This paper develops a version of Natural Logic -- an inference system that works directly on natural language syntactic representations, with no intermediate translation to logical formulae. Following work by Sanchez (1991), we develop a small fragment that computes semantic order relations between derivation trees in Categorial Grammar. Unlike previous works, the proposed system has the following new characteristics: (i) It uses orderings between derivation trees as purely syntactic units, derivable by a formal calculus. (ii) The system is extended for conjunctive phenomena like coordination and relative clauses. This allows a simple account of non-monotonic expressions that are reducible to conjunctions of monotonic ones. (iii) A preliminary proof search algorithm based on a tree generating regular system is developed for Sanchez' smaller fragment of Natural Logic.