Systems of Natural Logic with Adjectives and Their Completeness

Abstract

In natural logic, the goal is to create a system of logic that is as sim- ilar to natural language as possible. In order to build a natural logic, simple sentence forms are considered, slowly incorporating more language throughout time. As background, we rst consider the fragments of the form All X are Y and then add No X are Y , where X and Y are nouns. We will look at the rules of logic and completeness of their proof sys- tems. Next, we will introduce the idea of intersecting adjectives which are adjectives with meaning separate from the noun being modifying. For example, the plural noun red cars means the intersection of everything red and everything that is a car. This paper will then present versions of the simple systems that contain intersecting adjectives and will discuss their completeness.