Intransitive verbs

In Syntactic categories, we said that intransitive verbs have the syntactic category \((np\backslash s)\): they take a noun phrase to their left, and together with it, they produce a sentence. And because they have this category, they have the semantic type \((e → t)\): they take an entity as their semantic argument, and they produce a truth value—that is, they denote characteristic functions.

Take the verb sleeps as an example.

  1. Po sleeps.

To derive the sentence in (1), we can assign Po and sleeps the lexical entries in (2).

  1. \(⟨\textit{Po}, \ct{p}⟩ ⊢ np\)
    \(⟨\textit{sleeps}, (λx.\ct{sleep}(x))⟩ ⊢ (np\backslash s)\)

As usual, \(\ct{p}\) is an entity—some element of the domain of whatever model we happen to be considering: \[\ct{p} ∈ D_{e}\] Meanwhile, \(\ct{sleep}\) is a characteristic function of a set of elements of this very same domain: \[\ct{sleep} : D_{e} → \{\true, \false\}\] It takes an entity, and it gives you back a truth value (\(\true\) or \(\false\)).

Which characteristic function is it? Who knows. Or rather, we’re just using the expression ‘\(\ct{sleep}\)’ to stand for \(I_{\mathcal{M}}(\textit{sleeps})\), for whatever the model \(\mathcal{M}\) happens to be.

Using the lexical entries for Po and sleeps, we can now derive the sentence in (1):

\[\begin{prooftree} \AxiomC{\(⟨\textit{Po}, \ct{p}⟩ ⊢ np\)} \AxiomC{\(⟨\textit{sleeps}, (λx.\ct{sleep}(x))⟩ ⊢ (np\backslash s)\)} \RightLabel{\(\backslash\)}\BinaryInfC{\(⟨\textit{Po sleeps}, (λx.\ct{sleep}(x))(\ct{p})⟩ ⊢ s\)} \end{prooftree}\] As we’d expect from the semantic type of \(s\), the meaning derived for the entire sentence,

\[(λx.\ct{sleep}(x))(\ct{p})\] \[⇒ \ct{sleep}(\ct{p})\]

is a truth value—\(\true\) or \(\false\), depending on which of these values \(\ct{sleep}\) maps \(\ct{p}\) to.