probability, and quantum mechanics, offering a new lens to view measurements as information-subtraction operations and to interpret quantum amplitudes as complex-valued properties of questions. Overall, the paper advances a novel mathematical account of questions that yields the two-state quantum formalism and suggests avenues for extending the program to higher-dimensional systems and broader epistemic interpretations.

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.

Figures (12)

• Figure 2.1: The unique valid non-trivial solution to the constraint, $x (a - x) (b - x) (1 - a - b + x) = a^2 b^2 (1 - a)^2 (1 - b)^2$, as specified by Wolfram Alpha. The quartic equation has four solutions, including trivial independence, $x = ab$, the above solution, and two other solutions of similar complexity, neither of which satisfy both $x \leq \min(a, b)$ and $x \geq 0$, which are requirements for $x$ to be equal to $P(AB)$, where $a = P(A)$ and $b = P(B)$. The solution above is the only non-trivial valid probability distribution that satisfies the constraint.
• Figure 2.2: Surface plots showing how $P(AB)$ depends on $P(A)$ and $P(B)$ in two cases. Independence, $P(AB) = P(A)P(B)$, is shown on the left, and the tilde relation is shown on the right. The surfaces are very similar, but not exactly identical. These two surfaces are the only solutions to the equation, $i(A,B) + i(A, \neg B) + i(\neg A, B) + i(\neg A, \neg B) = 0$.
• Figure 2.3: The discrepancy, $P(AB) - P(A)P(B)$, between the $\sim$ relation and independence, as a function of $P(A)$ and $P(B)$. The discrepancy is exactly 0 when either $P(A)$ or $P(B)$ is equal to 0, $1/2$, or 1. When both $P(A)$ and $P(B)$ are greater than $1/2$, or both are less than $1/2$, $P(AB)$ exceeds $P(A)P(B)$ by an amount whose maximum value is about 0.0674. When $P(A)$ and $P(B)$ are on opposite sides of $1/2$, $P(AB)$ is smaller than $P(A)P(B)$ by a corresponding amount. The entire surface forms a smooth, symmetric wave over the unit square of probabilities of $A$ and $B$.
• Figure 2.4: Surface plots of $P(B|A)$ as a function of $P(A)$ and $P(B)$ for the tilde relation. They show that $P(B|A) = P(B)$ at $P(A) = 1$, and $P(B|A)$ approaches $P(\neg B)$ as $P(A)$ approaches 0, except at $P(B) = 0$ or 1. The value of $P(B|A)$ at $P(A) = 1/2$ is exactly equal to $P(B)$. The surface is a cubic interpolation between the three discrete rules that specify the values of $P(B|A)$ at $P(A) = 0$, 1 and $1/2$.
• Figure 7.1: The shape formed in the complex plane by the values of $V$ for every possible value of $P(A)$ and $P(B)$. The unit square of possible probabilities is mapped by $V$ to a curved triangular shape in the complex plane. The real co-ordinate of a point inside this shape plays the role of a probability gap in the equation, $x - a \cdot b = (\text{Re}(V) - \text{gap}(A?) \cdot \text{gap}(B?))/3$.