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.

翻译：我们证明优选的有条件逻辑是完全的,符合欧几里德平面有限点数的精度。如果能够解释前些点的所有极端点都符合这一结果,则在一定的一组点数中确定为有条件。同样,如果前几个点数包含在满足前些点数和随后各点数的圆柱数中,则条件是完全的。因此,我们的结果是,每一个没有嵌入条件的一致公式都可以在基于该平面有限点数的模型中作比较。证据依据的是Richter和Rogers的结果显示,每一有限的抽象方位数的几何方法都可以由平面上的方形方形来代表。