Transitive verbs

Transitive verbs like punches increase the number of syntactic and semantic arguments of intransitives by one. To analyze the sentence in (1),

Tigress punches Po.

we can assign a lexical entry like the one in (2) to this verb.

\(⟨\textit{punches}, (λx.(λy.\ct{punch}(y, x)))⟩ ⊢ ((np\backslash s)/np)\)

In this case, we have a function of the following type:

\[(λx.(λy.\ct{punch}(y, x))) : D_{e} → D_{e} → \{\true, \false\}\]

This function is defined in terms of a relation on entities, \(\ct{punch}\). That is, given two entities \(x\) and \(y\), \(\ct{punch}(y, x)\) is some truth value.

Then, given the lexical entry for Tigress where it denotes \(\ct{ti}\), we can derive the meaning and syntactic category of the sentence in (1):

\[\begin{prooftree} \AxiomC{\(⟨\textit{Tigress}, \ct{ti}⟩ ⊢ np\)} \AxiomC{\(⟨\textit{punches}, (λx.(λy.\ct{punch}(y, x)))⟩ ⊢ (np\backslash s)/np\)} \AxiomC{\(⟨\textit{Po}, \ct{p}⟩ ⊢ np\)} \RightLabel{\(/\)}\BinaryInfC{\(⟨\textit{punches Po}, (λx.(λy.\ct{punch}(y, x)))(\ct{p})⟩ ⊢ (np\backslash s)\)} \RightLabel{\(\backslash\)}\BinaryInfC{\(⟨\textit{Tigress punches Po}, (λx.(λy.\ct{punch}(y, x)))(\ct{p})(\ct{ti})⟩ ⊢ s\)} \end{prooftree}\]

Since \((λx.(λy.\ct{punch}(y, x)))\) takes two entities in order, we can get a function takes one entity by applying it first to \(\ct{p}\):

\[(λx.(λy.\ct{punch}(y, x)))(\ct{p})\] \[⇒ (λy.\ct{punch}(y, \ct{p}))\]

This function is itself a characteristic function of a set of entities in \(D_{e}\). As a result, if we apply it to two entities, we get a truth value:

\[(λx.(λy.\ct{punch}(y, x)))(\ct{p})(\ct{ti})\] \[⇒ (λy.\ct{punch}(x, \ct{p}))(\ct{ti})\] \[⇒ \ct{punch}(\ct{ti}, \ct{p})\]

This result, \(\ct{punch}(\ct{ti}, \ct{p})\), will be \(\true\) or \(\false\), depending on the model.

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title: "Transitive verbs"
bibliography: ../../semantics.bib
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::: {.hidden}
$$
\newcommand{\expr}[3]{\begin{array}{c}
#1 \\
\bbox[lightblue,5px]{#2}
\end{array} ⊢ #3}
\newcommand{\ct}[1]{\bbox[font-size: 0.8em]{\mathsf{#1}}}
\newcommand{\abbr}[1]{\bbox[transform: scale(0.95)]{\mathtt{#1}}}
\def\true{\ct{T}}
\def\false{\ct{F}}
$$
:::

Transitive verbs like *punches* increase the number of syntactic and semantic arguments of intransitives by one.
To analyze the sentence in (@ex-punches),

(@ex-punches) Tigress punches Po.

we can assign a lexical entry like the one in (@ex-lexicon) to this verb.

(@ex-lexicon) $⟨\textit{punches}, (λx.(λy.\ct{punch}(y, x)))⟩ ⊢ ((np\backslash s)/np)$

In this case, we have a function of the following type:

$$(λx.(λy.\ct{punch}(y, x))) : D_{e} → D_{e} → \{\true, \false\}$$

This function is defined in terms of a relation on entities, $\ct{punch}$.
That is, given two entities $x$ and $y$, $\ct{punch}(y, x)$ is some truth value.

Then, given the lexical entry for *Tigress* where it denotes $\ct{ti}$, we can derive the meaning and syntactic category of the sentence in (@ex-punches):

$$\begin{prooftree}
\AxiomC{\(⟨\textit{Tigress}, \ct{ti}⟩ ⊢ np\)}
\AxiomC{\(⟨\textit{punches}, (λx.(λy.\ct{punch}(y, x)))⟩ ⊢ (np\backslash s)/np\)}
\AxiomC{\(⟨\textit{Po}, \ct{p}⟩ ⊢ np\)}
\RightLabel{\(/\)}\BinaryInfC{\(⟨\textit{punches Po}, (λx.(λy.\ct{punch}(y, x)))(\ct{p})⟩ ⊢ (np\backslash s)\)}
\RightLabel{\(\backslash\)}\BinaryInfC{\(⟨\textit{Tigress punches Po}, (λx.(λy.\ct{punch}(y, x)))(\ct{p})(\ct{ti})⟩ ⊢ s\)}
\end{prooftree}$$

Since $(λx.(λy.\ct{punch}(y, x)))$ takes two entities in order, we can get a function takes *one* entity by applying it first to $\ct{p}$:

$$(λx.(λy.\ct{punch}(y, x)))(\ct{p})$$
$$⇒ (λy.\ct{punch}(y, \ct{p}))$$

This function is itself a characteristic function of a set of entities in $D_{e}$.
As a result, if we apply it to *two* entities, we get a truth value:

$$(λx.(λy.\ct{punch}(y, x)))(\ct{p})(\ct{ti})$$
$$⇒ (λy.\ct{punch}(x, \ct{p}))(\ct{ti})$$
$$⇒ \ct{punch}(\ct{ti}, \ct{p})$$

This result, $\ct{punch}(\ct{ti}, \ct{p})$, will be $\true$ or $\false$, depending on the model.