Degree of Satisfiability in Heyting Algebras

TL;DR

This paper studies the probability that randomly chosen elements in Heyting algebras satisfy certain logical equations, classifies equations based on their satisfiability gap, and explores implications for logic and topology.

Contribution

It classifies all one-variable equations in Heyting algebras by their finite satisfiability gap and analyzes classical principles in multi-variable contexts.

Findings

In finite non-Boolean Heyting algebras, the probability of satisfying $x \/\ eg x = \top$ is at most 2/3.

Classifies all one-variable equations in Heyting algebras by their finite satisfiability gap.

Extends results to infinite Heyting algebras with applications to topology and logic.

Abstract

Given a finite structure $M$ and property $p$, it is a natural to study the degree of satisfiability of $p$ in $M$; i.e. to ask: what is the probability that uniformly randomly chosen elements in $M$ satisfy $p$? In group theory, a well-known result of Gustafson states that the equation $xy=yx$ has a finite satisfiability gap: its degree of satisfiability is either $1$ (in Abelian groups) or no larger than $\frac{5}{8}$. Degree of satisfiability has proven useful in the study of (finite and infinite) group-like and ring-like algebraic structures, but finite satisfiability gap questions have not been considered in lattice-like, order-theoretic settings yet. Here we investigate degree of satisfiability questions in the context of Heyting algebras and intuitionistic logic. We classify all equations in one free variable with respect to finite satisfiability gap, and determine which common principles of classical logic in multiple free variables have finite satisfiability gap. In particular we prove that, in a finite non-Boolean Heyting algebra, the probability that a randomly chosen element satisfies $x \vee \neg x = \top$ is no larger than $\frac{2}{3}$. Finally, we generalize our results to infinite Heyting algebras, and present their applications to point-set topology, black-box algebras, and the philosophy of logic.