The objective of this chapter is to demonstrate that degrees should be formalized as intervals on a scale rather than as points, as traditionally assumed. Using cross-polar anomaly as the empirical basis for my claims, I will argue that gradable adjectives denote functions from objects to intervals on a scale, or extents, and assume an ontology which distinguishes between two sorts of extents: positive extents and negative extents (as in Seuren 1978, von Stechow 1984b, and Lobner 1990). I characterize the difference between positive and negative adjectives as a sortal one: positive adjectives denote functions from objects to positive extents, and negative adjectives denote functions from objects to negative extents. After setting this analysis into the semantic framework developed in chapter 2, I show that cross-polar anomaly can be explained in the same way as incommensurability: a consequence of the sortal characterization of adjectival polarity is that the ranges of positive and negative adjectives are disjoint; as a result, the comparison relation in examples of cross-polar anomaly is undefined. Section 3.2 continues with an examination of comparison

of deviation constructions, which at first glance appear to be counterexamples to the proposals made in section 3.1, but upon closer examination turn out to fit in naturally with the analysis of degree constructions developed in chapter 2. Finally, section 3.3 demonstrates that the algebra of extents, in conjunction with the semantic analysis of gradable adjectives as measure phrases, has the additional positive result of providing the basis for an insightful explanation of the monotonicity properties of gradable adjectives.