Factor phrases - modifiers such as "twice" or "three times" - are objects that show up cross-categorially, and yet their semantics remains largely neglected, with virtually no work done on them. This dissertation thus redresses the balance, examining factor phrases in three different domains: the verbal domain ("Floyd walked the dog three times"), the adjectival domain ("Floyd is three times as tall as Clyde"), and the nominal domain ("Floyd has twice Clyde's wisdom"). I argue that while the verbal form is a type of event counter, as described by Landman (2004), the adjectival and nominal cases are instead modifiers of type : a kind of modifier called a ratio degree. I show how these ratio degrees interact with degree morphemes such as "as" and "-er", including explaining why, in English, "Floyd is three times as tall as Clyde" and "Floyd is three times taller than Clyde" mean the same thing, despite the meanings of "as" and "-er" not being equivalent. This multiplicative use also extends into the verbal domain, with sentences such as "Floyd walked the dog three times as many times as Clyde did", and we see that ratio degrees also work successfully in this domain. The use of ratio degrees is then extended to the nominal domain, with a number of novel observations about the various kinds of nouns that can occur (mass, count, etc.) and their interactions with factor phrases. The dissertation concludes with a discussion of basic arithmetic, including a detailed syntactic and semantic analysis of basic arithmetic phrases such as "Three times seven is twenty-one" and "Fourteen divided by two is seven" -- an area that has been almost completely neglected. I show that these sentences still follow the same basic syntactic rules as the rest of natural language, and that consequently we can use the tools of natural language to provide a semantics for these basic arithmetic operations.
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