This paper presents a part of work in progress on axiomatizing a spatial logic with convexity and inclusion predicates (hereinafter called convexity logic), with some intended interpretation over the real plane. More formally, let L_conv,≤ be a language of first order logic and two non-logical primitives: conv (interpreted as a property of a set of being convex) and ≤ (interpreted as the set inclusion relation). We let variables range over regular open rational polygons in the real plane (denoted ROQ(R²)). We call the tuple M = 〈ROQ, conv, ≤〉 — where primitives are defined as indicated above — a standard model. We propose an axiomatization of the theory of M and prove soundness and completeness for this axiomatization.