Homework 5: Quantificational Determiners

Words like no, every and some are called 'quantificational determiners,' and are used to express various kinds of generalizations in sentences like the following:

• No doctor smokes.
• Every doctor smokes.
• Some doctor smokes.

Today in class, we considered the possibility that these words have (extensional) denotations that are type <<e,t>,<<e,t>,t>>: functions from functions from individuals to truth values (i.e., noun meanings) to functions from functions from individuals to truth values (i.e., VP meanings) to truth values (i.e., sentence meanings). But what exactly are such functions, and can they capture the different truth conditions that come from the use of no, every, some and other determiners?

In order to build an answer to this question, I want you to first think about something else we discussed at the end of class. We said that, in general, any expression with a denotation of type <x,<x,t>> (where x is any type) has a RELATIONAL meaning: it takes two things of type x and returns TRUE or FALSE depending on whether certain conditions obtain. We also observed that expressions with denotations of type <e,t> (like nouns and VPs) can be thought of either as functions from individuals to truth values or as SETS: the set of individuals that the function is true of. Putting these two ideas together, we can observe that if determiners have meanings of type <<e,t>,<<e,t>,t>>, then it should be possible to characterize their meanings as relations between sets.

PART ONE: Let D represent the set of doctors, and let S represent the set of smokers. For each of the following sentences, say what relation between D and S must obtain if the sentence is true.

1. No doctor smokes.
2. Every doctor smokes.
3. Some doctor smokes.
4. Three doctors smoke.
5. Most doctors smoke.
6. Few doctors smoke.

PART TWO: Given our Principle of Function Application, if determiners like no, every and some denote functions of type <<e,t>,<<e,t>,t>> and nouns like doctor denote functions of type <e,t>, then NPs of the form Det N must denote functions of type <<e,t>,t>: functions from functions from individuals to truth values (VP meanings) to truth values (S meanings). Taking your observations in Part One as a starting point, say what functions the following three NPs denote:

• no doctor
• every doctor
• some doctor