Ditransitive verbs

Let’s consider the two sentences in (1), which, arguably, entail each other.

1. Po showed noodles to Tigress.
2. Po showed Tigress noodles.

To capture this mutual entailment, we might assign the ditransitive verb showed two distinct lexical entries, as in (2).

$⟨\textit{showed}_{1}, (λx.(λy.(λz.\ct{show}(z, x, y))))⟩ ⊢ (((np\backslash s)/pp)/np)$
$⟨\textit{showed}_{2}, (λy.(λx.(λz.\ct{show}(z, x, y))))⟩ ⊢ (((np\backslash s)/np)/np)$

We can also assign a lexical entry to the preposition to, according to which it is interpreted as the identity function $(λx.x)$.

$⟨\textit{to}, (λx.x)⟩ ⊢ (pp/np)$

According to the first of these lexical entries, the verb showed combines with an object noun phrase; then it combines with a prepositional phrase; and then it combines with a subject noun phrase, yielding, e.g., (1-a). According to the second, it combines with an indirect-object noun phrase; then a direct-object noun phrase; and then the subject, yielding, e.g., (1-b).

A derivation of the (1-a) sentence would go:

\[\begin{prooftree} \AxiomC{$⟨\textit{Po}, \ct{p}⟩ ⊢ np⟩$} \AxiomC{$⟨\textit{showed}_{1}, (λx.(λy.(λz.\ct{show}(z, x, y))))⟩ ⊢ (((np\backslash s)/pp)/np)$} \AxiomC{$⟨\textit{noodles}, \ct{n}⟩ ⊢ np$} \RightLabel{$/$}\BinaryInfC{$⟨\textit{showed\(_{1}$ noodles}, (λx.(λy.(λz.\ct{show}(z, x, y))))(\ct{n})⟩ ⊢ ((np\backslash s)/pp)\)} \AxiomC{$⟨\textit{to}, (λx.x)⟩ ⊢ (pp/np)$} \AxiomC{$\textit{Tigress}, \ct{ti} ⟩ ⊢ np$} \RightLabel{$/$}\BinaryInfC{$⟨\textit{to Tigress}, (λx.x)(\ct{ti})⟩ ⊢ pp$} \RightLabel{$/$}\BinaryInfC{$⟨\textit{showed\(_{1}$ noodles to Tigress}, (λx.(λy.(λz.\ct{show}(z, x, y))))(\ct{n})((λx.x)(\ct{ti}))⟩ ⊢ (np\backslash s)\)} \RightLabel{$\backslash$}\BinaryInfC{$⟨\textit{Po showed\(_{1}$ noodles to Tigress}, (λx.(λy.(λz.\ct{show}(z, x, y))))(\ct{n})((λx.x)(\ct{ti}))(\ct{p})⟩ ⊢ s\)} \end{prooftree}\]

At the end of the day, we end up with the following truth value as the meaning of this sentence:

\[(λx.(λy.(λz.\ct{show}(z, x, y))))(\ct{n})((λx.x)(\ct{ti}))(\ct{p})\] \[⇒ (λy.(λz.\ct{show}(z, \ct{n}, y))))((λx.x)(\ct{ti}))(\ct{p})\] \[⇒ (λz.\ct{show}(z, \ct{n}, (λx.x)(\ct{ti})))(\ct{p})\] \[⇒ \ct{show}(\ct{p}, \ct{n}, (λx.x)(\ct{ti}))\] \[⇒ \ct{show}(\ct{p}, \ct{n}, \ct{ti})\]

(It can be helpful to check and make sure you understand each line as the above metalanguage expression is evaluated.)

