Conditional Logic is Complete for Convexity in the Plane

TL;DR

This paper proves that preferential conditional logic is complete for convexity in the Euclidean plane, showing that all consistent non-nested formulas can be modeled with finite point sets, linking logic and convex geometry.

Contribution

It establishes the completeness of preferential conditional logic for convexity in the plane, connecting logical satisfiability with geometric convexity representations.

Findings

Every consistent non-nested formula is satisfiable in a finite planar convex model.

The proof uses the representation of convex geometries by convex polygons in the plane.

The result bridges logical reasoning and geometric convexity in finite point sets.

Abstract

We prove completeness of preferential conditional logic with respect to convexity over finite sets of points in the Euclidean plane. A conditional is defined to be true in a finite set of points if all extreme points of the set interpreting the antecedent satisfy the consequent. Equivalently, a conditional is true if the antecedent is contained in the convex hull of the points that satisfy both the antecedent and consequent. Our result is then that every consistent formula without nested conditionals is satisfiable in a model based on a finite set of points in the plane. The proof relies on a result by Richter and Rogers showing that every finite abstract convex geometry can be represented by convex polygons in the plane.