Overview

In class, we began talking about quantifiers—specifically, quantifiers in subject position. Take (1), for example.

  1. Some dog slept.

What’s interesting about quantifiers like some dog is that while they have the distribution of noun phrases, they apparently do not pick out entities. What (1) says, for example, is not that there is a particular dog, say Fido, who slept. Instead, the claim that (1) makes is much weaker than that. All it requires is that some dog slept—could be Fido, could be Bella, could be Lassie, or yet some other dog—and it is true just in case any one of these dogs slept.

One way of thinking about what the noun phrase some dog does, semantically, is by considering it to denote a property of verb phrase meanings (i.e., characteristic functions of sets of entities). Specifically, it is true of any verb phrase meaning that is itself true of some or other dog, and it is false of any verb phrase meaning that is not true of any dogs. If we think of the meaning of some dog this way, then we can see how applying it to a verb phrase will give rise to a true sentence just in case the verb phrase is true of some dog—exactly what we want.

To carry this out, we might assume that some dog is kind of like a coordination of two noun phrases (e.g., Po and Tigress), insofar as it acts on a verb phrase, rather than allowing a verb phrase to act on it (see Coordinating subject noun phrases for discussion). Thus we can assign it the meaning and syntactic category in (2).

  1. \(⟨\textit{some dog}, (λf.\ct{dog}_{SET} ∩ f_{SET} ≠ \varnothing)⟩ ⊢ (s/(np\backslash s))\)

Here, we use the notation ‘\(f_{SET}\)’ to refer to the set characterized by the function \(f\). Formally:

\[f_{SET} ≝ \{x ∈ D_{e} ∣ f(x) = \true\}\]

So, for example, if we have some set of entities \(S\), then \((S_{CF})_{SET} = S\): turning that set into its characteristic function and then back into a set again gives you back the original set. And if we have a characteristic function of a set of entities \(f\), then \((f_{SET})_{CF} = f\): turning it into the set it characterizes and then back into a characteristic function also ultimately has no effect. In this sense, we are really always talking about “the same thing” whenever we talk about a set versus its characteristic function—the difference between them is practically only notational.

So what does (2) say about the meaning of some dog? Basically, that, whatever verb phrase it combines with, the set of dogs intersected with the set characterized by that verb phrase is not the empty set. For example, if that verb phrase is slept, then some dog slept will be true just in case the intersection of dogs with sleeping things isn’t empty.

Expressions like some dog are what are known as generalized quantifiers. They are a bit like the “universal” quantifier \(∀\) (‘for all’) and the “existential” quantifier ∃ (‘there exists’) that one sees in predicate logic; but the semantic type of generalized quantifiers—\(((e → t) → t)\)—allows one to encode many more meanings than just those pertaining to universality and existence.