Meanwhile, a derivation of (1-b) would go:

\[\begin{prooftree} \AxiomC{$⟨\textit{Po}, \ct{p}⟩ ⊢ np⟩$} \AxiomC{$⟨\textit{showed}_{2}, (λy.(λx.(λz.\ct{show}(z, x, y))))⟩ ⊢ (((np\backslash s)/np)/np)$} \AxiomC{$⟨\textit{Tigress}, \ct{ti}⟩ ⊢ np$} \RightLabel{$/$}\BinaryInfC{$⟨\textit{showed\(_{2}$ Tigress}, (λy.(λx.(λz.\ct{show}(z, x, y))))(\ct{ti})⟩ ⊢ ((np\backslash s)/np)\)} \AxiomC{$⟨\textit{noodles}, \ct{n} ⟩ ⊢ np$} \RightLabel{$/$}\BinaryInfC{$⟨\textit{showed\(_{2}$ Tigress noodles}, (λy.(λx.(λz.\ct{show}(z, x, y))))(\ct{ti})(\ct{n})⟩ ⊢ (np\backslash s)\)} \RightLabel{$\backslash$}\BinaryInfC{$⟨\textit{Po showed\(_{2}$ Tigress noodles}, (λy.(λx.(λz.\ct{show}(z, x, y))))(\ct{ti})(\ct{n})(\ct{p})⟩ ⊢ s\)} \end{prooftree}\]

If we evaluate the result, we get:

\[(λy.(λx.(λz.\ct{show}(z, x, y))))(\ct{ti})(\ct{n})(\ct{p})\] \[⇒ (λx.(λz.\ct{show}(z, x, \ct{ti})))(\ct{n})(\ct{p})\] \[⇒ (λz.\ct{show}(z, \ct{n}, \ct{ti}))(\ct{p})\] \[⇒ \ct{show}(\ct{p}, \ct{n}, \ct{ti})\]

(Again, check each line! :)

Thankfully, the resulting interpretation is the same in both cases. Why “thankfully”? Well, because as we saw, the two sentences entail each other!

And importantly, our analysis derives this result precisely because of how we designed the lexical enries for prepositional-dative show and double-object dative show. In particular, how we designed their meanings.

