Degree of Satisfiability in Heyting Algebras

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