The goal of natural logic is to present and study logical systems for reasoning with sentences of (or which are reasonably close to) ordinary language. This paper explores simple systems of natural logic which make use of intersecting adjectives; these are adjectives whose interpretation does not vary with the noun they modify. Our project in this paper is to take one of the simplest syllogistic fragments, that of
, and to add intersecting adjectives. There are two ways to do this, depending on whether one allows iteration or prefers a “flat” structure of at most one adjective. We present rules of inference for both types of syntax, and these differ. The main results are four completeness theorems: for each of the two types of syntax we have completeness for the
fragment and for the full language of this paper.