--- title: "Ditransitive verbs" bibliography: ../../semantics.bib format: html: css: ../styles.css html-math-method: mathjax mathjax-config: loader: {load: ['[tex]/bussproofs','[tex]/bbox','[tex]/colorbox']} tex: packages: {'[+]': ['bussproofs','bbox','colorbox']} --- ::: {.hidden} $$ \newcommand{\expr}[3]{\begin{array}{c} #1 \\ \bbox[lightblue,5px]{#2} \end{array} ⊢ #3} \newcommand{\ct}[1]{\bbox[font-size: 0.8em]{\mathsf{#1}}} \newcommand{\abbr}[1]{\bbox[transform: scale(0.95)]{\mathtt{#1}}} \def\true{\ct{T}} \def\false{\ct{F}} $$ ::: Let's consider the two sentences in (@ex-showed), which, arguably, entail each other. (@ex-showed) a. Po showed noodles to Tigress. <br> b. Po showed Tigress noodles. To capture this mutual entailment, we might assign the ditransitive verb *showed* two distinct lexical entries, as in (@ex-showed-lexicon). (@ex-showed-lexicon) $⟨\textit{showed}_{1}, (λx.(λy.(λz.\ct{show}(z, x, y))))⟩ ⊢ (((np\backslash s)/pp)/np)$ <br> $⟨\textit{showed}_{2}, (λy.(λx.(λz.\ct{show}(z, x, y))))⟩ ⊢ (((np\backslash s)/np)/np)$ We can also assign a lexical entry to the preposition *to*, according to which it is interpreted as the identity function $(λx.x)$. (@ex-to-lexicon) $⟨\textit{to}, (λx.x)⟩ ⊢ (pp/np)$ According to the first of these lexical entries, the verb *showed* combines with an object noun phrase; then it combines with a prepositional phrase; and then it combines with a subject noun phrase, yielding, e.g., (@ex-showed\-a). According to the second, it combines with an indirect-object noun phrase; then a direct-object noun phrase; and then the subject, yielding, e.g., (@ex-showed\-b). A derivation of the (@ex-showed\-a) sentence would go: $$\begin{prooftree} \AxiomC{$⟨\textit{Po}, \ct{p}⟩ ⊢ np⟩$} \AxiomC{$⟨\textit{showed}_{1}, (λx.(λy.(λz.\ct{show}(z, x, y))))⟩ ⊢ (((np\backslash s)/pp)/np)$} \AxiomC{$⟨\textit{noodles}, \ct{n}⟩ ⊢ np$} \RightLabel{$/$}\BinaryInfC{$⟨\textit{showed\(_{1}$ noodles}, (λx.(λy.(λz.\ct{show}(z, x, y))))(\ct{n})⟩ ⊢ ((np\backslash s)/pp)\)} \AxiomC{$⟨\textit{to}, (λx.x)⟩ ⊢ (pp/np)$} \AxiomC{$\textit{Tigress}, \ct{ti} ⟩ ⊢ np$} \RightLabel{$/$}\BinaryInfC{$⟨\textit{to Tigress}, (λx.x)(\ct{ti})⟩ ⊢ pp$} \RightLabel{$/$}\BinaryInfC{$⟨\textit{showed\(_{1}$ noodles to Tigress}, (λx.(λy.(λz.\ct{show}(z, x, y))))(\ct{n})((λx.x)(\ct{ti}))⟩ ⊢ (np\backslash s)\)} \RightLabel{$\backslash$}\BinaryInfC{$⟨\textit{Po showed\(_{1}$ noodles to Tigress}, (λx.(λy.(λz.\ct{show}(z, x, y))))(\ct{n})((λx.x)(\ct{ti}))(\ct{p})⟩ ⊢ s\)} \end{prooftree}$$ At the end of the day, we end up with the following truth value as the meaning of this sentence: $$(λx.(λy.(λz.\ct{show}(z, x, y))))(\ct{n})((λx.x)(\ct{ti}))(\ct{p})$$ $$⇒ (λy.(λz.\ct{show}(z, \ct{n}, y))))((λx.x)(\ct{ti}))(\ct{p})$$ $$⇒ (λz.\ct{show}(z, \ct{n}, (λx.x)(\ct{ti})))(\ct{p})$$ $$⇒ \ct{show}(\ct{p}, \ct{n}, (λx.x)(\ct{ti}))$$ $$⇒ \ct{show}(\ct{p}, \ct{n}, \ct{ti})$$ (It can be helpful to check and make sure you understand each line as the above metalanguage expression is evaluated.) Meanwhile, a derivation of (@ex-showed\-b) would go: $$\begin{prooftree} \AxiomC{$⟨\textit{Po}, \ct{p}⟩ ⊢ np⟩$} \AxiomC{$⟨\textit{showed}_{2}, (λy.(λx.(λz.\ct{show}(z, x, y))))⟩ ⊢ (((np\backslash s)/np)/np)$} \AxiomC{$⟨\textit{Tigress}, \ct{ti}⟩ ⊢ np$} \RightLabel{$/$}\BinaryInfC{$⟨\textit{showed\(_{2}$ Tigress}, (λy.(λx.(λz.\ct{show}(z, x, y))))(\ct{ti})⟩ ⊢ ((np\backslash s)/np)\)} \AxiomC{$⟨\textit{noodles}, \ct{n} ⟩ ⊢ np$} \RightLabel{$/$}\BinaryInfC{$⟨\textit{showed\(_{2}$ Tigress noodles}, (λy.(λx.(λz.\ct{show}(z, x, y))))(\ct{ti})(\ct{n})⟩ ⊢ (np\backslash s)\)} \RightLabel{$\backslash$}\BinaryInfC{$⟨\textit{Po showed\(_{2}$ Tigress noodles}, (λy.(λx.(λz.\ct{show}(z, x, y))))(\ct{ti})(\ct{n})(\ct{p})⟩ ⊢ s\)} \end{prooftree}$$ If we evaluate the result, we get: $$(λy.(λx.(λz.\ct{show}(z, x, y))))(\ct{ti})(\ct{n})(\ct{p})$$ $$⇒ (λx.(λz.\ct{show}(z, x, \ct{ti})))(\ct{n})(\ct{p})$$ $$⇒ (λz.\ct{show}(z, \ct{n}, \ct{ti}))(\ct{p})$$ $$⇒ \ct{show}(\ct{p}, \ct{n}, \ct{ti})$$ (Again, check each line! :) Thankfully, the resulting interpretation is the same in both cases. Why "thankfully"? Well, because as we saw, the two sentences entail each other! And importantly, our analysis derives this result precisely because of how we designed the lexical enries for prepositional-dative *show* and double-object dative *show*. In particular, how we designed their meanings.